From 45d27046f0871720eb0bc48c49174b098d4df8d6 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sun, 14 May 2023 21:52:54 +0800 Subject: [PATCH 01/53] first draft --- src/dp/convex_hull_optimization.md | 195 +++++++++++++++++++++++++++++ 1 file changed, 195 insertions(+) create mode 100644 src/dp/convex_hull_optimization.md diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md new file mode 100644 index 00000000..b9da2e69 --- /dev/null +++ b/src/dp/convex_hull_optimization.md @@ -0,0 +1,195 @@ +# Convex Hull Optimization + +## Introduction + +中文稱作斜率優化,是其中一種常見的 DP 優化。 + +先考慮以下例題: + +> [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) +> +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 + +考慮 \\(O(n^2)\\) 的做法,令 \\(dp[i]\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 + +轉移式為:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\) + +但是這個複雜度無法解決此題。 + +### 斜率單調 & 查詢單調 +我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f[j]\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f[j_1]s[i_1]+dp[j_1]>f[j_2]s[i_1]+dp[j_2]\\),那麼對於任一個 \\(i_2>i_1\\),\\(f[j_1]s[i_2]+dp[j_1]>f[j_2]s[i_2]+dp[j_2]\\)也是成立的(因為\\(s[i]\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 + +我們再觀察一次轉移式:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: + +\\[y=dp[i]\\] +\\[x=s[i]\\] +\\[a=f[j]\\] +\\[b=dp[j]\\] + +因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。這時我們的題目便轉變成:給定一個 \\(x\\),我們要在一個線集中找出最小的 \\(y\\)。 + +接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集: + +// 對範例輸入畫個圖 + +可以發現這是一個斜率不斷遞減的線集(從算式中也能看出,作為斜率的 \\(f[j]\\) 是遞減的)。對於每個 \\(x\\),我們都能在這個線集中找到對應的最小值,如果我們將它們通通標出來,可以發現會形成一個上凸包: + +// 畫圖標記上凸包 + +對任何 \\(x'\\) 而言,\\(x=x'\\) 與這個上凸包的交點便是所求的最小值。要找到這個交點,我們必須維護這個凸包,並支援兩種操作: + +- 查詢:對於 \\(x=s[i]\\),我們要能找出其與凸包的交點。 + + // 查最左邊的線,並和下一條比較,下一條比較小就 pop + + // 因為查詢具單調性 + +- 插入:插入一條新的線 + + // 往最右邊插入斜率最小的線,如果有線不在凸包上就 pop 掉 + + // 因為斜率具單調性 + +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 dp[i],便將 pair \\((f[i], dp[i])\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\);在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入 + +// 補上實作細節 + +這麼一來複雜度便降到 \\(O(n)\\),以下是參考的程式碼: + +// 貼 code + +#### 小結 +// 步驟總結 +1. 將初始值放入 deque +2. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線。 +3. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 + +// 要注意的地方 +- 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(應在頭或尾插入等) +- 注意題目是否會有轉移範圍的限制 +- 轉移式改成取 max 也可以做,只是變成建下凸包 + +#### 另一種思路 +// 補上讓 j 為點、i 為線的想法與實作方法 + +### 斜率單調 & 查詢不單調 +我們再來看另一個題目: + +// 一個斜率單調但查詢不單調的題目 + +// 目前只有在 POJ 上找到這類題目,希望可以在其他平台上找到 + +查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。於是我們需要一個可以支援插入、刪除以及二分搜的資料結構,可以改用 set 來維護,複雜度 \\(O(n\log n)\\)。 + +這題的範例 code: + +// 貼 code + +### 斜率不單調 +斜率不單調的情況更為複雜,我們不能再使用 deque 或是 set 來維護凸包,以下會介紹幾個解決此問題的方法。 + +#### CDQ 分治 +直接來看一個題目。 +> [CSES - Monster Game II](https://cses.fi/problemset/task/2085) +> +> // 補題敘 + +題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 + +// 將下列整理 + +- 遞迴的時候要對斜率做排序:先排左半再排右半,最後合併。 +- 取到左半區間時,將直線丟入 set 中 +- 取到右半區間時,在 set 中詢問最佳解,像斜率單調那時一樣 +- 時間複雜度 \\(O(n\log^2 n)\\) + +#### 李超線段樹 +看題目之前,先來介紹這個資料結構。 + +假設現在我們想在平面座標上做兩個操作(強制在線): +1. 加入一條形式為 \\(y=ax+b\\) 的直線 +2. 詢問 \\(x=k\\) 與所有直線交點的 \\(y\\) 的最大值 + +看起來像是區間修改、單點查詢的題目。我們考慮線段樹的作法,令左右界為值域,每個節點存的東西是一條直線。接著我們來思考如何插入與查詢。 + +**插入** + +如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 + +如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\),這兩條直線可能會有以下四種情況: +1. 在 \\(\[L, R)\\) 中,\\(f\\) 上的點皆大於 \\(g\\) 上的點。 +此情況下 \\(g\\) 上的點便不可能成為答案了,我們直接把 \\(g\\) 拿掉改成 \\(f\\)。 + + 判斷方式:如果 \\(f\\) 在左右端點處都大於 \\(g\\) 則成立。 + +2. 在 \\(\[L, R)\\) 中,\\(g\\) 上的點皆大於 \\(f\\) 上的點。 +與上一個情況相似,此情況下 \\(f\\) 上的點便不可能成為答案了,我們不做更改。 + + 判斷方式:如果 \\(g\\) 在左右端點處都大於 \\(f\\) 則成立。 + +3. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 +我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 + + // 補圖 + + 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(f\\) 在中點也大於 \\(g\\) 則成立。我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + +4. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 +與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 + + 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(g\\) 在中點大於 \\(f\\) 則成立。一樣,我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + +由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 + +// 貼插入的範例 code + +**查詢** + +類似一般線段樹上的區間修改、單點查詢一樣,我們只看包含 \\(k\\) 的那些區間,並從這些線段中取位於 \\(k\\) 的最大值。 + +// 貼查詢的範例 code + +--- + +現在我們來看題目,一樣是 [CSES - Monster Game II](https://cses.fi/problemset/task/2085),但這次我們考慮李超線段樹的作法。 + +// 從 1~n 依序查詢最小值再插入新直線 + +// 貼 code + +#### 動態凸包 +// 希望這一part可以日後再補Q + +### 總結 +// 幾個重點,需要再整理重寫、添加 +- 斜率優化只是能優化特定形式轉移式的轉移複雜度 +- 斜率是否單調會影響凸包維護的方式;查詢是否單調會影響尋找轉移來源的方式。 + +## Exercises +// 題敘與 code 待補 + +> [CF 1083E - The Fair Nut and Rectangles](https://codeforces.com/contest/1083/problem/E) + +> [CF 311B - Cats Transport](https://codeforces.com/contest/311/problem/B) + +> [CF 319C - Kalila and Dimna in the Logging Industry]( +https://codeforces.com/problemset/problem/319/C) + +> [CF 1715E - Long Way Home](https://codeforces.com/problemset/problem/1715/E) + +> [CSES - Subarray Squares](https://cses.fi/problemset/task/2086) + +> [ICPC WF 2011](https://onlinejudge.org/index.php?option=onlinejudge&Itemid=8&page=show_problem&problem=3547) + +> [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) + +## References +- [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) +- [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) +- [Algorithms for competetive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) +- [[Tutorial] Convex Hull Trick — Geometry being useful](https://codeforces.com/blog/entry/63823) +- [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) +- [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) +- [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) +- [Oi wiki - 李超線段樹](https://oi-wiki.org/ds/li-chao-tree/) +- [A Simple Introduction to Li-Chao Segment Tree](https://robert1003.github.io/2020/02/06/li-chao-segment-tree.html) From 8265e3c4076511ba1ac797bfcd3e3b34ba7a462b Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sun, 14 May 2023 21:52:54 +0800 Subject: [PATCH 02/53] first draft --- src/dp/convex_hull_optimization.md | 195 +++++++++++++++++++++++++++++ 1 file changed, 195 insertions(+) create mode 100644 src/dp/convex_hull_optimization.md diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md new file mode 100644 index 00000000..b9da2e69 --- /dev/null +++ b/src/dp/convex_hull_optimization.md @@ -0,0 +1,195 @@ +# Convex Hull Optimization + +## Introduction + +中文稱作斜率優化,是其中一種常見的 DP 優化。 + +先考慮以下例題: + +> [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) +> +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 + +考慮 \\(O(n^2)\\) 的做法,令 \\(dp[i]\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 + +轉移式為:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\) + +但是這個複雜度無法解決此題。 + +### 斜率單調 & 查詢單調 +我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f[j]\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f[j_1]s[i_1]+dp[j_1]>f[j_2]s[i_1]+dp[j_2]\\),那麼對於任一個 \\(i_2>i_1\\),\\(f[j_1]s[i_2]+dp[j_1]>f[j_2]s[i_2]+dp[j_2]\\)也是成立的(因為\\(s[i]\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 + +我們再觀察一次轉移式:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: + +\\[y=dp[i]\\] +\\[x=s[i]\\] +\\[a=f[j]\\] +\\[b=dp[j]\\] + +因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。這時我們的題目便轉變成:給定一個 \\(x\\),我們要在一個線集中找出最小的 \\(y\\)。 + +接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集: + +// 對範例輸入畫個圖 + +可以發現這是一個斜率不斷遞減的線集(從算式中也能看出,作為斜率的 \\(f[j]\\) 是遞減的)。對於每個 \\(x\\),我們都能在這個線集中找到對應的最小值,如果我們將它們通通標出來,可以發現會形成一個上凸包: + +// 畫圖標記上凸包 + +對任何 \\(x'\\) 而言,\\(x=x'\\) 與這個上凸包的交點便是所求的最小值。要找到這個交點,我們必須維護這個凸包,並支援兩種操作: + +- 查詢:對於 \\(x=s[i]\\),我們要能找出其與凸包的交點。 + + // 查最左邊的線,並和下一條比較,下一條比較小就 pop + + // 因為查詢具單調性 + +- 插入:插入一條新的線 + + // 往最右邊插入斜率最小的線,如果有線不在凸包上就 pop 掉 + + // 因為斜率具單調性 + +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 dp[i],便將 pair \\((f[i], dp[i])\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\);在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入 + +// 補上實作細節 + +這麼一來複雜度便降到 \\(O(n)\\),以下是參考的程式碼: + +// 貼 code + +#### 小結 +// 步驟總結 +1. 將初始值放入 deque +2. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線。 +3. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 + +// 要注意的地方 +- 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(應在頭或尾插入等) +- 注意題目是否會有轉移範圍的限制 +- 轉移式改成取 max 也可以做,只是變成建下凸包 + +#### 另一種思路 +// 補上讓 j 為點、i 為線的想法與實作方法 + +### 斜率單調 & 查詢不單調 +我們再來看另一個題目: + +// 一個斜率單調但查詢不單調的題目 + +// 目前只有在 POJ 上找到這類題目,希望可以在其他平台上找到 + +查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。於是我們需要一個可以支援插入、刪除以及二分搜的資料結構,可以改用 set 來維護,複雜度 \\(O(n\log n)\\)。 + +這題的範例 code: + +// 貼 code + +### 斜率不單調 +斜率不單調的情況更為複雜,我們不能再使用 deque 或是 set 來維護凸包,以下會介紹幾個解決此問題的方法。 + +#### CDQ 分治 +直接來看一個題目。 +> [CSES - Monster Game II](https://cses.fi/problemset/task/2085) +> +> // 補題敘 + +題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 + +// 將下列整理 + +- 遞迴的時候要對斜率做排序:先排左半再排右半,最後合併。 +- 取到左半區間時,將直線丟入 set 中 +- 取到右半區間時,在 set 中詢問最佳解,像斜率單調那時一樣 +- 時間複雜度 \\(O(n\log^2 n)\\) + +#### 李超線段樹 +看題目之前,先來介紹這個資料結構。 + +假設現在我們想在平面座標上做兩個操作(強制在線): +1. 加入一條形式為 \\(y=ax+b\\) 的直線 +2. 詢問 \\(x=k\\) 與所有直線交點的 \\(y\\) 的最大值 + +看起來像是區間修改、單點查詢的題目。我們考慮線段樹的作法,令左右界為值域,每個節點存的東西是一條直線。接著我們來思考如何插入與查詢。 + +**插入** + +如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 + +如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\),這兩條直線可能會有以下四種情況: +1. 在 \\(\[L, R)\\) 中,\\(f\\) 上的點皆大於 \\(g\\) 上的點。 +此情況下 \\(g\\) 上的點便不可能成為答案了,我們直接把 \\(g\\) 拿掉改成 \\(f\\)。 + + 判斷方式:如果 \\(f\\) 在左右端點處都大於 \\(g\\) 則成立。 + +2. 在 \\(\[L, R)\\) 中,\\(g\\) 上的點皆大於 \\(f\\) 上的點。 +與上一個情況相似,此情況下 \\(f\\) 上的點便不可能成為答案了,我們不做更改。 + + 判斷方式:如果 \\(g\\) 在左右端點處都大於 \\(f\\) 則成立。 + +3. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 +我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 + + // 補圖 + + 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(f\\) 在中點也大於 \\(g\\) 則成立。我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + +4. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 +與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 + + 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(g\\) 在中點大於 \\(f\\) 則成立。一樣,我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + +由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 + +// 貼插入的範例 code + +**查詢** + +類似一般線段樹上的區間修改、單點查詢一樣,我們只看包含 \\(k\\) 的那些區間,並從這些線段中取位於 \\(k\\) 的最大值。 + +// 貼查詢的範例 code + +--- + +現在我們來看題目,一樣是 [CSES - Monster Game II](https://cses.fi/problemset/task/2085),但這次我們考慮李超線段樹的作法。 + +// 從 1~n 依序查詢最小值再插入新直線 + +// 貼 code + +#### 動態凸包 +// 希望這一part可以日後再補Q + +### 總結 +// 幾個重點,需要再整理重寫、添加 +- 斜率優化只是能優化特定形式轉移式的轉移複雜度 +- 斜率是否單調會影響凸包維護的方式;查詢是否單調會影響尋找轉移來源的方式。 + +## Exercises +// 題敘與 code 待補 + +> [CF 1083E - The Fair Nut and Rectangles](https://codeforces.com/contest/1083/problem/E) + +> [CF 311B - Cats Transport](https://codeforces.com/contest/311/problem/B) + +> [CF 319C - Kalila and Dimna in the Logging Industry]( +https://codeforces.com/problemset/problem/319/C) + +> [CF 1715E - Long Way Home](https://codeforces.com/problemset/problem/1715/E) + +> [CSES - Subarray Squares](https://cses.fi/problemset/task/2086) + +> [ICPC WF 2011](https://onlinejudge.org/index.php?option=onlinejudge&Itemid=8&page=show_problem&problem=3547) + +> [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) + +## References +- [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) +- [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) +- [Algorithms for competetive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) +- [[Tutorial] Convex Hull Trick — Geometry being useful](https://codeforces.com/blog/entry/63823) +- [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) +- [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) +- [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) +- [Oi wiki - 李超線段樹](https://oi-wiki.org/ds/li-chao-tree/) +- [A Simple Introduction to Li-Chao Segment Tree](https://robert1003.github.io/2020/02/06/li-chao-segment-tree.html) From e8ce01a391323012694f48c47f516fb797d14e39 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Mon, 15 May 2023 15:41:10 +0800 Subject: [PATCH 03/53] modify content for li chao tree - insert --- src/dp/convex_hull_optimization.md | 22 ++++++++-------------- 1 file changed, 8 insertions(+), 14 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index b9da2e69..3e79a88a 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -116,28 +116,22 @@ 如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 -如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\),這兩條直線可能會有以下四種情況: -1. 在 \\(\[L, R)\\) 中,\\(f\\) 上的點皆大於 \\(g\\) 上的點。 -此情況下 \\(g\\) 上的點便不可能成為答案了,我們直接把 \\(g\\) 拿掉改成 \\(f\\)。 +如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\) (如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: - 判斷方式:如果 \\(f\\) 在左右端點處都大於 \\(g\\) 則成立。 - -2. 在 \\(\[L, R)\\) 中,\\(g\\) 上的點皆大於 \\(f\\) 上的點。 -與上一個情況相似,此情況下 \\(f\\) 上的點便不可能成為答案了,我們不做更改。 - - 判斷方式:如果 \\(g\\) 在左右端點處都大於 \\(f\\) 則成立。 - -3. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 +1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 // 補圖 - 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(f\\) 在中點也大於 \\(g\\) 則成立。我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + 判斷方式:\\(f\\) 在 \\(mid\\) 的值大於 \\(g\\) 則成立。 -4. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 +2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 - 判斷方式:若 \\(f\\) 只在其中一個端點大於 \\(g\\),且 \\(g\\) 在中點大於 \\(f\\) 則成立。一樣,我們可以觀察 \\(f\\) 在左或右端點大於 \\(g\\) 來判斷要向下推的是左區間還是右區間。 + 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 + + +若 \\(f\\) 在 \\(mid\\) 的值等於 \\(g\\),可以歸入上述任一種情況做操作。 由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 From 514ecc7dc43ff0628b01c5d24a2cb286c668d510 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sun, 21 May 2023 22:11:10 +0800 Subject: [PATCH 04/53] add sample code, add description for cdq and some of the remaining parts --- src/dp/convex_hull_optimization.md | 458 ++++++++++++++++++++++++++--- 1 file changed, 415 insertions(+), 43 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 3e79a88a..17fdded4 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -7,8 +7,8 @@ 先考慮以下例題: > [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) -> -> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值,其中 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 考慮 \\(O(n^2)\\) 的做法,令 \\(dp[i]\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 @@ -17,6 +17,7 @@ 但是這個複雜度無法解決此題。 ### 斜率單調 & 查詢單調 + 我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f[j]\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f[j_1]s[i_1]+dp[j_1]>f[j_2]s[i_1]+dp[j_2]\\),那麼對於任一個 \\(i_2>i_1\\),\\(f[j_1]s[i_2]+dp[j_1]>f[j_2]s[i_2]+dp[j_2]\\)也是成立的(因為\\(s[i]\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 我們再觀察一次轉移式:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: @@ -40,73 +41,336 @@ - 查詢:對於 \\(x=s[i]\\),我們要能找出其與凸包的交點。 - // 查最左邊的線,並和下一條比較,下一條比較小就 pop + 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,我們取的直線斜率一定會越來越小。因此我們可以只考慮目前最左邊的直線,並確認目前直線與 \\(x=s[i]\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線與 \\(x=s[i]\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前最左邊的直線丟掉。反之,最左邊的直線與 \\(x=s[i]\\) 的交點就是我們要找的最小值。 - // 因為查詢具單調性 + // 補圖說明 - 插入:插入一條新的線 + 插入似乎相對直觀,只是畫一條新的線在凸包上。由於斜率單調,插入直線時我們只要將直線插入最右(左)邊就好。但我們考慮下列情況: + + // 畫圖,有直線不在凸包上 + + 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 + +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(dp[i]\\),便將 pair \\((f[i], dp[i])\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 + +// 畫圖 + +在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 + +這麼一來複雜度便降到 \\(O(n)\\)。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef pair pii; +typedef long long ll; +typedef pair pll; +#define X first +#define Y second +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define setpr setprecision +#define ef emplace_front +#define eb emplace_back +#define pub push_back +#define em emplace +#define pb pop_back +#define pf pop_front +#define lowb lower_bound +#define uppb upper_bound +#define lowbit(x) ((x) & -(x)) +#define mk make_pair +const ll M = 1e9 + 7; +const ll INF = 1e14; +# define N 200005 + +ll s[N], f[N]; +ll dp[N]; + +ll cal(ll x, pll line){ + return x * line.X + line.Y; +} + +bool cmp(pll line1, pll line2, pll line3){ + return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); +} + +int main(){ + int n; + cin >> n >> f[0]; + for(int i = 1; i <= n; i++){ + cin >> s[i]; + } + for(int i = 1; i <= n; i++){ + cin >> f[i]; + } + deque dq; + dq.eb(f[0], 0); + for(int i = 1; i <= n; i++){ + while(dq.size() >= 2 && cal(s[i], dq[0]) > cal(s[i], dq[1])) dq.pf(); + dp[i] = cal(s[i], dq[0]); + pll line(f[i], dp[i]); + while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); + dq.eb(line); + } + cout << dp[n] << "\n"; + return 0; +} +``` + +
- // 往最右邊插入斜率最小的線,如果有線不在凸包上就 pop 掉 - - // 因為斜率具單調性 - -綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 dp[i],便將 pair \\((f[i], dp[i])\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\);在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入 - -// 補上實作細節 - -這麼一來複雜度便降到 \\(O(n)\\),以下是參考的程式碼: +#### 小結 -// 貼 code +// 需再統整或添加 -#### 小結 // 步驟總結 + 1. 將初始值放入 deque 2. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線。 3. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 // 要注意的地方 + - 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(應在頭或尾插入等) - 注意題目是否會有轉移範圍的限制 - 轉移式改成取 max 也可以做,只是變成建下凸包 #### 另一種思路 + // 補上讓 j 為點、i 為線的想法與實作方法 ### 斜率單調 & 查詢不單調 -我們再來看另一個題目: - -// 一個斜率單調但查詢不單調的題目 - -// 目前只有在 POJ 上找到這類題目,希望可以在其他平台上找到 - -查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。於是我們需要一個可以支援插入、刪除以及二分搜的資料結構,可以改用 set 來維護,複雜度 \\(O(n\log n)\\)。 - -這題的範例 code: -// 貼 code +上面的題目由於查詢單調,我們才能保證可以只看 deque 前端的線段就找到最小值。那麼,要是查詢不單調該怎麼辦? + +查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。我們一樣維護一個斜率有單調性的序列,不過這次我們不再取最前端的直線來計算答案,而是改為使用二分搜。這時我們不再需要 pop 最前段的線,但是在 insert 新的直線到序列最尾端時,我們一樣要將不可能是答案的直線 pop 掉。 + +至於二分搜要搜什麼呢?注意到我們維護的是一個凸包,凸包上的點的 \\(x\\) 座標是單調的。而每當我們要查詢的時候,我們想找的那條線上的兩個頂點會在當前查詢的 \\(x\\) 座標的左右兩側: + +// 一個凸包的圖,標記兩側頂點 + +因此我們在二分搜時,可以比較當前線段與下一條線段交點的 \\(x\\) 座標,以及當前要查詢的 \\(x\\) 座標,若前者較大則右界改為當前 \\(mid\\)、反之更改左界。 + +將查詢改為二分搜會讓複雜度多一個 \\(\log\\),總複雜度是 \\(O(n\log n)\\)。 + +以下是用此方法解 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 的範例 code。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef pair pii; +typedef long long ll; +typedef pair pll; +#define X first +#define Y second +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define setpr setprecision +#define ef emplace_front +#define eb emplace_back +#define pub push_back +#define em emplace +#define pb pop_back +#define pf pop_front +#define lowb lower_bound +#define uppb upper_bound +#define lowbit(x) ((x) & -(x)) +#define mk make_pair +const ll M = 1e9 + 7; +const ll INF = 1e14; +# define N 200005 + +ll s[N], f[N]; +ll dp[N]; + +ll cal(ll x, pll line){ + return x * line.X + line.Y; +} + +ll cmp2(pll p1, pll p2, ll x){ + return (p2.Y - p1.Y) > (p1.X - p2.X) * x; +} + +pll find_line(vector v, ll x){ + int l = 0, r = v.size() - 2, mid, ans = max(r + 1, 0); + while(l <= r){ + mid = (l + r) >> 1; + if(cmp2(v[mid], v[mid + 1], x)){ + r = mid - 1; + ans = mid; + } + else l = mid + 1; + } + return v[ans]; +} + +bool cmp(pll line1, pll line2, pll line3){ + return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); +} + +int main(){ + int n; + cin >> n >> f[0]; + for(int i = 1; i <= n; i++){ + cin >> s[i]; + } + for(int i = 1; i <= n; i++){ + cin >> f[i]; + } + vector v; + v.eb(f[0], 0); + for(int i = 1; i <= n; i++){ + pll fl = find_line(v, s[i]); + dp[i] = cal(s[i], fl); + pll line(f[i], dp[i]); + while(v.size() >= 2 && cmp(v[v.size() - 2], v[v.size() - 1], line)) v.pb(); + v.eb(line); + } + cout << dp[n] << "\n"; + return 0; +} +``` + +
### 斜率不單調 + 斜率不單調的情況更為複雜,我們不能再使用 deque 或是 set 來維護凸包,以下會介紹幾個解決此問題的方法。 #### CDQ 分治 + 直接來看一個題目。 > [CSES - Monster Game II](https://cses.fi/problemset/task/2085) -> -> // 補題敘 +> +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 -// 將下列整理 - -- 遞迴的時候要對斜率做排序:先排左半再排右半,最後合併。 -- 取到左半區間時,將直線丟入 set 中 -- 取到右半區間時,在 set 中詢問最佳解,像斜率單調那時一樣 -- 時間複雜度 \\(O(n\log^2 n)\\) +我們一樣要在遞迴過程中,將序列分左半邊與右半邊處理。前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 + +由此,我們在做完遞迴完一個區間時,我們可以得到該區間的所有直線,且依照斜率排序。那麼接下來的問題就是該如何計算 dp 值。如果今天我們左半邊的區間已經處理好了(dp 值皆計算完成且直線依斜率排好),那我們便可以用左半區間去更新右半區間的答案。更新完之後,左半邊的資訊便沒有用了,我們就可以往下遞迴右半區間,重複一樣的動作。 + +更新的時候就跟斜率單調、查詢不單調時的情況一樣。我們已經有左半邊排序好的序列,現在對右半邊的每個查詢(不單調),我們都要在左半邊的序列中找到最佳解並更新。由於這時斜率是單調的,一樣可以使用二分搜。 + +總結一下重點: + +- 先遞迴左半邊,再來用左半邊的資訊更新右半邊的 dp 值,最後遞迴右半邊。 +- 遞迴的時候要一邊對斜率做排序。 +- 更新右半區間時,二分搜左半區間詢問最佳解,像斜率單調那時一樣。 +- 時間複雜度 \\(O(n\log^2 n)\\)。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef pair pii; +typedef long long ll; +typedef pair pll; +#define X first +#define Y second +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define setpr setprecision +#define ef emplace_front +#define eb emplace_back +#define pub push_back +#define em emplace +#define pb pop_back +#define pf pop_front +#define lowb lower_bound +#define uppb upper_bound +#define lowbit(x) ((x) & -(x)) +#define mk make_pair +const ll M = 1e9 + 7; +const ll INF = 1e14; +#define N 200005 + +ll dp[N], s[N], f[N]; + +ll cal(pll line, ll x){ + return x * line.X + line.Y; +} + +ll cmp2(pll p1, pll p2, ll x){ + return (p2.Y - p1.Y) > (p1.X - p2.X) * x; +} + +bool cmp(pll line1, pll line2, pll line3){ + return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); +} + +pll find_line(vector v, ll x){ + int l = 0, r = v.size() - 2, mid, ans = max(0, r + 1); + while(l <= r){ + mid = (l + r) >> 1; + if(cmp2(v[mid], v[mid + 1], x)){ + r = mid - 1; + ans = mid; + } + else l = mid + 1; + } + return v[ans]; +} + +vector solve(int l, int r){ + vector v; + if(l == r){ + v.eb(f[l], dp[l]); + return v; + } + int mid = (l + r) >> 1; + vector v1 = solve(l, mid); + // calculate dp value for (mid, r] + for(int i = mid + 1; i <= r; i++){ + pll fl = find_line(v1, s[i]); + dp[i] = min(dp[i], cal(fl, s[i])); + } + vector v2 = solve(mid + 1, r); + int idx1 = 0, idx2 = 0; + while(idx1 < v1.size() || idx2 < v2.size()){ + pll pl; + if(idx2 >= v2.size() || (idx1 < v1.size() && v1[idx1] >= v2[idx2])) pl = v1[idx1++]; + else pl = v2[idx2++]; + while(v.size() >= 2 && cmp(v[v.size() - 2], v.back(), pl)) v.pb(); + v.eb(pl); + } + return v; +} + +int main(){ + int n; + cin >> n >> f[0]; + for(int i = 1; i <= n; i++){ + cin >> s[i]; + } + for(int i = 1; i <= n; i++){ + cin >> f[i]; + dp[i] = INF; + } + solve(0, n); + cout << dp[n] << "\n"; + return 0; +} +``` + +
#### 李超線段樹 + 看題目之前,先來介紹這個資料結構。 假設現在我們想在平面座標上做兩個操作(強制在線): + 1. 加入一條形式為 \\(y=ax+b\\) 的直線 2. 詢問 \\(x=k\\) 與所有直線交點的 \\(y\\) 的最大值 @@ -130,36 +394,143 @@ 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 - 若 \\(f\\) 在 \\(mid\\) 的值等於 \\(g\\),可以歸入上述任一種情況做操作。 由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 -// 貼插入的範例 code +```cpp +void insert(int l, int r, int id, pll line){ + if(l == r){ + if(cal(line, l) < cal(seg[id], l)) seg[id] = line; + return; + } + int mid = (l + r) >> 1; + if(line.X > seg[id].X) swap(line, seg[id]); + if(cal(line, mid) <= cal(seg[id], mid)){ + insert(l, mid, id * 2, seg[id]); + seg[id] = line; + } + else{ + insert(mid + 1, r, id * 2 + 1, line); + } +} +``` **查詢** 類似一般線段樹上的區間修改、單點查詢一樣,我們只看包含 \\(k\\) 的那些區間,並從這些線段中取位於 \\(k\\) 的最大值。 -// 貼查詢的範例 code +```cpp +ll query(int l, int r, int id, ll x){ + if(x < l || x > r) return INF; + if(l == r) return cal(seg[id], x); + int mid = (l + r) >> 1; + ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); + return min(val, cal(seg[id], x)); +} +``` --- -現在我們來看題目,一樣是 [CSES - Monster Game II](https://cses.fi/problemset/task/2085),但這次我們考慮李超線段樹的作法。 - -// 從 1~n 依序查詢最小值再插入新直線 - -// 貼 code +現在我們來看題目,一樣是 [CSES - Monster Game II](https://cses.fi/problemset/task/2085),但這次我們考慮李超線段樹的作法。想法很直觀,我們只要先在李超樹上查詢最小值以得到當前 dp 值,再將新的線插入樹中就好。時間複雜度 \\(O(n\log n)\\)。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef pair pii; +typedef long long ll; +typedef pair pll; +#define X first +#define Y second +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define setpr setprecision +#define ef emplace_front +#define eb emplace_back +#define pub push_back +#define em emplace +#define pb pop_back +#define pf pop_front +#define lowb lower_bound +#define uppb upper_bound +#define lowbit(x) ((x) & -(x)) +#define mk make_pair +const ll M = 1e9 + 7; +const ll INF = 1e18; +#define N 1000005 + +ll s[N], f[N]; +ll dp[N]; +pll seg[N * 4]; + +ll cal(pll line, ll x){ + return line.X * x + line.Y; +} + +void insert(int l, int r, int id, pll line){ + if(l == r){ + if(cal(line, l) < cal(seg[id], l)) seg[id] = line; + return; + } + int mid = (l + r) >> 1; + if(line.X > seg[id].X) swap(line, seg[id]); + if(cal(line, mid) <= cal(seg[id], mid)){ + insert(l, mid, id * 2, seg[id]); + seg[id] = line; + } + else{ + insert(mid + 1, r, id * 2 + 1, line); + } +} + +ll query(int l, int r, int id, ll x){ + if(x < l || x > r) return INF; + if(l == r) return cal(seg[id], x); + int mid = (l + r) >> 1; + ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); + return min(val, cal(seg[id], x)); +} + +int main(){ + int n, x; + cin >> n >> x; + for(int i = 1; i <= n; i++){ + cin >> s[i]; + } + for(int i = 1; i <= n; i++){ + cin >> f[i]; + } + for(int i = 1; i <= 4e6; i++){ + seg[i] = pll(0, INF); + } + insert(1, 1e6, 1, pll(x, 0)); + for(int i = 1; i <= n; i++){ + dp[i] = query(1, 1e6, 1, s[i]); + insert(1, 1e6, 1, pll(f[i], dp[i])); + } + cout << dp[n] << "\n"; + return 0; +} +``` + +
#### 動態凸包 + // 希望這一part可以日後再補Q ### 總結 + // 幾個重點,需要再整理重寫、添加 + - 斜率優化只是能優化特定形式轉移式的轉移複雜度 - 斜率是否單調會影響凸包維護的方式;查詢是否單調會影響尋找轉移來源的方式。 ## Exercises + // 題敘與 code 待補 > [CF 1083E - The Fair Nut and Rectangles](https://codeforces.com/contest/1083/problem/E) @@ -178,10 +549,11 @@ https://codeforces.com/problemset/problem/319/C) > [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) ## References + - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) - [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) -- [Algorithms for competetive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) -- [[Tutorial] Convex Hull Trick — Geometry being useful](https://codeforces.com/blog/entry/63823) +- [Algorithms for competitive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) +- [[Tutorial] Convex Hull Trick — Geometry being useful]() - [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) - [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) - [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) From 65a6239b3ea974ae450e7a39bd81e4e13d086b6a Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 09:57:25 +0800 Subject: [PATCH 05/53] fixed dp[i] -> F(i) --- src/dp/convex_hull_optimization.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 17fdded4..d660ce8d 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -10,22 +10,22 @@ > > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值,其中 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 -考慮 \\(O(n^2)\\) 的做法,令 \\(dp[i]\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 +考慮 \\(O(n^2)\\) 的做法,令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 -轉移式為:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\) +轉移式為:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\) 但是這個複雜度無法解決此題。 ### 斜率單調 & 查詢單調 -我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f[j]\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f[j_1]s[i_1]+dp[j_1]>f[j_2]s[i_1]+dp[j_2]\\),那麼對於任一個 \\(i_2>i_1\\),\\(f[j_1]s[i_2]+dp[j_1]>f[j_2]s[i_2]+dp[j_2]\\)也是成立的(因為\\(s[i]\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 +我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f_j\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f_{j_1}s_{i_1}+F(j_1)>f_{j_2}s_{i_1}+F(j_2)\\),那麼對於任一個 \\(i_2>i_1\\),\\(f_{j_1}s_{i_2}+F(j_1)>f_{j_2}s_{i_2}+F(j_2)\\)也是成立的(因為\\(s_i\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 -我們再觀察一次轉移式:\\(dp[i]=\underset{i>j}{\min}(dp[j]+f[j]\times s[i])\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: +我們再觀察一次轉移式:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: -\\[y=dp[i]\\] -\\[x=s[i]\\] -\\[a=f[j]\\] -\\[b=dp[j]\\] +\\[y=F(i)\\] +\\[x=s_i\\] +\\[a=f_j\\] +\\[b=F(j)\\] 因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。這時我們的題目便轉變成:給定一個 \\(x\\),我們要在一個線集中找出最小的 \\(y\\)。 @@ -33,15 +33,15 @@ // 對範例輸入畫個圖 -可以發現這是一個斜率不斷遞減的線集(從算式中也能看出,作為斜率的 \\(f[j]\\) 是遞減的)。對於每個 \\(x\\),我們都能在這個線集中找到對應的最小值,如果我們將它們通通標出來,可以發現會形成一個上凸包: +可以發現這是一個斜率不斷遞減的線集(從算式中也能看出,作為斜率的 \\(f_j\\) 是遞減的)。對於每個 \\(x\\),我們都能在這個線集中找到對應的最小值,如果我們將它們通通標出來,可以發現會形成一個上凸包: // 畫圖標記上凸包 對任何 \\(x'\\) 而言,\\(x=x'\\) 與這個上凸包的交點便是所求的最小值。要找到這個交點,我們必須維護這個凸包,並支援兩種操作: -- 查詢:對於 \\(x=s[i]\\),我們要能找出其與凸包的交點。 +- 查詢:對於 \\(x=s_i\\),我們要能找出其與凸包的交點。 - 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,我們取的直線斜率一定會越來越小。因此我們可以只考慮目前最左邊的直線,並確認目前直線與 \\(x=s[i]\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線與 \\(x=s[i]\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前最左邊的直線丟掉。反之,最左邊的直線與 \\(x=s[i]\\) 的交點就是我們要找的最小值。 + 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,我們取的直線斜率一定會越來越小。因此我們可以只考慮目前最左邊的直線,並確認目前直線與 \\(x=s_i\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線與 \\(x=s_i\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前最左邊的直線丟掉。反之,最左邊的直線與 \\(x=s_i\\) 的交點就是我們要找的最小值。 // 補圖說明 @@ -52,7 +52,7 @@ 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 -綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(dp[i]\\),便將 pair \\((f[i], dp[i])\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(F(i)\\),便將 pair \\((f_i, F(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 // 畫圖 From 67e9709b0eebb6822965bacf5f1b399459e20692 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 10:09:58 +0800 Subject: [PATCH 06/53] remove useless template code --- src/dp/convex_hull_optimization.md | 63 +++--------------------------- 1 file changed, 6 insertions(+), 57 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index d660ce8d..37941447 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -64,7 +64,6 @@ ```cpp #include -#pragma GCC optimize("O2") using namespace std; typedef pair pii; @@ -72,21 +71,10 @@ typedef long long ll; typedef pair pll; #define X first #define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define setpr setprecision -#define ef emplace_front -#define eb emplace_back -#define pub push_back -#define em emplace +#define pf pop_front #define pb pop_back -#define pf pop_front -#define lowb lower_bound -#define uppb upper_bound -#define lowbit(x) ((x) & -(x)) -#define mk make_pair -const ll M = 1e9 + 7; -const ll INF = 1e14; -# define N 200005 +#define eb emplace_back +#define N 200005 ll s[N], f[N]; ll dp[N]; @@ -94,7 +82,6 @@ ll dp[N]; ll cal(ll x, pll line){ return x * line.X + line.Y; } - bool cmp(pll line1, pll line2, pll line3){ return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); } @@ -164,7 +151,6 @@ int main(){ ```cpp #include -#pragma GCC optimize("O2") using namespace std; typedef pair pii; @@ -172,21 +158,10 @@ typedef long long ll; typedef pair pll; #define X first #define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define setpr setprecision -#define ef emplace_front #define eb emplace_back -#define pub push_back -#define em emplace #define pb pop_back #define pf pop_front -#define lowb lower_bound -#define uppb upper_bound -#define lowbit(x) ((x) & -(x)) -#define mk make_pair -const ll M = 1e9 + 7; -const ll INF = 1e14; -# define N 200005 +#define N 200005 ll s[N], f[N]; ll dp[N]; @@ -194,11 +169,9 @@ ll dp[N]; ll cal(ll x, pll line){ return x * line.X + line.Y; } - ll cmp2(pll p1, pll p2, ll x){ return (p2.Y - p1.Y) > (p1.X - p2.X) * x; } - pll find_line(vector v, ll x){ int l = 0, r = v.size() - 2, mid, ans = max(r + 1, 0); while(l <= r){ @@ -279,21 +252,10 @@ typedef long long ll; typedef pair pll; #define X first #define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define setpr setprecision -#define ef emplace_front #define eb emplace_back -#define pub push_back -#define em emplace #define pb pop_back -#define pf pop_front -#define lowb lower_bound -#define uppb upper_bound -#define lowbit(x) ((x) & -(x)) -#define mk make_pair -const ll M = 1e9 + 7; -const ll INF = 1e14; #define N 200005 +const ll INF = 1e14; ll dp[N], s[N], f[N]; @@ -446,21 +408,8 @@ typedef long long ll; typedef pair pll; #define X first #define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define setpr setprecision -#define ef emplace_front -#define eb emplace_back -#define pub push_back -#define em emplace -#define pb pop_back -#define pf pop_front -#define lowb lower_bound -#define uppb upper_bound -#define lowbit(x) ((x) & -(x)) -#define mk make_pair -const ll M = 1e9 + 7; -const ll INF = 1e18; #define N 1000005 +const ll INF = 1e18; ll s[N], f[N]; ll dp[N]; From 2c9b4d9bcb809de7250496bb3d778a3c2d8a77ea Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 10:12:50 +0800 Subject: [PATCH 07/53] add the range of n for problems --- src/dp/convex_hull_optimization.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 37941447..e198dc60 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -9,6 +9,7 @@ > [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) > > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值,其中 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> - \\(1\leq n\leq 2\times 10^5\\) 考慮 \\(O(n^2)\\) 的做法,令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 @@ -224,6 +225,7 @@ int main(){ > [CSES - Monster Game II](https://cses.fi/problemset/task/2085) > > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> - \\(1\leq n\leq 2\times 10^5\\) 題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 From d2673b2b2444a56ce4281e6579c0f93f2d688c4d Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 10:19:06 +0800 Subject: [PATCH 08/53] content fixed (set->vector) --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index e198dc60..9e2e9ff4 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -217,7 +217,7 @@ int main(){ ### 斜率不單調 -斜率不單調的情況更為複雜,我們不能再使用 deque 或是 set 來維護凸包,以下會介紹幾個解決此問題的方法。 +斜率不單調的情況更為複雜,我們不能再使用 deque 或是 vector 來維護凸包,以下會介紹幾個解決此問題的方法。 #### CDQ 分治 From 05a92b2fe6277e0236749232326ac8a152fee53d Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 10:21:35 +0800 Subject: [PATCH 09/53] add formula for cdq(monster game II) --- src/dp/convex_hull_optimization.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 9e2e9ff4..26cb144f 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -227,6 +227,10 @@ int main(){ > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 > - \\(1\leq n\leq 2\times 10^5\\) +令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 + +轉移式為:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\) + 題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 我們一樣要在遞迴過程中,將序列分左半邊與右半邊處理。前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 From 2602eda1bb60b5577bfcec82afe46891c05c1425 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 23 May 2023 10:44:41 +0800 Subject: [PATCH 10/53] add content for cdq --- src/dp/convex_hull_optimization.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 26cb144f..7bd5a1c9 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -233,7 +233,11 @@ int main(){ 題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 -我們一樣要在遞迴過程中,將序列分左半邊與右半邊處理。前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 +CDQ 分治的過程就是對於一個序列 \\([l, r]\\),我們先遞迴解決 \\([l, mid]\\) 的子問題,再處理 \\([l, mid]\\) 對 \\([mid + 1, r]\\) 的貢獻,最後再遞迴處理 \\([mid + 1, r]\\)。 + +以這題來說,我們會先遞迴左半區間,接著用計算好的左半區間,去更新右半區間的 dp 值,最後再往下遞迴右半區間。 + +前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 由此,我們在做完遞迴完一個區間時,我們可以得到該區間的所有直線,且依照斜率排序。那麼接下來的問題就是該如何計算 dp 值。如果今天我們左半邊的區間已經處理好了(dp 值皆計算完成且直線依斜率排好),那我們便可以用左半區間去更新右半區間的答案。更新完之後,左半邊的資訊便沒有用了,我們就可以往下遞迴右半區間,重複一樣的動作。 From 207281bcdd2de877d6b9045282739afa17c2c03b Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 27 May 2023 00:17:50 +0800 Subject: [PATCH 11/53] add comment for reference --- src/dp/convex_hull_optimization.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 7bd5a1c9..d6efb989 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -508,13 +508,13 @@ https://codeforces.com/problemset/problem/319/C) > [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) ## References - +底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為前兩個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 +- [[Tutorial] Convex Hull Trick — Geometry being useful]() +- [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) - [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) - [Algorithms for competitive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) -- [[Tutorial] Convex Hull Trick — Geometry being useful]() - [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) - [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) -- [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 李超線段樹](https://oi-wiki.org/ds/li-chao-tree/) - [A Simple Introduction to Li-Chao Segment Tree](https://robert1003.github.io/2020/02/06/li-chao-segment-tree.html) From b14cf33588a0b38c434eb2d6c11a29bdcbf4214a Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 27 May 2023 02:32:36 +0800 Subject: [PATCH 12/53] add summary --- src/dp/convex_hull_optimization.md | 29 ++++++++++++++++------------- 1 file changed, 16 insertions(+), 13 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index d6efb989..b7b634d0 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -114,19 +114,21 @@ int main(){ #### 小結 -// 需再統整或添加 +以上就是斜率優化的例題與概念,我們再複習一下大致的步驟: +1. 將初始值放入 deque +2. 跑迴圈,從 1 到 n,做下面兩個步驟: -// 步驟總結 + i. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線,不斷重複直到最前端的線有最好的答案,或是 deque 中只剩一條線。 -1. 將初始值放入 deque -2. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線。 -3. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 + ii. 利用 deque 最前端的線計算 \\(dp[i]\\)。 + + iii. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 -// 要注意的地方 -- 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(應在頭或尾插入等) -- 注意題目是否會有轉移範圍的限制 -- 轉移式改成取 max 也可以做,只是變成建下凸包 +雖然步驟都大同小異,但不同題目還是有一些差異,在實作前要特別注意: +- 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(比較線段時要用大於還是小於等等)。 +- 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉。 +- 轉移式改成取 max 只是從建上凸包變成建下凸包,方法是差不多的。 #### 另一種思路 @@ -483,10 +485,11 @@ int main(){ ### 總結 -// 幾個重點,需要再整理重寫、添加 - -- 斜率優化只是能優化特定形式轉移式的轉移複雜度 -- 斜率是否單調會影響凸包維護的方式;查詢是否單調會影響尋找轉移來源的方式。 +以上就是斜率優化的簡單介紹,最後這裡做個重點整理: +- 斜率優化的功能在於優化特定形式轉移式的轉移複雜度。在這之前,觀察題目並列出、整理轉移式也是很重要的技巧。 +- 斜率是否單調會影響凸包維護的方式:單調時可以用 deque、vector;不單調時需要李超線段樹或是動態凸包等等。 +- 查詢是否單調會影響尋找轉移來源的方式:單調時可以用單調隊列就好;不單調時需要使用二分搜。 +- 許多 dp 優化的題目可以用很多不同的優化方式來做,斜率優化只是其中一種。可以依據實作難度、想法複不複雜等等來決定要使用哪一個。 ## Exercises From b45d9ea5d33b111556b0abaf5998ae469a9ce39d Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 20:32:03 +0800 Subject: [PATCH 13/53] add more complicate problem for monotonous slope and query --- src/dp/convex_hull_optimization.md | 172 ++++++++++++++++++++++++++++- 1 file changed, 171 insertions(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index b7b634d0..8127f9f8 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -112,6 +112,176 @@ int main(){ +
+ +--- + +我們來看一個稍微複雜的例題。 + +> [TIOJ 1676 - 烏龜疊疊樂](https://tioj.ck.tp.edu.tw/problems/1676) +> +> 給定一個長度為 \\(n\\) 的陣列與一個整數 \\(k\\),你可以將陣列切成數段連續的區間,區間大小不能超過 \\(k\\),求切完之後的最大價值。 +> 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \Sigma^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 +> - \\(1 \leq k \leq n \leq 500000\\) + +一樣,我們先試著列出轉移式。令 \\(F(i)\\) 為前 \\(i\\) 個數字貢獻的價值(同時第 \\(i\\) 個元素為區間右界)。 + +轉移式為:\\(F(i)=\underset{i>j}{\max}(F(j)-(i-j)^2+suf(i+1))\\) + +其中 \\(suf(k)\\) 是後綴和,範圍從 \\(k\\) 到 \\(n\\)。 + +除了 \\(suf(i+1)\\) 這一項外,其他部分還蠻直觀的。而 \\(suf(i+1)\\) 這一項對應到題目的 \\((m-1)\times \Sigma^r_{i=l}a_i\\),我們看一張示意圖以更好理解 \\(suf(i+1)\\) 的意義: + +// 示意圖 + +上面的示意圖對每個的區間畫出了其 \\(suf(i+1)\\) 的值。我們可以將它們重新分割變成下面這樣: + +// 重新著色的示意圖 + +可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\) ,這個部分的總和。 + +瞭解轉移式的推導後,我們回來觀察這個轉移式。它看起來與 \\(y=ax+b\\) 的形式不太像,但我們試著將它展開: +\\(F(i)=\underset{i>j>i-k}{\max}(F(j)-(i-j)^2+suf(i+1))\\) +\\(=\underset{i>j>i-k}{\max}(F(j)-i^2+2ij-j^2+suf(i+1))\\) +\\(=\underset{i>j>i-k}{\max}(F(j)-j^2+2ij)-i^2+suf(i+1)\\) + +可以發現 \\(max\\) 裡面的部分變成\\(y=ax+b\\)的形式了: +\\[y=F(i)\\] +\\[x=2i\\] +\\[a=j\\] +\\[b=F(j)-j^2\\] + +這麼一來就可以使用剛剛提到的技巧了!但仔細觀察一下,這個轉移式與上一個例題有兩個不同的地方: +1. 上一題是取 \\(min\\)、這一題是取 \\(max\\),同時斜率從遞增變為遞減。 + 其實概念是差不多的,我們一樣把線畫出來觀察一下: + + // 畫兩張:所有直線、標記下凸包 + + 可以發現我們只是從在上凸包中尋找答案,改為在下凸包中尋找答案而已。實作上來說,我們只要在比較哪個答案比較好時,改成越大的答案越好就好。也就是下面這份 code 的第二行,後面比較時要使用小於(這邊的 \\(x\\)、\\(a\\)、\\(b\\),意義同 \\(y=ax+b\\) 中的 \\(x\\)、\\(a\\)、\\(b\\))。 + + ```cpp + for(ll i = 1; i <= n; i++){ + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); + while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); + dq.eb(line); + } + ``` + +2. 轉移範圍有限制,只能從往前 \\(k-1\\) 個轉移點來轉移。 + 看起來也不是什麼大問題,好像只需要在 pop deque 前端的直線時,將過期的線也 pop 掉就好。 + + 這邊假設 deque 裡面的每個元素由 \\(a\\)、\\(b\\)、\\(idx\\) 組成,\\(a\\)、\\(b\\) 代表直線的資訊,\\(idx\\) 代表轉移點的 index。 + ```cpp + for(ll i = 1; i <= n; i++){ + while(dq.size() >= 1 && dq[0].idx < i - k) dq.pf(); + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); + while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); + dq.eb(line); + } + ``` + + 但這樣真的解決了嗎?我們考慮一下下面這種情況: + + // 一張圖,兩條直線 + + 我們現在要求最大值、且斜率遞增。假設現在我們要插入一條新直線(X色): + + // 一張圖,兩條直線跟一條新直線 + + 依照前一個例題的概念,藍色線上的所有點都不在凸包上,我們應該要移除它。這邊將它變虛線表示移除。 + + // 一張圖,倒數第二條線變虛線 + + 接下來我們要進行下一個查詢: + + // 一張圖,加上 x=x' 示意查詢 + + 假設這時最左邊的線過期了,於是我們將它移除: + + // 一張圖,移除最左邊的線 + + 這時我們發現其實剛剛被我們移除掉的藍色線才能提供最好答案,但我們已經將它移除了。從這個例子我們發現,在加了直線會過期的限制後,我們除了要在直線過期時移除它之外,我們也要更改一下移除不在凸包上直線的策略。 + + 注意到在綠色線被移除之前,藍色線都不可能提供答案。因此我們要多判斷的是藍色線在綠色線被移除之後,會不會成為可能的轉移來源。具體而言,在我們放入紅色線前,要決定藍色線是否要被移除,而方法是多判斷在綠色線被 pop 時的那一個查詢,藍色線會不會提供比紅色線更好的答案。 + + 直接來看 code 比較好理解,這邊以這題的情況,假設線過期的條件是 \\(j < i-k\\) 則過期。 + + ```cpp + for(ll i = 1; i <= n; i++){ + while(dq.size() >= 1 && dq[0].idx < i - k) dq.pf(); + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); + while(dq.size() >= 2 + && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line) + && cal(x[dq[dq.size() - 2].idx], dq[dq.size() - 1]) <= cal(x[dq[dq.size() - 2].idx], line)) dq.pb(); + dq.eb(line); + } + ``` + + 在加線之前判斷是否要 pop 掉 deque 最後方的線那一部分,我們多加了一個條件判斷 deque 最後方的線是否有可能在它前方的線被 pop 掉後提供答案。由此我們可以保證每一條在 deque 裡面的線都是可能的轉移來源。 + +其他部分便與上一個例題概念相同,上述做法複雜度一樣為 \\(O(n)\\)。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef long long ll; +typedef pair pll; +#define X first +#define Y second +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define eb emplace_back +#define pb pop_back +#define pf pop_front +#define N 500005 + +ll a[N], suf[N], dp[N]; + +ll cal(ll x, pll line){ + return x * line.X + line.Y; +} +bool cmp(pll line1, pll line2, pll line3){ + return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); +} + +int main(){ + io + ll n, k; + cin >> n >> k; + for(int i = 1; i <= n; i++){ + cin >> a[i]; + } + for(int i = n; i >= 1; i--){ + suf[i] = a[i] + suf[i + 1]; + } + deque dq; + dq.eb(0, 0); + for(ll i = 1; i <= n; i++){ + while(dq.size() >= 1 && dq[0].X < i-k) dq.pf(); + while(dq.size() >= 2 && cal(2 * i, dq[0]) <= cal(2 * i, dq[1])) dq.pf(); + dp[i] = cal(2 * i, dq[0]) - i * i + suf[i + 1]; + pll line(i, dp[i] - i * i); + while(dq.size() >= 2 + && cal(2 * (dq[dq.size() - 2].X + k), dq.back()) <= cal(2 * (dq[dq.size() - 2].X + k), line) + && cmp(dq[dq.size() - 2], dq.back(), line)) dq.pb(); + dq.eb(line); + } + cout << dp[n] << "\n"; + return 0; +} +``` + +
+ #### 小結 以上就是斜率優化的例題與概念,我們再複習一下大致的步驟: @@ -128,7 +298,7 @@ int main(){ 雖然步驟都大同小異,但不同題目還是有一些差異,在實作前要特別注意: - 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(比較線段時要用大於還是小於等等)。 - 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉。 -- 轉移式改成取 max 只是從建上凸包變成建下凸包,方法是差不多的。 +- 轉移式取 max 與取 min 的差別在於建下凸包或是建上凸包,方法是差不多的。 #### 另一種思路 From ba9d7f442b836be79474798ead1ebb780471dc8b Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 20:36:03 +0800 Subject: [PATCH 14/53] summary update --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 8127f9f8..5058ac61 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -297,7 +297,7 @@ int main(){ 雖然步驟都大同小異,但不同題目還是有一些差異,在實作前要特別注意: - 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(比較線段時要用大於還是小於等等)。 -- 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉。 +- 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉,同時拔掉不在凸包上的點的條件也要做更改。 - 轉移式取 max 與取 min 的差別在於建下凸包或是建上凸包,方法是差不多的。 #### 另一種思路 From afc2684cf8ec731a5db25ebafec9dba2f0f10783 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 20:55:17 +0800 Subject: [PATCH 15/53] reference comment update --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 5058ac61..4b43a2e9 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -681,7 +681,7 @@ https://codeforces.com/problemset/problem/319/C) > [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) ## References -底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為前兩個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 +底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為第一個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 - [[Tutorial] Convex Hull Trick — Geometry being useful]() - [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) From 29ce8d40b66f44782b5ab7d6655a5fe070c285da Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 20:57:50 +0800 Subject: [PATCH 16/53] typo and indentation fixed --- src/dp/convex_hull_optimization.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 4b43a2e9..6b40fb99 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -669,8 +669,7 @@ int main(){ > [CF 311B - Cats Transport](https://codeforces.com/contest/311/problem/B) -> [CF 319C - Kalila and Dimna in the Logging Industry]( -https://codeforces.com/problemset/problem/319/C) +> [CF 319C - Kalila and Dimna in the Logging Industry](https://codeforces.com/problemset/problem/319/C) > [CF 1715E - Long Way Home](https://codeforces.com/problemset/problem/1715/E) @@ -681,6 +680,7 @@ https://codeforces.com/problemset/problem/319/C) > [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) ## References + 底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為第一個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 - [[Tutorial] Convex Hull Trick — Geometry being useful]() - [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) From 8e5475d0547245b2903c692f425c23a7a06df384 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 23:17:03 +0800 Subject: [PATCH 17/53] add img name --- src/dp/convex_hull_optimization.md | 15 +++++++++++++++ 1 file changed, 15 insertions(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 6b40fb99..bfbb94b0 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -33,10 +33,12 @@ 接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集: // 對範例輸入畫個圖 + 可以發現這是一個斜率不斷遞減的線集(從算式中也能看出,作為斜率的 \\(f_j\\) 是遞減的)。對於每個 \\(x\\),我們都能在這個線集中找到對應的最小值,如果我們將它們通通標出來,可以發現會形成一個上凸包: // 畫圖標記上凸包 + 對任何 \\(x'\\) 而言,\\(x=x'\\) 與這個上凸包的交點便是所求的最小值。要找到這個交點,我們必須維護這個凸包,並支援兩種操作: @@ -45,17 +47,20 @@ 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,我們取的直線斜率一定會越來越小。因此我們可以只考慮目前最左邊的直線,並確認目前直線與 \\(x=s_i\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線與 \\(x=s_i\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前最左邊的直線丟掉。反之,最左邊的直線與 \\(x=s_i\\) 的交點就是我們要找的最小值。 // 補圖說明 + - 插入:插入一條新的線 插入似乎相對直觀,只是畫一條新的線在凸包上。由於斜率單調,插入直線時我們只要將直線插入最右(左)邊就好。但我們考慮下列情況: // 畫圖,有直線不在凸包上 + 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(F(i)\\),便將 pair \\((f_i, F(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 // 畫圖 + 在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 @@ -133,10 +138,12 @@ int main(){ 除了 \\(suf(i+1)\\) 這一項外,其他部分還蠻直觀的。而 \\(suf(i+1)\\) 這一項對應到題目的 \\((m-1)\times \Sigma^r_{i=l}a_i\\),我們看一張示意圖以更好理解 \\(suf(i+1)\\) 的意義: // 示意圖 + 上面的示意圖對每個的區間畫出了其 \\(suf(i+1)\\) 的值。我們可以將它們重新分割變成下面這樣: // 重新著色的示意圖 + 可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\) ,這個部分的總和。 @@ -156,6 +163,8 @@ int main(){ 其實概念是差不多的,我們一樣把線畫出來觀察一下: // 畫兩張:所有直線、標記下凸包 + + 可以發現我們只是從在上凸包中尋找答案,改為在下凸包中尋找答案而已。實作上來說,我們只要在比較哪個答案比較好時,改成越大的答案越好就好。也就是下面這份 code 的第二行,後面比較時要使用小於(這邊的 \\(x\\)、\\(a\\)、\\(b\\),意義同 \\(y=ax+b\\) 中的 \\(x\\)、\\(a\\)、\\(b\\))。 @@ -187,22 +196,27 @@ int main(){ 但這樣真的解決了嗎?我們考慮一下下面這種情況: // 一張圖,兩條直線 + 我們現在要求最大值、且斜率遞增。假設現在我們要插入一條新直線(X色): // 一張圖,兩條直線跟一條新直線 + 依照前一個例題的概念,藍色線上的所有點都不在凸包上,我們應該要移除它。這邊將它變虛線表示移除。 // 一張圖,倒數第二條線變虛線 + 接下來我們要進行下一個查詢: // 一張圖,加上 x=x' 示意查詢 + 假設這時最左邊的線過期了,於是我們將它移除: // 一張圖,移除最左邊的線 + 這時我們發現其實剛剛被我們移除掉的藍色線才能提供最好答案,但我們已經將它移除了。從這個例子我們發現,在加了直線會過期的限制後,我們除了要在直線過期時移除它之外,我們也要更改一下移除不在凸包上直線的策略。 @@ -530,6 +544,7 @@ int main(){ 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 // 補圖 + 判斷方式:\\(f\\) 在 \\(mid\\) 的值大於 \\(g\\) 則成立。 From 0d25d3fe6be3a8cebe7d34e80260de490b5f2637 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 30 May 2023 23:18:29 +0800 Subject: [PATCH 18/53] indentation fixed --- src/dp/convex_hull_optimization.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index bfbb94b0..54a05e69 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -541,7 +541,8 @@ int main(){ 如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\) (如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: 1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 -我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 + + 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 // 補圖 @@ -549,7 +550,8 @@ int main(){ 判斷方式:\\(f\\) 在 \\(mid\\) 的值大於 \\(g\\) 則成立。 2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 -與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 + + 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 From e343be5d50ef7b3ca7bd2301270548de0c93d579 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 31 May 2023 21:58:53 +0800 Subject: [PATCH 19/53] add example for non-monotonic query --- src/dp/convex_hull_optimization.md | 90 ++++++++++++++++++++---------- 1 file changed, 59 insertions(+), 31 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 54a05e69..05d01439 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -140,7 +140,7 @@ int main(){ // 示意圖 -上面的示意圖對每個的區間畫出了其 \\(suf(i+1)\\) 的值。我們可以將它們重新分割變成下面這樣: +上面的示意圖對每個的區間畫出了其 \\((m-1)\times \Sigma^r_{i=l}a_i\\) 的值。我們可以將它們重新分割變成下面這樣: // 重新著色的示意圖 @@ -198,7 +198,7 @@ int main(){ // 一張圖,兩條直線 - 我們現在要求最大值、且斜率遞增。假設現在我們要插入一條新直線(X色): + 我們現在要求最大值、且斜率遞增。假設現在我們要插入一條新直線(紅色): // 一張圖,兩條直線跟一條新直線 @@ -322,7 +322,38 @@ int main(){ 上面的題目由於查詢單調,我們才能保證可以只看 deque 前端的線段就找到最小值。那麼,要是查詢不單調該怎麼辦? -查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。我們一樣維護一個斜率有單調性的序列,不過這次我們不再取最前端的直線來計算答案,而是改為使用二分搜。這時我們不再需要 pop 最前段的線,但是在 insert 新的直線到序列最尾端時,我們一樣要將不可能是答案的直線 pop 掉。 +我們來看另一個例題: +> [P5785 任務安排](https://www.luogu.com.cn/problem/P5785) +> +> 有 \\(n\\) 個任務,第 \\(i\\) 個任務單獨完成所需的時間為 \\(T_i\\)、費用為 \\(C_i\\)。現在我們可以將任務分批,每批包含相鄰的若干個任務。任務被依序執行,完成一批任務所需的時間是各個任務需要時間的總和,同一批任務會在同一時刻完成。另外,在每批任務開始前,機器需要啟動時間 \\(s\\)。一個任務的 cost 被定義為它的完成時刻乘以它的費用。起始時間為 \\(0\\),求最小的 total cost。 +> - \\(1 \leq n \leq 300000\\) +> - \\(-2^8 \leq T_i \leq 2^8\\) +> - \\(0 \leq C_i \leq 2^8\\) +> + +先列轉移式。令 \\(F(i)\\) 為完成前 \\(i\\) 個任務的最小 cost。 + +轉移式為:\\(F(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+F(j))\\) + +其中,\\(s \times \Sigma_{k=j+1}^n C_k\\) 這一項是為了計算 \\(s\\) 所影響的任務完成時刻。由於影響的時間與總共切幾個區間有關,這裡用類似前面例題 [TIOJ 1676 - 烏龜疊疊樂](https://tioj.ck.tp.edu.tw/problems/1676),使用後綴和的想法,在每多切一個區間出來時就加上後綴因此而產生的 cost。 + +接下來,我們將轉移式展開: + +\\(F(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+F(j))\\) +\\(=\underset{i>j}{\min}(pre_t(i) \times (pre_c(i)-pre_c(j))+s \times (pre_c(n)-pre_c(j))+F(j))\\) +\\(=\underset{i>j}{\min}(-pre_t(i) \times pre_c(j)-s \times pre_c(j)+F(j))+pre_c(i) \times pre_t(i)+s \times pre_c(n)\\) + +其中 \\(pre_t(i)\\) 為 \\(T\\) 的前綴和,即 \\(\Sigma_{k=1}^i T_k\\);\\(pre_c(i)\\) 為 \\(C\\) 的前綴和,即 \\(\Sigma_{k=1}^i C_k\\)。 + +我們可以得到: +\\[y=F(i)\\] +\\[x=-pre_t(i)\\] +\\[a=pre_c(j)\\] +\\[b=F(j)-s \times pre_c(j)\\] + +\\(pre_c(j)\\) 是遞增的,沒什麼問題。但 \\(pre_t(i)\\) 就不是了。由於 \\(T_i\\) 有可能小於 \\(0\\),\\(T\\) 的前綴和不具單調性。也就是說,我們的查詢不再單調了。 + +查詢不再單調,意味著轉移來源不單調,因此我們查詢的時候不能再取最左(右)邊的線來做比較。但是,斜率依然是單調的,因此我們可以改為使用二分搜來尋找答案。我們一樣維護一個斜率有單調性的序列,不過這次我們不再取最前端的直線來計算答案,而是改為使用二分搜。這時我們不再需要 pop 最前段的線,但是在 insert 新的直線到序列最尾端時,我們一樣要將不可能是答案的直線 pop 掉。 至於二分搜要搜什麼呢?注意到我們維護的是一個凸包,凸包上的點的 \\(x\\) 座標是單調的。而每當我們要查詢的時候,我們想找的那條線上的兩個頂點會在當前查詢的 \\(x\\) 座標的左右兩側: @@ -332,35 +363,34 @@ int main(){ 將查詢改為二分搜會讓複雜度多一個 \\(\log\\),總複雜度是 \\(O(n\log n)\\)。 -以下是用此方法解 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 的範例 code。 -
Solution Code +此題時間較緊,需要壓一下常數,因此這邊使用陣列來實作。一般情況下,只要可以 push_back 與 pop_back 的資料結構都可以,像是 vector。 + ```cpp #include using namespace std; -typedef pair pii; typedef long long ll; typedef pair pll; #define X first #define Y second -#define eb emplace_back -#define pb pop_back -#define pf pop_front -#define N 200005 +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define N 300005 -ll s[N], f[N]; +ll pret[N], prec[N]; ll dp[N]; +pll v[N]; +ll sz; -ll cal(ll x, pll line){ +inline ll cal(ll x, pll line){ return x * line.X + line.Y; } -ll cmp2(pll p1, pll p2, ll x){ +inline bool cmp2(pll p1, pll p2, ll x){ return (p2.Y - p1.Y) > (p1.X - p2.X) * x; } -pll find_line(vector v, ll x){ - int l = 0, r = v.size() - 2, mid, ans = max(r + 1, 0); +inline pll find_line(ll x){ + int l = 0, r = sz - 2, mid, ans = r + 1; while(l <= r){ mid = (l + r) >> 1; if(cmp2(v[mid], v[mid + 1], x)){ @@ -371,28 +401,26 @@ pll find_line(vector v, ll x){ } return v[ans]; } - -bool cmp(pll line1, pll line2, pll line3){ - return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); +inline bool cmp(pll line1, pll line2, pll line3){ + return (__int128)(line3.Y - line1.Y) * (__int128)(line1.X - line2.X) <= (__int128)(line2.Y - line1.Y) * (__int128)(line1.X - line3.X); } int main(){ - int n; - cin >> n >> f[0]; - for(int i = 1; i <= n; i++){ - cin >> s[i]; - } + io + ll n, s; + cin >> n >> s; for(int i = 1; i <= n; i++){ - cin >> f[i]; + cin >> pret[i] >> prec[i]; + pret[i] += pret[i - 1]; + prec[i] += prec[i - 1]; } - vector v; - v.eb(f[0], 0); + v[sz++] = pll(0, 0); for(int i = 1; i <= n; i++){ - pll fl = find_line(v, s[i]); - dp[i] = cal(s[i], fl); - pll line(f[i], dp[i]); - while(v.size() >= 2 && cmp(v[v.size() - 2], v[v.size() - 1], line)) v.pb(); - v.eb(line); + pll fl = find_line(pret[i]); + dp[i] = cal(pret[i], fl) + prec[i] * pret[i] + s * prec[n]; + pll line(-prec[i], dp[i] - s * prec[i]); + while(sz >= 2 && cmp(v[sz - 2], v[sz - 1], line)) sz--; + v[sz++] = line; } cout << dp[n] << "\n"; return 0; From 708ebbf72f38bfe8ea42bbaa25f5ef7cca95baa5 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 20:58:24 +0800 Subject: [PATCH 20/53] add dynamic hull --- src/dp/convex_hull_optimization.md | 74 +++++++++++++++++++++++++++++- 1 file changed, 73 insertions(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 05d01439..f0d191d3 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -696,7 +696,78 @@ int main(){ #### 動態凸包 -// 希望這一part可以日後再補Q +動態凸包,顧名思義就是動態維護的凸包,支援的操作有插入直線與詢問在 \\(x\\) 處的最大值。這個東西有模板可以用,應該蠻好找的,因此這邊不討論如何實作,只討論如何使用它。 + +插入直線與查詢最大值恰好就是我們需要的兩個操作,因此就直接使用即可。如果轉移式是要求最小值,我們就只要把所有東西都加負號並改取最大值就好。 + +這邊提供使用動態凸包解 [CSES - Monster Game II](https://cses.fi/problemset/task/2085) 的範例 code。模板是使用 kth-competitive-programming 的 [LineContainer](https://github.com/kth-competitive-programming/kactl/blob/main/content/data-structures/LineContainer.h)。時間複雜度一樣是 \\(O(n\log n)\\)。 + +
Solution Code + +```cpp +#include +#pragma GCC optimize("O2") +using namespace std; + +typedef long long ll; +#define io ios_base::sync_with_stdio(0); cin.tie(0); +#define N 200005 + +struct Line { + mutable ll k, m, p; + bool operator<(const Line& o) const { return k < o.k; } + bool operator<(ll x) const { return p < x; } +}; + +struct LineContainer : multiset> { + // (for doubles, use inf = 1/.0, div(a,b) = a/b) + static const ll inf = LLONG_MAX; + ll div(ll a, ll b) { // floored division + return a / b - ((a ^ b) < 0 && a % b); } + bool isect(iterator x, iterator y) { + if (y == end()) return x->p = inf, 0; + if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; + else x->p = div(y->m - x->m, x->k - y->k); + return x->p >= y->p; + } + void add(ll k, ll m) { + auto z = insert({k, m, 0}), y = z++, x = y; + while (isect(y, z)) z = erase(z); + if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); + while ((y = x) != begin() && (--x)->p >= y->p) + isect(x, erase(y)); + } + ll query(ll x) { + assert(!empty()); + auto l = *lower_bound(x); + return l.k * x + l.m; + } +}; + +ll s[N], f[N], dp[N]; + +int main(){ + io + ll n, x; + cin >> n >> x; + for(int i = 1; i <= n; i++){ + cin >> s[i]; + } + for(int i = 1; i <= n; i++){ + cin >> f[i]; + } + LineContainer lc; + lc.add(-x, 0); + for(int i = 1; i <= n; i++){ + dp[i] = -lc.query(s[i]); + lc.add(-f[i], -dp[i]); + } + cout << dp[n] << "\n"; + return 0; +} +``` + +
### 總結 @@ -735,4 +806,5 @@ int main(){ - [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) - [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) - [Oi wiki - 李超線段樹](https://oi-wiki.org/ds/li-chao-tree/) +- [DP optimization - Convex Hull Optimization](https://robert1003.github.io/2020/02/17/dp-opt-convex-hull-trick.html#convex-hull-trick-cht) - [A Simple Introduction to Li-Chao Segment Tree](https://robert1003.github.io/2020/02/06/li-chao-segment-tree.html) From 8eb57f667847899c142af5628dc8f194ba287082 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 21:17:06 +0800 Subject: [PATCH 21/53] fix indentation test --- src/dp/convex_hull_optimization.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index f0d191d3..08eae4a7 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -9,6 +9,7 @@ > [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) > > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值,其中 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> > - \\(1\leq n\leq 2\times 10^5\\) 考慮 \\(O(n^2)\\) 的做法,令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 From 737b556e989283ae6fd585388f260e4833abd291 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 21:47:29 +0800 Subject: [PATCH 22/53] markdownlint fixed --- src/dp/convex_hull_optimization.md | 149 +++++++++++++++-------------- 1 file changed, 79 insertions(+), 70 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 08eae4a7..9ddfffec 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -128,6 +128,7 @@ int main(){ > > 給定一個長度為 \\(n\\) 的陣列與一個整數 \\(k\\),你可以將陣列切成數段連續的區間,區間大小不能超過 \\(k\\),求切完之後的最大價值。 > 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \Sigma^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 +> > - \\(1 \leq k \leq n \leq 500000\\) 一樣,我們先試著列出轉移式。令 \\(F(i)\\) 為前 \\(i\\) 個數字貢獻的價值(同時第 \\(i\\) 個元素為區間右界)。 @@ -149,8 +150,11 @@ int main(){ 可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\) ,這個部分的總和。 瞭解轉移式的推導後,我們回來觀察這個轉移式。它看起來與 \\(y=ax+b\\) 的形式不太像,但我們試著將它展開: + \\(F(i)=\underset{i>j>i-k}{\max}(F(j)-(i-j)^2+suf(i+1))\\) + \\(=\underset{i>j>i-k}{\max}(F(j)-i^2+2ij-j^2+suf(i+1))\\) + \\(=\underset{i>j>i-k}{\max}(F(j)-j^2+2ij)-i^2+suf(i+1)\\) 可以發現 \\(max\\) 裡面的部分變成\\(y=ax+b\\)的形式了: @@ -160,9 +164,10 @@ int main(){ \\[b=F(j)-j^2\\] 這麼一來就可以使用剛剛提到的技巧了!但仔細觀察一下,這個轉移式與上一個例題有兩個不同的地方: + 1. 上一題是取 \\(min\\)、這一題是取 \\(max\\),同時斜率從遞增變為遞減。 其實概念是差不多的,我們一樣把線畫出來觀察一下: - + // 畫兩張:所有直線、標記下凸包 @@ -183,6 +188,7 @@ int main(){ 看起來也不是什麼大問題,好像只需要在 pop deque 前端的直線時,將過期的線也 pop 掉就好。 這邊假設 deque 裡面的每個元素由 \\(a\\)、\\(b\\)、\\(idx\\) 組成,\\(a\\)、\\(b\\) 代表直線的資訊,\\(idx\\) 代表轉移點的 index。 + ```cpp for(ll i = 1; i <= n; i++){ while(dq.size() >= 1 && dq[0].idx < i - k) dq.pf(); @@ -239,7 +245,7 @@ int main(){ ``` 在加線之前判斷是否要 pop 掉 deque 最後方的線那一部分,我們多加了一個條件判斷 deque 最後方的線是否有可能在它前方的線被 pop 掉後提供答案。由此我們可以保證每一條在 deque 裡面的線都是可能的轉移來源。 - + 其他部分便與上一個例題概念相同,上述做法複雜度一樣為 \\(O(n)\\)。
Solution Code @@ -300,17 +306,18 @@ int main(){ #### 小結 以上就是斜率優化的例題與概念,我們再複習一下大致的步驟: + 1. 將初始值放入 deque 2. 跑迴圈,從 1 到 n,做下面兩個步驟: i. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線,不斷重複直到最前端的線有最好的答案,或是 deque 中只剩一條線。 ii. 利用 deque 最前端的線計算 \\(dp[i]\\)。 - - iii. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 + iii. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 雖然步驟都大同小異,但不同題目還是有一些差異,在實作前要特別注意: + - 要注意斜率以及查詢的單調性是遞增或遞減,會影響實作(比較線段時要用大於還是小於等等)。 - 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉,同時拔掉不在凸包上的點的條件也要做更改。 - 轉移式取 max 與取 min 的差別在於建下凸包或是建上凸包,方法是差不多的。 @@ -326,7 +333,8 @@ int main(){ 我們來看另一個例題: > [P5785 任務安排](https://www.luogu.com.cn/problem/P5785) > -> 有 \\(n\\) 個任務,第 \\(i\\) 個任務單獨完成所需的時間為 \\(T_i\\)、費用為 \\(C_i\\)。現在我們可以將任務分批,每批包含相鄰的若干個任務。任務被依序執行,完成一批任務所需的時間是各個任務需要時間的總和,同一批任務會在同一時刻完成。另外,在每批任務開始前,機器需要啟動時間 \\(s\\)。一個任務的 cost 被定義為它的完成時刻乘以它的費用。起始時間為 \\(0\\),求最小的 total cost。 +> 有 \\(n\\) 個任務,第 \\(i\\) 個任務單獨完成所需的時間為 \\(T_i\\)、費用為 \\(C_i\\)。現在我們可以將任務分批,每批包含相鄰的若干個任務。任務被依序執行,完成一批任務所需的時間是各個任務需要時間的總和,同一批任務會在同一時刻完成。另外,在每批任務開始前,機器需要啟動時間 \\(s\\)。一個任務的 cost 被定義為它的完成時刻乘以它的費用。起始時間為 \\(0\\),求最小的 total cost。 +> > - \\(1 \leq n \leq 300000\\) > - \\(-2^8 \leq T_i \leq 2^8\\) > - \\(0 \leq C_i \leq 2^8\\) @@ -440,6 +448,7 @@ int main(){ > [CSES - Monster Game II](https://cses.fi/problemset/task/2085) > > 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> > - \\(1\leq n\leq 2\times 10^5\\) 令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 @@ -563,62 +572,62 @@ int main(){ 看起來像是區間修改、單點查詢的題目。我們考慮線段樹的作法,令左右界為值域,每個節點存的東西是一條直線。接著我們來思考如何插入與查詢。 -**插入** +- **插入** -如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 + 如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 -如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\) (如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: + 如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\) (如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: -1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 + 1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 - 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 + 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 - // 補圖 - + // 補圖 + - 判斷方式:\\(f\\) 在 \\(mid\\) 的值大於 \\(g\\) 則成立。 + 判斷方式:\\(f\\) 在 \\(mid\\) 的值大於 \\(g\\) 則成立。 -2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 + 2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 - 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 + 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 - 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 + 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 -若 \\(f\\) 在 \\(mid\\) 的值等於 \\(g\\),可以歸入上述任一種情況做操作。 + 若 \\(f\\) 在 \\(mid\\) 的值等於 \\(g\\),可以歸入上述任一種情況做操作。 -由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 + 由於我們向下推的時候只會推左或右其中一個子區間,這麼做的複雜度為 \\(O(\log C)\\),其中 \\(C\\) 是值域大小。 -```cpp -void insert(int l, int r, int id, pll line){ - if(l == r){ - if(cal(line, l) < cal(seg[id], l)) seg[id] = line; - return; - } - int mid = (l + r) >> 1; - if(line.X > seg[id].X) swap(line, seg[id]); - if(cal(line, mid) <= cal(seg[id], mid)){ - insert(l, mid, id * 2, seg[id]); - seg[id] = line; - } - else{ - insert(mid + 1, r, id * 2 + 1, line); + ```cpp + void insert(int l, int r, int id, pll line){ + if(l == r){ + if(cal(line, l) < cal(seg[id], l)) seg[id] = line; + return; + } + int mid = (l + r) >> 1; + if(line.X > seg[id].X) swap(line, seg[id]); + if(cal(line, mid) <= cal(seg[id], mid)){ + insert(l, mid, id * 2, seg[id]); + seg[id] = line; + } + else{ + insert(mid + 1, r, id * 2 + 1, line); + } } -} -``` + ``` -**查詢** +- **查詢** -類似一般線段樹上的區間修改、單點查詢一樣,我們只看包含 \\(k\\) 的那些區間,並從這些線段中取位於 \\(k\\) 的最大值。 + 類似一般線段樹上的區間修改、單點查詢一樣,我們只看包含 \\(k\\) 的那些區間,並從這些線段中取位於 \\(k\\) 的最大值。 -```cpp -ll query(int l, int r, int id, ll x){ - if(x < l || x > r) return INF; - if(l == r) return cal(seg[id], x); - int mid = (l + r) >> 1; - ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); - return min(val, cal(seg[id], x)); -} -``` + ```cpp + ll query(int l, int r, int id, ll x){ + if(x < l || x > r) return INF; + if(l == r) return cal(seg[id], x); + int mid = (l + r) >> 1; + ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); + return min(val, cal(seg[id], x)); + } + ``` --- @@ -715,34 +724,34 @@ typedef long long ll; #define N 200005 struct Line { - mutable ll k, m, p; - bool operator<(const Line& o) const { return k < o.k; } - bool operator<(ll x) const { return p < x; } + mutable ll k, m, p; + bool operator<(const Line& o) const { return k < o.k; } + bool operator<(ll x) const { return p < x; } }; struct LineContainer : multiset> { - // (for doubles, use inf = 1/.0, div(a,b) = a/b) - static const ll inf = LLONG_MAX; - ll div(ll a, ll b) { // floored division - return a / b - ((a ^ b) < 0 && a % b); } - bool isect(iterator x, iterator y) { - if (y == end()) return x->p = inf, 0; - if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; - else x->p = div(y->m - x->m, x->k - y->k); - return x->p >= y->p; - } - void add(ll k, ll m) { - auto z = insert({k, m, 0}), y = z++, x = y; - while (isect(y, z)) z = erase(z); - if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); - while ((y = x) != begin() && (--x)->p >= y->p) - isect(x, erase(y)); - } - ll query(ll x) { - assert(!empty()); - auto l = *lower_bound(x); - return l.k * x + l.m; - } + // (for doubles, use inf = 1/.0, div(a,b) = a/b) + static const ll inf = LLONG_MAX; + ll div(ll a, ll b) { // floored division + return a / b - ((a ^ b) < 0 && a % b); } + bool isect(iterator x, iterator y) { + if (y == end()) return x->p = inf, 0; + if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; + else x->p = div(y->m - x->m, x->k - y->k); + return x->p >= y->p; + } + void add(ll k, ll m) { + auto z = insert({k, m, 0}), y = z++, x = y; + while (isect(y, z)) z = erase(z); + if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); + while ((y = x) != begin() && (--x)->p >= y->p) + isect(x, erase(y)); + } + ll query(ll x) { + assert(!empty()); + auto l = *lower_bound(x); + return l.k * x + l.m; + } }; ll s[N], f[N], dp[N]; From aa682ebf8488d94b16326011c32b2b71cf0e5876 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 21:53:25 +0800 Subject: [PATCH 23/53] markdownlint fixed --- src/dp/convex_hull_optimization.md | 28 ++++++++++++++++++++++++++-- 1 file changed, 26 insertions(+), 2 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 9ddfffec..021f3c49 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -228,7 +228,7 @@ int main(){ 這時我們發現其實剛剛被我們移除掉的藍色線才能提供最好答案,但我們已經將它移除了。從這個例子我們發現,在加了直線會過期的限制後,我們除了要在直線過期時移除它之外,我們也要更改一下移除不在凸包上直線的策略。 注意到在綠色線被移除之前,藍色線都不可能提供答案。因此我們要多判斷的是藍色線在綠色線被移除之後,會不會成為可能的轉移來源。具體而言,在我們放入紅色線前,要決定藍色線是否要被移除,而方法是多判斷在綠色線被 pop 時的那一個查詢,藍色線會不會提供比紅色線更好的答案。 - + 直接來看 code 比較好理解,這邊以這題的情況,假設線過期的條件是 \\(j < i-k\\) 則過期。 ```cpp @@ -782,6 +782,7 @@ int main(){ ### 總結 以上就是斜率優化的簡單介紹,最後這裡做個重點整理: + - 斜率優化的功能在於優化特定形式轉移式的轉移複雜度。在這之前,觀察題目並列出、整理轉移式也是很重要的技巧。 - 斜率是否單調會影響凸包維護的方式:單調時可以用 deque、vector;不單調時需要李超線段樹或是動態凸包等等。 - 查詢是否單調會影響尋找轉移來源的方式:單調時可以用單調隊列就好;不單調時需要使用二分搜。 @@ -793,21 +794,44 @@ int main(){ > [CF 1083E - The Fair Nut and Rectangles](https://codeforces.com/contest/1083/problem/E) +
Solution + +
+ > [CF 311B - Cats Transport](https://codeforces.com/contest/311/problem/B) +
Solution + +
+ > [CF 319C - Kalila and Dimna in the Logging Industry](https://codeforces.com/problemset/problem/319/C) +
Solution + +
+ > [CF 1715E - Long Way Home](https://codeforces.com/problemset/problem/1715/E) +
Solution + +
+ > [CSES - Subarray Squares](https://cses.fi/problemset/task/2086) +
Solution + +
+ > [ICPC WF 2011](https://onlinejudge.org/index.php?option=onlinejudge&Itemid=8&page=show_problem&problem=3547) -> [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) +
Solution + +
## References 底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為第一個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 + - [[Tutorial] Convex Hull Trick — Geometry being useful]() - [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) From 905d5b2279dbc53d786f9f84c16224240297f217 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 22:01:26 +0800 Subject: [PATCH 24/53] textlint fixed --- src/dp/convex_hull_optimization.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 021f3c49..ca1599f4 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -147,7 +147,7 @@ int main(){ // 重新著色的示意圖 -可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\) ,這個部分的總和。 +可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\),這個部分的總和。 瞭解轉移式的推導後,我們回來觀察這個轉移式。它看起來與 \\(y=ax+b\\) 的形式不太像,但我們試著將它展開: @@ -157,7 +157,7 @@ int main(){ \\(=\underset{i>j>i-k}{\max}(F(j)-j^2+2ij)-i^2+suf(i+1)\\) -可以發現 \\(max\\) 裡面的部分變成\\(y=ax+b\\)的形式了: +可以發現 \\(max\\) 裡面的部分變成 \\(y=ax+b\\) 的形式了: \\[y=F(i)\\] \\[x=2i\\] \\[a=j\\] @@ -166,6 +166,7 @@ int main(){ 這麼一來就可以使用剛剛提到的技巧了!但仔細觀察一下,這個轉移式與上一個例題有兩個不同的地方: 1. 上一題是取 \\(min\\)、這一題是取 \\(max\\),同時斜率從遞增變為遞減。 + 其實概念是差不多的,我們一樣把線畫出來觀察一下: // 畫兩張:所有直線、標記下凸包 @@ -185,6 +186,7 @@ int main(){ ``` 2. 轉移範圍有限制,只能從往前 \\(k-1\\) 個轉移點來轉移。 + 看起來也不是什麼大問題,好像只需要在 pop deque 前端的直線時,將過期的線也 pop 掉就好。 這邊假設 deque 裡面的每個元素由 \\(a\\)、\\(b\\)、\\(idx\\) 組成,\\(a\\)、\\(b\\) 代表直線的資訊,\\(idx\\) 代表轉移點的 index。 @@ -832,8 +834,8 @@ int main(){ 底下的資源都是在介紹斜率優化,但講解方式不盡相同,可以多參考幾個以更深入了解斜率優化的概念。其中我認為第一個 reference 的講解最清楚詳細,而最後一個 reference 對於李超線段樹的說明也很清楚,建議可以去看看。Codeforces 上面還有更多關於斜率優化的 tutorial 與題單,有興趣的讀者可以自己去搜尋。 -- [[Tutorial] Convex Hull Trick — Geometry being useful]() -- [【學習筆記】動態規劃—斜率優化DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) +- [[Tutorial] Convex Hull Trick - Geometry being useful](https://codeforces.com/blog/entry/63823) +- [【學習筆記】動態規劃—斜率優化 DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) - [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) - [Algorithms for competitive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) From e94310f864967e586981dd84e0c61ef5bbd4b495 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 1 Jun 2023 22:03:51 +0800 Subject: [PATCH 25/53] textlint fixed --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index ca1599f4..4736bc9e 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -578,7 +578,7 @@ int main(){ 如果當前的節點沒有存直線,那我們就直接將要插入的直線存進去。 - 如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\) (如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: + 如果當前的節點已經存直線了,那我們要想辦法將他們合併起來。假設當前節點的區間是 \\(\[L, R)\\)、中點為 \\(mid=\frac{L+R}{2}\\)。要加進去的直線為 \\(f\\)、在節點中的直線為 \\(g\\)。不失一般性,我們假設 \\(g\\) 的斜率大於 \\(f\\)(如果不是的話就交換,其餘做法相同)。這兩條直線可能會有以下兩種情況: 1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 From f0f07c7131f7d6964ec29ad0c7b17e65393b0ae9 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Fri, 2 Jun 2023 00:32:04 +0800 Subject: [PATCH 26/53] add image dir --- src/dp/images/cses2084_1.png | Bin 0 -> 142685 bytes src/dp/images/cses2084_2.png | Bin 0 -> 118761 bytes src/dp/images/cses2084_3.png | Bin 0 -> 73313 bytes src/dp/images/cses2084_4.png | Bin 0 -> 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    -> [TIOJ 1921 - 吐鈔機2](https://tioj.ck.tp.edu.tw/problems/1921) +> [TIOJ 1921 - 吐鈔機 2](https://tioj.ck.tp.edu.tw/problems/1921) > > 有 \\(N\\) 台機器,每台機器都有 \\(D_i, P_i, G_i, R_i\\):售出的日期、購入的價格、每天產出的金額、賣出的價格。給定 \\(C, D\\):第 \\(0\\) 天持有的金額,停止交易的日期。 > 問在停止交易當天所能持有的最大金額是多少。 From 0f0d8b0c78af7b556bbb7a48eca33d2be27dc158 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 22 Jun 2023 10:27:04 +0800 Subject: [PATCH 32/53] add md file to SUMMARY.md --- src/SUMMARY.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 2a4d3da4..ab98c3e4 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -10,6 +10,7 @@ - [Basic Greedy](greedy/basic.md) - [Dynamic Programming]() - [Basic DP]() + - [Convex hull optimization](dp/convex_hull_optimization.md) - [Flow]() - [Sqrt Technique](sqrt/intro.md) - [Square Root Decomposition](sqrt/sqrt_decomposition.md) From 26aebf7ce6cd9882ec182984849b3be3dffb2a7d Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 22 Jun 2023 10:30:35 +0800 Subject: [PATCH 33/53] change F(i)->f(i) --- src/dp/convex_hull_optimization.md | 104 ++++++++++++++--------------- 1 file changed, 52 insertions(+), 52 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 54bfa513..789e3437 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -2,7 +2,7 @@ ## Introduction -中文稱作斜率優化,是其中一種常見的 DP 優化。 +中文稱作斜率優化或凸包優化,是其中一種常見的 DP 優化。 先考慮以下例題: @@ -14,20 +14,20 @@ 考慮 \\(O(n^2)\\) 的做法,令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 -轉移式為:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\) +轉移式為:\\(f(i)=\underset{i>j}{\min}(f(j)+f_j\times s_i)\\) 但是這個複雜度無法解決此題。 ### 斜率單調 & 查詢單調 -我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f_j\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f_{j_1}s_{i_1}+F(j_1)>f_{j_2}s_{i_1}+F(j_2)\\),那麼對於任一個 \\(i_2>i_1\\),\\(f_{j_1}s_{i_2}+F(j_1)>f_{j_2}s_{i_2}+F(j_2)\\)也是成立的(因為\\(s_i\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 +我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f_j\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f_{j_1}s_{i_1}+f(j_1)>f_{j_2}s_{i_1}+f(j_2)\\),那麼對於任一個 \\(i_2>i_1\\),\\(f_{j_1}s_{i_2}+f(j_1)>f_{j_2}s_{i_2}+f(j_2)\\)也是成立的(因為\\(s_i\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 -我們再觀察一次轉移式:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: +我們再觀察一次轉移式:\\(f(i)=\underset{i>j}{\min}(f(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: -\\[y=F(i)\\] +\\[y=f(i)\\] \\[x=s_i\\] \\[a=f_j\\] -\\[b=F(j)\\] +\\[b=f(j)\\] 因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。這時我們的題目便轉變成:給定一個 \\(x\\),我們要在一個線集中找出最小的 \\(y\\)。 @@ -71,7 +71,7 @@ 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 -綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(F(i)\\),便將 pair \\((f_i, F(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(f(i)\\),便將 pair \\((f_i, f(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 此圖中,\\(line1\\) 為 deque 中的倒數第二條線;\\(line2\\) 為 deque 中的最後一條線;\\(line3\\) 為新加入的直線。在這裡,\\(line1\\) 與 \\(line3\\) 的交點有更小的 \\(x\\) 座標,因此 \\(line2\\) 要被刪掉: @@ -145,9 +145,9 @@ int main(){ > > - \\(1 \leq k \leq n \leq 500000\\) -一樣,我們先試著列出轉移式。令 \\(F(i)\\) 為前 \\(i\\) 個數字貢獻的價值(同時第 \\(i\\) 個元素為區間右界)。 +一樣,我們先試著列出轉移式。令 \\(f(i)\\) 為前 \\(i\\) 個數字貢獻的價值(同時第 \\(i\\) 個元素為區間右界)。 -轉移式為:\\(F(i)=\underset{i>j}{\max}(F(j)-(i-j)^2+suf(i+1))\\) +轉移式為:\\(f(i)=\underset{i>j}{\max}(f(j)-(i-j)^2+suf(i+1))\\) 其中 \\(suf(k)\\) 是後綴和,範圍從 \\(k\\) 到 \\(n\\)。 @@ -163,17 +163,17 @@ int main(){ 瞭解轉移式的推導後,我們回來觀察這個轉移式。它看起來與 \\(y=ax+b\\) 的形式不太像,但我們試著將它展開: -\\(F(i)=\underset{i>j>i-k}{\max}(F(j)-(i-j)^2+suf(i+1))\\) +\\(f(i)=\underset{i>j>i-k}{\max}(f(j)-(i-j)^2+suf(i+1))\\) -\\(=\underset{i>j>i-k}{\max}(F(j)-i^2+2ij-j^2+suf(i+1))\\) +\\(=\underset{i>j>i-k}{\max}(f(j)-i^2+2ij-j^2+suf(i+1))\\) -\\(=\underset{i>j>i-k}{\max}(F(j)-j^2+2ij)-i^2+suf(i+1)\\) +\\(=\underset{i>j>i-k}{\max}(f(j)-j^2+2ij)-i^2+suf(i+1)\\) 可以發現 \\(max\\) 裡面的部分變成 \\(y=ax+b\\) 的形式了: -\\[y=F(i)\\] +\\[y=f(i)\\] \\[x=2i\\] \\[a=j\\] -\\[b=F(j)-j^2\\] +\\[b=f(j)-j^2\\] 這麼一來就可以使用剛剛提到的技巧了!但仔細觀察一下,這個轉移式與上一個例題有兩個不同的地方: @@ -350,25 +350,25 @@ int main(){ > - \\(0 \leq C_i \leq 2^8\\) > -先列轉移式。令 \\(F(i)\\) 為完成前 \\(i\\) 個任務的最小 cost。 +先列轉移式。令 \\(f(i)\\) 為完成前 \\(i\\) 個任務的最小 cost。 -轉移式為:\\(F(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+F(j))\\) +轉移式為:\\(f(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+f(j))\\) 其中,\\(s \times \Sigma_{k=j+1}^n C_k\\) 這一項是為了計算 \\(s\\) 所影響的任務完成時刻。由於影響的時間與總共切幾個區間有關,這裡用類似前面例題 [TIOJ 1676 - 烏龜疊疊樂](https://tioj.ck.tp.edu.tw/problems/1676),使用後綴和的想法,在每多切一個區間出來時就加上後綴因此而產生的 cost。 接下來,我們將轉移式展開: -\\(F(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+F(j))\\) -\\(=\underset{i>j}{\min}(pre_t(i) \times (pre_c(i)-pre_c(j))+s \times (pre_c(n)-pre_c(j))+F(j))\\) -\\(=\underset{i>j}{\min}(-pre_t(i) \times pre_c(j)-s \times pre_c(j)+F(j))+pre_c(i) \times pre_t(i)+s \times pre_c(n)\\) +\\(f(i)=\underset{i>j}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+f(j))\\) +\\(=\underset{i>j}{\min}(pre_t(i) \times (pre_c(i)-pre_c(j))+s \times (pre_c(n)-pre_c(j))+f(j))\\) +\\(=\underset{i>j}{\min}(-pre_t(i) \times pre_c(j)-s \times pre_c(j)+f(j))+pre_c(i) \times pre_t(i)+s \times pre_c(n)\\) 其中 \\(pre_t(i)\\) 為 \\(T\\) 的前綴和,即 \\(\Sigma_{k=1}^i T_k\\);\\(pre_c(i)\\) 為 \\(C\\) 的前綴和,即 \\(\Sigma_{k=1}^i C_k\\)。 我們可以得到: -\\[y=F(i)\\] +\\[y=f(i)\\] \\[x=-pre_t(i)\\] \\[a=pre_c(j)\\] -\\[b=F(j)-s \times pre_c(j)\\] +\\[b=f(j)-s \times pre_c(j)\\] \\(pre_c(j)\\) 是遞增的,沒什麼問題。但 \\(pre_t(i)\\) 就不是了。由於 \\(T_i\\) 有可能小於 \\(0\\),\\(T\\) 的前綴和不具單調性。也就是說,我們的查詢不再單調了。 @@ -463,7 +463,7 @@ int main(){ 令 \\(f(i)\\) 為殺掉第 \\(i\\) 關怪物所需花的最少時間。 -轉移式為:\\(F(i)=\underset{i>j}{\min}(F(j)+f_j\times s_i)\\) +轉移式為:\\(f(i)=\underset{i>j}{\min}(f(j)+f_j\times s_i)\\) 題目大致與 [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) 相同,但此題不再保證 \\(f\\) 與 \\(s\\) 的單調性,意即斜率與查詢都不再單調。不過透過 CDQ 分治,我們可以「人為」創造出斜率的單調性。 @@ -590,7 +590,7 @@ int main(){ 1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 - 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(F\\) 則保留下來存在當前節點。 + 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(f\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(f\\) 則保留下來存在當前節點。 @@ -598,7 +598,7 @@ int main(){ 2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 - 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(F\\) 向下推。 + 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(f\\) 向下推。 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 @@ -809,18 +809,18 @@ int main(){ 先將所有矩形依照 \\(x\\) 座標由大至小 sort,這時 \\(y\\) 座標也會由小至大排好。 -令 \\F(i\\) 為考慮前 \\(i\\) 個矩形,且要取第 \\(i\\) 個矩形,所能得到的最大值。 +令 \\f(i\\) 為考慮前 \\(i\\) 個矩形,且要取第 \\(i\\) 個矩形,所能得到的最大值。 -轉移式為:\\(F(i)=\underset{i>j}{\max}(x_i(y_i-y_j)+F(j)-a_i)\\) +轉移式為:\\(f(i)=\underset{i>j}{\max}(x_i(y_i-y_j)+f(j)-a_i)\\) -\\(=\underset{i>j}{\max}(F(j)-x_iy_j)+x_iy_i-a_i\\) +\\(=\underset{i>j}{\max}(f(j)-x_iy_j)+x_iy_i-a_i\\) 我們得到: -\\[y=F(i)\\] +\\[y=f(i)\\] \\[x=x_i\\] \\[a=y_j\\] -\\[b=F(j)\\] +\\[b=f(j)\\] 斜率、查詢皆單調,使用單調隊列做斜率優化。此題數字較大,要特別注意 overflow 的問題(可能需要使用 `__int128` 之類的)。 @@ -840,20 +840,20 @@ int main(){ 接著將貓照處理過的 \\(t_i\\) 由小至大 sort,題目便轉成如何將序列切成 \\(p\\) 個部分,使得 cost 最小。 -令 \\(F(i, j)\\) 為前 \\(i\\) 個 feeder 帶走前 \\(j\\) 隻貓所需的最小 cost。 +令 \\(f(i, j)\\) 為前 \\(i\\) 個 feeder 帶走前 \\(j\\) 隻貓所需的最小 cost。 -則轉移式為:\\(F(i, j)=\underset{j>k}{\min}(F(i-1, k)-\Sigma_{p=k+1}^j\space t_p+t_j \times (j-k))\\) +則轉移式為:\\(f(i, j)=\underset{j>k}{\min}(f(i-1, k)-\Sigma_{p=k+1}^j\space t_p+t_j \times (j-k))\\) -\\(=\underset{j>k}{\min}(F(i-1, k)-suf(k+1)+suf(j+1)+t_{j} \times (j-k))\\) +\\(=\underset{j>k}{\min}(f(i-1, k)-suf(k+1)+suf(j+1)+t_{j} \times (j-k))\\) 其中 \\(suf(i)\\) 為後綴和(從 \\(i\\) 加到 \\(n\\))。 由轉移式我們得到: -\\[y=F(i, j)\\] +\\[y=f(i, j)\\] \\[x=t_j\\] \\[a=-k\\] -\\[b=F(i-1, k)-suf(k+1)\\] +\\[b=f(i-1, k)-suf(k+1)\\] 斜率、查詢皆單調,可以使用單調隊列做斜率優化。 @@ -870,16 +870,16 @@ int main(){ 由於 \\(b_n=0\\),在砍掉最後一棵樹後,砍所有樹都不需要成本了。因此題目轉變成:砍掉最後一棵樹的最小成本。 -令 \\(F(i)\\) 為只砍 index 小於 \\(i\\) 的樹的情況下,砍掉第 \\(i\\) 棵樹的最小 cost。 +令 \\(f(i)\\) 為只砍 index 小於 \\(i\\) 的樹的情況下,砍掉第 \\(i\\) 棵樹的最小 cost。 -\\(F(i)=\underset{i>j}{\min}(F(j)+b_j \times a_i)\\) +\\(f(i)=\underset{i>j}{\min}(f(j)+b_j \times a_i)\\) 由轉移式我們得到: -\\[y=F(i)\\] +\\[y=f(i)\\] \\[x=a_i\\] \\[a=b_j\\] -\\[b=F(j)\\] +\\[b=f(j)\\] 斜率、查詢皆單調,可以使用單調隊列做斜率優化。 @@ -894,18 +894,18 @@ int main(){
    Solution -令 \\(F(i, j)\\) 代表從城市 \\(1\\) 用 \\(j\\) 步走到城市 \\(i\\) 的最小成本,則: +令 \\(f(i, j)\\) 代表從城市 \\(1\\) 用 \\(j\\) 步走到城市 \\(i\\) 的最小成本,則: -\\(F(i, j) = \underset{1 \le k \le n}{\min}(F(k, j-1)+i^2+k^2-2ik)\\) +\\(f(i, j) = \underset{1 \le k \le n}{\min}(f(k, j-1)+i^2+k^2-2ik)\\) -\\(= \underset{1 \le k \le n}{\min}(F(k, j-1)+k^2-2ik)+i^2\\) +\\(= \underset{1 \le k \le n}{\min}(f(k, j-1)+k^2-2ik)+i^2\\) 由轉移式我們得到: -\\[y=F(i, j)\\] +\\[y=f(i, j)\\] \\[x=i\\] \\[a=-2k\\] -\\[b=F(k, j-1)+k^2\\] +\\[b=f(k, j-1)+k^2\\] 斜率、查詢皆單調,可以使用單調隊列做斜率優化。 @@ -920,20 +920,20 @@ int main(){
    Solution -令 \\(F(i, j)\\) 為將前 \\(j\\) 個 elements 分成 \\(i\\) 個 subarrays 的最小 cost。 +令 \\(f(i, j)\\) 為將前 \\(j\\) 個 elements 分成 \\(i\\) 個 subarrays 的最小 cost。 -\\(F(i, j)=\underset{j>k}{\min}(F(i-1, k)+(\Sigma_{t=k+1}^{j}x_t)^2)\\) +\\(f(i, j)=\underset{j>k}{\min}(f(i-1, k)+(\Sigma_{t=k+1}^{j}x_t)^2)\\) -\\(=\underset{j>k}{\min}(F(i-1, k)+(pre_{j}-pre_k)^2)\\) +\\(=\underset{j>k}{\min}(f(i-1, k)+(pre_{j}-pre_k)^2)\\) -\\(=\underset{j>k}{\min}(F(i-1, k)+pre_j^2+pre_k^2-2\times pre_j\times pre_k)\\) +\\(=\underset{j>k}{\min}(f(i-1, k)+pre_j^2+pre_k^2-2\times pre_j\times pre_k)\\) 由轉移式我們得到: -\\[y=F(i, j)\\] +\\[y=f(i, j)\\] \\[x=pre_j\\] \\[a=-2\times pre_k\\] -\\[b=F(i-1, k)+pre_k^2\\] +\\[b=f(i-1, k)+pre_k^2\\] 斜率、查詢皆單調,可以使用單調隊列做斜率優化。 @@ -950,11 +950,11 @@ int main(){
    Solution -考慮 \\(F(i, j)\\) 代表能使用的機器只有前 \\(i\\) 種,第 \\(j\\) 天能得到的最大收益。 +考慮 \\(f(i, j)\\) 代表能使用的機器只有前 \\(i\\) 種,第 \\(j\\) 天能得到的最大收益。 -考慮\\(F(i)\\) 到 \\(F(i+1)\\)的轉移會發現,如果 \\(F(i, D_{i+1}) \ge P_{i+1}\\),\\(F(i+1)\\) 的函數圖形會是 \\(F(i)\\) 與直線 \\(y = G_{i+1} \times (j-D_{i+1}-1) - P_{i+1} + R_{i+1}\\) 形成的凸包。 +考慮\\(f(i)\\) 到 \\(f(i+1)\\)的轉移會發現,如果 \\(f(i, D_{i+1}) \ge P_{i+1}\\),\\(f(i+1)\\) 的函數圖形會是 \\(f(i)\\) 與直線 \\(y = G_{i+1} \times (j-D_{i+1}-1) - P_{i+1} + R_{i+1}\\) 形成的凸包。 -不然我們不可能構入此機器,\\(F(i+1) = F(i)\\)。 +不然我們不可能構入此機器,\\(f(i+1) = f(i)\\)。 於是我們只要一個能夠支援加入直線、詢問單點值的資料結構:即動態凸包。 From a392a3bc5cdb455cff2cf5a5fcbd73218c56555b Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 22 Jun 2023 14:33:43 +0800 Subject: [PATCH 34/53] resolve simple problems --- src/dp/convex_hull_optimization.md | 31 +++++++++++++----------------- 1 file changed, 13 insertions(+), 18 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 789e3437..10e77a13 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -8,7 +8,7 @@ > [CSES - Monster Game I](https://cses.fi/problemset/task/2084/) > -> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值,其中 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。每關有兩個數字 \\(s_i\\) 與 \\(f_i\\),且 \\(s\\) 隨 \\(i\\) 遞增、\\(f\\) 隨 \\(i\\) 遞減。你會有一個起始的能力值,對於除了第 \\(n\\) 關以外的怪物,你可以選擇打或不打,打了就能將能力值更新成 \\(f_i\\)。而打敗一關的怪物需消耗 \\(s_i \times f\\) 的 cost,其中 \\(f\\) 為當前能力值。求打敗第 \\(n\\) 關怪物的最小 cost。 > > - \\(1\leq n\leq 2\times 10^5\\) @@ -20,16 +20,14 @@ ### 斜率單調 & 查詢單調 -我們觀察一下可以發現,其實並不是所有的候選人 \\(j\\) 都有可能成為轉移來源。由於 \\(f_j\\) 是遞減的,因此對於\\(j_2>j_1\\),若有一個 \\(i_1\\) 使得 \\(f_{j_1}s_{i_1}+f(j_1)>f_{j_2}s_{i_1}+f(j_2)\\),那麼對於任一個 \\(i_2>i_1\\),\\(f_{j_1}s_{i_2}+f(j_1)>f_{j_2}s_{i_2}+f(j_2)\\)也是成立的(因為\\(s_i\\)遞增),意即在 \\(i_1\\) 之後 \\(j_1\\) 就再也不會被取到了,因為至少有 \\(j_2\\) 做為更好的轉移來源。 - -我們再觀察一次轉移式:\\(f(i)=\underset{i>j}{\min}(f(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: +我們觀察一下轉移式:\\(f(i)=\underset{i>j}{\min}(f(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: \\[y=f(i)\\] \\[x=s_i\\] \\[a=f_j\\] \\[b=f(j)\\] -因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。這時我們的題目便轉變成:給定一個 \\(x\\),我們要在一個線集中找出最小的 \\(y\\)。 +因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。在求 \\(f(i)\\) 時,我們的題目便轉變成:給定一個 \\(x\\),也就是給定 \\(s_i\\) ,求他帶入一個直線集合中,哪一條能有最小的 \\(y\\),也就是哪個 \\(j\\) 能有最小的 \\(f(j) + f_j * s_i\\) 接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集,應該會長類似這個樣子: @@ -39,7 +37,7 @@ -將不在凸包上的線拿掉後: +為了說明方便先將不再凸包的線拿掉,事實上每次插入一條線我們都能判斷哪些線將不在凸包上,後面會講解: @@ -47,7 +45,7 @@ - 查詢:對於 \\(x=s_i\\),我們要能找出其與凸包的交點。 - 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,我們取的直線斜率一定會越來越小。因此我們可以只考慮目前斜率最大的直線,並確認目前直線與 \\(x=s_i\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線(斜率第二大的直線)與 \\(x=s_i\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前斜率最大的直線丟掉。反之,斜率最大的直線與 \\(x=s_i\\) 的交點就是我們要找的最小值。 + 由於查詢具單調性,以此題而言,查詢遞增且我們要找最小值,最小值所在的直線的斜率一定會越來越小。因此我們可以只考慮目前斜率最大的直線,並確認目前直線與 \\(x=s_i\\) 的交點是否真的是最小值。確認的方法便是比較當前交點以及下一條直線(斜率第二大的直線)與 \\(x=s_i\\) 的交點,若下一條直線能讓我們得到更好的答案,我們就要把當前斜率最大的直線丟掉。反之,斜率最大的直線與 \\(x=s_i\\) 的交點就是我們要找的最小值。 比較兩條直線提供的答案: @@ -77,7 +75,7 @@ -在查詢最小值時,我們不斷比較隊列中第一個與第二個元素,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 +在查詢最小值時,我們不斷比較隊列中第一個與第二個直線,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 這麼一來複雜度便降到 \\(O(n)\\)。 @@ -319,11 +317,11 @@ int main(){ 以上就是斜率優化的例題與概念,我們再複習一下大致的步驟: 1. 將初始值放入 deque -2. 跑迴圈,從 1 到 n,做下面兩個步驟: +2. 根據填表順序計算 \\(f(i)\\): i. 不斷比較 deque 最前端的兩條線。若第二條線有較好的答案就 pop 掉最前端的線,不斷重複直到最前端的線有最好的答案,或是 deque 中只剩一條線。 - ii. 利用 deque 最前端的線計算 \\(dp[i]\\)。 + ii. 利用 deque 最前端的線計算 \\(f(i)\\)。 iii. 不斷比較 deque 最尾端的線與新的直線。若前者在新的線加入後便不在凸包中就 pop 掉,最後插入新的直線。 @@ -333,9 +331,6 @@ int main(){ - 注意題目是否會有轉移範圍的限制。如果有的話,在計算 dp 值之前,過期的線段也要 pop 掉,同時拔掉不在凸包上的點的條件也要做更改。 - 轉移式取 max 與取 min 的差別在於建下凸包或是建上凸包,方法是差不多的。 - - ### 斜率單調 & 查詢不單調 上面的題目由於查詢單調,我們才能保證可以只看 deque 前端的線段就找到最小值。那麼,要是查詢不單調該怎麼辦? @@ -384,7 +379,7 @@ int main(){
    Solution Code -此題時間較緊,需要壓一下常數,因此這邊使用陣列來實作。一般情況下,只要可以 push_back 與 pop_back 的資料結構都可以,像是 vector。 +此題時間較緊,需要壓一下常數,因此這邊使用陣列來實作。一般情況下,deque 常數較大,且因為我們不需要 pop_front,可以使用 vector 來實作。 ```cpp #include @@ -457,7 +452,7 @@ int main(){ 直接來看一個題目。 > [CSES - Monster Game II](https://cses.fi/problemset/task/2085) > -> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。打敗一關的怪物需消耗 \\(sf\\) 的 cost,\\(s\\) 為怪物血量,\\(f\\) 為能力值。你會有一個起始的能力值,每到一關,你可以選擇是否打敗該關怪物,並將能力值更新為 \\(f_i\\)。求打敗第 \\(n\\) 關怪物的最小 cost。 +> 有 \\(n\\) 個關卡,每關有一個怪物,通關時必須從第一關開始依序通關。每關有兩個數字 \\(s_i\\) 與 \\(f_i\\)。你會有一個起始的能力值,對於除了第 \\(n\\) 關以外的怪物,你可以選擇打或不打,打了就能將能力值更新成 \\(f_i\\)。而打敗一關的怪物需消耗 \\(s_i \times f\\) 的 cost,其中 \\(f\\) 為當前能力值。求打敗第 \\(n\\) 關怪物的最小 cost。 > > - \\(1\leq n\leq 2\times 10^5\\) @@ -580,7 +575,7 @@ int main(){ 1. 加入一條形式為 \\(y=ax+b\\) 的直線 2. 詢問 \\(x=k\\) 與所有直線交點的 \\(y\\) 的最大值 -看起來像是區間修改、單點查詢的題目。我們考慮線段樹的作法,令左右界為值域,每個節點存的東西是一條直線。接著我們來思考如何插入與查詢。 +看起來像是區間修改、單點查詢的題目。我們考慮線段樹的作法,用線段樹維護值域,每個節點存的是在 \\(x\\) 等於該節點對應的值時,能獲的最大值的直線是哪一條。接著我們來思考如何插入與查詢。 - **插入** @@ -715,7 +710,7 @@ int main(){ #### 動態凸包 -動態凸包,顧名思義就是動態維護的凸包,支援的操作有插入直線與詢問在 \\(x\\) 處的最大值。這個東西有模板可以用,應該蠻好找的,因此這邊不討論如何實作,只討論如何使用它。 +動態凸包,顧名思義就是動態維護的凸包,支援的操作有插入直線與詢問在 \\(x\\) 處的最大值。這個東西有模板可以用,應該蠻好找的,因此這邊不討論如何實作,只討論如何使用它。不過非常建議將動態凸包的模板放到 codebook 內,當遇到斜率優化的題目且斜率不單調時可以直接套。 插入直線與查詢最大值恰好就是我們需要的兩個操作,因此就直接使用即可。如果轉移式是要求最小值,我們就只要把所有東西都加負號並改取最大值就好。 @@ -967,7 +962,7 @@ int main(){ - [[Tutorial] Convex Hull Trick - Geometry being useful](https://codeforces.com/blog/entry/63823) - [【學習筆記】動態規劃—斜率優化 DP(超詳細)](https://www.cnblogs.com/Xing-Ling/p/11210179.html) - [Oi wiki - 斜率優化](https://oi-wiki.org/dp/opt/slope/) -- [TIOJ 建中培訓講義](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) +- [2016 建中培訓講義 - 分治、DP 進階](https://tioj.ck.tp.edu.tw/uploads/attachment/5/27/5.pdf) - [Algorithms for competitive programming - Convex hull trick](https://cp-algorithms.com/geometry/convex_hull_trick.html) - [PEGWiki - Convex hull trick](https://wcipeg.com/wiki/Convex_hull_trick) - [USACO Guide - Convex Hull Trick](https://usaco.guide/plat/convex-hull-trick?lang=cpp) From fb83546c842e403db52dfc6cb65e653ad991769d Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Tue, 11 Jul 2023 10:44:55 +0800 Subject: [PATCH 35/53] image modify (cses2084_query_1) --- src/dp/convex_hull_optimization.md | 4 ++-- .../cses2084_query_1_v2.png | Bin 0 -> 80041 bytes .../cses2084_query_2_v2.png | Bin 0 -> 71032 bytes 3 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 src/dp/image/convex_hull_optimization/cses2084_query_1_v2.png create mode 100644 src/dp/image/convex_hull_optimization/cses2084_query_2_v2.png diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 10e77a13..281cda93 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -49,11 +49,11 @@ 比較兩條直線提供的答案: - + 下一條直線提供的答案更小,因此我們刪除斜率最大的直線: - + - 插入:插入一條新的線 diff --git a/src/dp/image/convex_hull_optimization/cses2084_query_1_v2.png 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11 Jul 2023 11:36:21 +0800 Subject: [PATCH 36/53] image modify (cses2084_insert_2) --- src/dp/convex_hull_optimization.md | 4 ++-- .../cses2084_insert_1_v2.png | Bin 0 -> 88013 bytes .../cses2084_insert_2_v2.png | Bin 0 -> 100883 bytes 3 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 src/dp/image/convex_hull_optimization/cses2084_insert_1_v2.png create mode 100644 src/dp/image/convex_hull_optimization/cses2084_insert_2_v2.png diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 281cda93..e59c8c50 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -61,11 +61,11 @@ 原本的線集: - + 加入新的直線: - + 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 diff --git a/src/dp/image/convex_hull_optimization/cses2084_insert_1_v2.png b/src/dp/image/convex_hull_optimization/cses2084_insert_1_v2.png new file mode 100644 index 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zb}g)yDhrTl{mgl0k?`h-Om7=DQYAKrbUGhvU-XqOw+$C<#J${ws4U8g?qHO@8$^T$>gd1PwU7~AHl(tm7g`yYXjU{4C(F<>_Rtl^>rh{!6dj8VJSysRPJ4$4oVDd@rc}%|DVY={s(((?fb9PtW`1 zPTRi|rBmG3(f-q3wPXnd7HF>YNLmT$E8g)NG;-7~Zpj+C@izt}?2b##P^zHxs zx@dBV>L38heH|{O^M4~>AFpa_Y(DkY6h*ZJe0yZ%dV*v_%HMk`JKX^OF4E=4fprvV zznay%Ki;UuZd&zoMg*?oNBa$}$OgF*NNrzkGTqo|lj(Yn&#DJVF_h!)$UK!QUB7T@ zh+@^WHxRf2?EB#Vy>;eGXa7v+g!!=1uH#0PB}PT3_~* zZE7wOXze%5m~!xa1vAWMlMYL3SIF9L4pPu1 kSt{^0$r$-tRmE9rGNiU6ny`eKK*@*UmX7Bt&iLK=KTK+%H2?qr literal 0 HcmV?d00001 From 42ef1f853d30dcdce68145d3cd9a4ab24d89e740 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Mon, 7 Aug 2023 09:16:28 +0800 Subject: [PATCH 37/53] fixed i>j to jj}{\min}(f(j)+f_j\times s_i)\\) +轉移式為:\\(f(i)=\underset{jj}{\min}(f(j)+f_j\times s_i)\\)。形式其實類似於 \\(y=ax+b\\),其中 \\(a\\) 與 \\(b\\) 只和 \\(j\\) 有關、\\(x\\) 與 \\(y\\) 只和 \\(i\\) 有關。以剛剛的轉移式來說,對應關係如下: +我們觀察一下轉移式:\\(f(i)=\underset{jj}{\max}(f(j)-(i-j)^2+suf(i+1))\\) +轉移式為:\\(f(i)=\underset{jj}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+f(j))\\) +轉移式為:\\(f(i)=\underset{jj}{\min}(\Sigma_{k=1}^i T_k \times \Sigma_{k=j+1}^i C_k+s \times \Sigma_{k=j+1}^n C_k+f(j))\\) -\\(=\underset{i>j}{\min}(pre_t(i) \times (pre_c(i)-pre_c(j))+s \times (pre_c(n)-pre_c(j))+f(j))\\) -\\(=\underset{i>j}{\min}(-pre_t(i) \times pre_c(j)-s \times pre_c(j)+f(j))+pre_c(i) \times pre_t(i)+s \times pre_c(n)\\) +\\(f(i)=\underset{jj}{\min}(f(j)+f_j\times s_i)\\) +轉移式為:\\(f(i)=\underset{jj}{\max}(x_i(y_i-y_j)+f(j)-a_i)\\) +轉移式為:\\(f(i)=\underset{jj}{\max}(f(j)-x_iy_j)+x_iy_i-a_i\\) +\\(=\underset{jj}{\min}(f(j)+b_j \times a_i)\\) +\\(f(i)=\underset{j Date: Mon, 7 Aug 2023 09:18:45 +0800 Subject: [PATCH 38/53] fixed suf(k) -> suf(i+1) --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index e91c614b..4662afac 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -147,7 +147,7 @@ int main(){ 轉移式為:\\(f(i)=\underset{j Date: Mon, 7 Aug 2023 09:50:41 +0800 Subject: [PATCH 39/53] add explaination for interval (cdq) --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 4662afac..e0cc7062 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -464,7 +464,7 @@ int main(){ CDQ 分治的過程就是對於一個序列 \\([l, r]\\),我們先遞迴解決 \\([l, mid]\\) 的子問題,再處理 \\([l, mid]\\) 對 \\([mid + 1, r]\\) 的貢獻,最後再遞迴處理 \\([mid + 1, r]\\)。 -以這題來說,我們會先遞迴左半區間,接著用計算好的左半區間,去更新右半區間的 dp 值,最後再往下遞迴右半區間。 +以這題來說,對於當前遞迴到的區間 \\([l, r]\\),我們會先遞迴左半區間 \\([l, mid]\\) ,接著用計算好的左半區間,去更新右半區間 \\([mid + 1, r]\\) 的 dp 值,最後再往下遞迴右半區間。 前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 From a492f9ff2b979824366419e0ab7ff6e5460e2803 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 16 Aug 2023 17:32:41 +0800 Subject: [PATCH 40/53] template code fixed (cses2084, tioj1676) --- src/dp/convex_hull_optimization.md | 115 +++++++++++++++++------------ 1 file changed, 67 insertions(+), 48 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index e0cc7062..8a94a94e 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -85,24 +85,19 @@ #include using namespace std; -typedef pair pii; -typedef long long ll; -typedef pair pll; -#define X first -#define Y second -#define pf pop_front -#define pb pop_back -#define eb emplace_back -#define N 200005 +using ll = long long; +using pll = pair; +const ll N = 200005; -ll s[N], f[N]; +int s[N], f[N]; ll dp[N]; ll cal(ll x, pll line){ - return x * line.X + line.Y; + return x * line.first + line.second; } + bool cmp(pll line1, pll line2, pll line3){ - return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); + return (line3.second - line1.second) * (line1.first - line2.first) <= (line2.second - line1.second) * (line1.first - line3.first); } int main(){ @@ -115,14 +110,21 @@ int main(){ cin >> f[i]; } deque dq; - dq.eb(f[0], 0); + dq.emplace_back(f[0], 0); + for(int i = 1; i <= n; i++){ - while(dq.size() >= 2 && cal(s[i], dq[0]) > cal(s[i], dq[1])) dq.pf(); + while(dq.size() >= 2 && cal(s[i], dq[0]) > cal(s[i], dq[1])) dq.pop_front(); dp[i] = cal(s[i], dq[0]); + pll line(f[i], dp[i]); - while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); - dq.eb(line); + int sz = dq.size(); + while(sz >= 2 && cmp(dq[sz - 2], dq[sz - 1], line)){ + dq.pop_back(); + sz--; + } + dq.emplace_back(line); } + cout << dp[n] << "\n"; return 0; } @@ -189,11 +191,16 @@ int main(){ ```cpp for(ll i = 1; i <= n; i++){ - while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pop_front(); dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); - while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); - dq.eb(line); + int sz = dq.size(); + while(sz >= 2 && cmp(dq[sz - 2], dq[sz - 1], line)){ + dq.pop_back(); + sz--; + } + dq.emplace_back(line); } ``` @@ -205,12 +212,17 @@ int main(){ ```cpp for(ll i = 1; i <= n; i++){ - while(dq.size() >= 1 && dq[0].idx < i - k) dq.pf(); - while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + while(dq.size() >= 1 && dq[0].idx < i - k) dq.pop_front(); + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pop_front(); dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); - while(dq.size() >= 2 && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line)) dq.pb(); - dq.eb(line); + int sz = dq.size(); + while(sz >= 2 && cmp(dq[sz - 2], dq[sz - 1], line)){ + dq.pop_back(); + sz--; + } + dq.emplace_back(line); } ``` @@ -242,14 +254,19 @@ int main(){ ```cpp for(ll i = 1; i <= n; i++){ - while(dq.size() >= 1 && dq[0].idx < i - k) dq.pf(); - while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pf(); + while(dq.size() >= 1 && dq[0].idx < i - k) dq.pop_front(); + while(dq.size() >= 2 && cal(x[i], dq[0]) < cal(x[i], dq[1])) dq.pop_front(); dp[i] = cal(x[i], dq[0]); + pll line(a[i], b[i]); - while(dq.size() >= 2 - && cmp(dq[dq.size() - 2], dq[dq.size() - 1], line) - && cal(x[dq[dq.size() - 2].idx], dq[dq.size() - 1]) <= cal(x[dq[dq.size() - 2].idx], line)) dq.pb(); - dq.eb(line); + int sz = dq.size(); + while(sz >= 2 + && cmp(dq[sz - 2], dq[sz - 1], line) + && cal(x[dq[sz - 2].idx], dq[sz - 1]) <= cal(x[dq[sz - 2].idx], line)){ + dq.pop_back(); + sz--; + } + dq.emplace_back(line); } ``` @@ -264,27 +281,22 @@ int main(){ #pragma GCC optimize("O2") using namespace std; -typedef long long ll; -typedef pair pll; -#define X first -#define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define eb emplace_back -#define pb pop_back -#define pf pop_front -#define N 500005 +using ll = long long; +using pll = pair; +const ll N = 500005; ll a[N], suf[N], dp[N]; ll cal(ll x, pll line){ - return x * line.X + line.Y; + return x * line.first + line.second; } + bool cmp(pll line1, pll line2, pll line3){ - return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); + return (line3.second - line1.second) * (line1.first - line2.first) <= (line2.second - line1.second) * (line1.first - line3.first); } int main(){ - io + ios_base::sync_with_stdio(0); cin.tie(0); ll n, k; cin >> n >> k; for(int i = 1; i <= n; i++){ @@ -293,17 +305,24 @@ int main(){ for(int i = n; i >= 1; i--){ suf[i] = a[i] + suf[i + 1]; } + deque dq; - dq.eb(0, 0); + dq.emplace_back(0, 0); for(ll i = 1; i <= n; i++){ - while(dq.size() >= 1 && dq[0].X < i-k) dq.pf(); - while(dq.size() >= 2 && cal(2 * i, dq[0]) <= cal(2 * i, dq[1])) dq.pf(); + while (dq.size() > 0 && dq[0].first < i-k) dq.pop_front(); + while(dq.size() >= 2 && cal(2 * i, dq[0]) <= cal(2 * i, dq[1])) dq.pop_front(); dp[i] = cal(2 * i, dq[0]) - i * i + suf[i + 1]; + pll line(i, dp[i] - i * i); - while(dq.size() >= 2 - && cal(2 * (dq[dq.size() - 2].X + k), dq.back()) <= cal(2 * (dq[dq.size() - 2].X + k), line) - && cmp(dq[dq.size() - 2], dq.back(), line)) dq.pb(); - dq.eb(line); + int sz = dq.size(); + while(sz >= 2 + && cal(2 * (dq[sz - 2].first + k), dq.back()) <= cal(2 * (dq[sz - 2].first + k), line) + && cmp(dq[sz - 2], dq.back(), line) + ){ + sz--; + dq.pop_back(); + } + dq.emplace_back(line); } cout << dp[n] << "\n"; return 0; From c52fcb6d5f29f6b36c075fa2f6d555442126253c Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 16 Aug 2023 18:13:36 +0800 Subject: [PATCH 41/53] rearrange content for explaination of deleting element from convex hull --- src/dp/convex_hull_optimization.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 8a94a94e..3965f613 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -69,11 +69,13 @@ 在我們加入新的直線後,有一些直線便不在凸包上了,意即它們不可能成為任何查詢的答案。這時我們要將它們移出凸包,否則會取到錯誤的答案。 -綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(f(i)\\),便將 pair \\((f_i, f(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們要檢查是否有直線在這次插入後,再也不會被當成答案。檢查的方式是看 deque 中的倒數第二條線與當前要插入的直線的交點,以及 deque 中的倒數第二條線與 deque 中的最後一條線的交點。若前者有更小的 \\(x\\) 座標,則 deque 中的最後一條線永遠不會被取到,要 pop 掉。 + 因此,在每次加入新的直線之前,我們要檢查是否有直線在這次加入後,再也不會被當成答案。檢查的方式是看凸包中倒數第二條線(根據加入順序)與當前要插入的直線的交點,以及凸包中倒數第二條線與最後一條線的交點。若前者有更小的 \\(x\\) 座標,則最後一條線不在凸包上,要拿掉。 -此圖中,\\(line1\\) 為 deque 中的倒數第二條線;\\(line2\\) 為 deque 中的最後一條線;\\(line3\\) 為新加入的直線。在這裡,\\(line1\\) 與 \\(line3\\) 的交點有更小的 \\(x\\) 座標,因此 \\(line2\\) 要被刪掉: + 舉例來說,下圖中 \\(line1\\) 為凸包中的倒數第二條線;\\(line2\\) 為凸包中最後一條線;\\(line3\\) 為新加入的直線。在這裡,\\(line1\\) 與 \\(line3\\) 的交點有更小的 \\(x\\) 座標,因此 \\(line2\\) 要被刪掉: - + + +綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(f(i)\\),便將 pair \\((f_i, f(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們檢查是否有直線在這次插入後,再也不會被當成答案,有的話就從後端 pop 掉。 在查詢最小值時,我們不斷比較隊列中第一個與第二個直線,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 From d460fa56dbb2d7469d19f36e54aa2328d02a61e6 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Thu, 17 Aug 2023 12:14:37 +0800 Subject: [PATCH 42/53] solution code fixed (cses 2085 - dynamic hull) --- src/dp/convex_hull_optimization.md | 62 +++++++++++++++--------------- 1 file changed, 32 insertions(+), 30 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 3965f613..c659533e 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -744,45 +744,47 @@ int main(){ #pragma GCC optimize("O2") using namespace std; -typedef long long ll; -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define N 200005 +using ll = long long; +const ll N = 200005; struct Line { - mutable ll k, m, p; - bool operator<(const Line& o) const { return k < o.k; } - bool operator<(ll x) const { return p < x; } + mutable ll k, m, p; + bool operator<(const Line& o) const { return k < o.k; } + bool operator<(ll x) const { return p < x; } }; struct LineContainer : multiset> { - // (for doubles, use inf = 1/.0, div(a,b) = a/b) - static const ll inf = LLONG_MAX; - ll div(ll a, ll b) { // floored division - return a / b - ((a ^ b) < 0 && a % b); } - bool isect(iterator x, iterator y) { - if (y == end()) return x->p = inf, 0; - if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; - else x->p = div(y->m - x->m, x->k - y->k); - return x->p >= y->p; - } - void add(ll k, ll m) { - auto z = insert({k, m, 0}), y = z++, x = y; - while (isect(y, z)) z = erase(z); - if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); - while ((y = x) != begin() && (--x)->p >= y->p) - isect(x, erase(y)); - } - ll query(ll x) { - assert(!empty()); - auto l = *lower_bound(x); - return l.k * x + l.m; - } + // (for doubles, use inf = 1/.0, div(a,b) = a/b) + static const ll inf = LLONG_MAX; + ll div(ll a, ll b) { // floored division + return a / b - ((a ^ b) < 0 && a % b); } + bool isect(iterator x, iterator y) { + if (y == end()) return x->p = inf, 0; + if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; + else x->p = div(y->m - x->m, x->k - y->k); + return x->p >= y->p; + } + void add(ll k, ll m) { + auto z = insert({k, m, 0}), y = z++, x = y; + while (isect(y, z)) z = erase(z); + if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); + while ((y = x) != begin() && (--x)->p >= y->p) + isect(x, erase(y)); + } + ll query(ll x) { + assert(!empty()); + auto l = *lower_bound(x); + return l.k * x + l.m; + } }; -ll s[N], f[N], dp[N]; +int s[N], f[N]; +ll dp[N]; int main(){ - io + ios_base::sync_with_stdio(0); + cin.tie(0); + ll n, x; cin >> n >> x; for(int i = 1; i <= n; i++){ From 62de19d62433d9256cee3ee7f581925ab03b213b Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Fri, 18 Aug 2023 10:52:00 +0800 Subject: [PATCH 43/53] fix rest of the solution code --- src/dp/convex_hull_optimization.md | 122 ++++++++++++++--------------- 1 file changed, 60 insertions(+), 62 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index c659533e..459290ad 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -406,12 +406,9 @@ int main(){ #include using namespace std; -typedef long long ll; -typedef pair pll; -#define X first -#define Y second -#define io ios_base::sync_with_stdio(0); cin.tie(0); -#define N 300005 +using ll = long long; +using pll = pair; +const ll N = 300005; ll pret[N], prec[N]; ll dp[N]; @@ -419,10 +416,10 @@ pll v[N]; ll sz; inline ll cal(ll x, pll line){ - return x * line.X + line.Y; + return x * line.first + line.second; } inline bool cmp2(pll p1, pll p2, ll x){ - return (p2.Y - p1.Y) > (p1.X - p2.X) * x; + return (p2.second - p1.second) > (p1.first - p2.first) * x; } inline pll find_line(ll x){ int l = 0, r = sz - 2, mid, ans = r + 1; @@ -436,12 +433,13 @@ inline pll find_line(ll x){ } return v[ans]; } + inline bool cmp(pll line1, pll line2, pll line3){ - return (__int128)(line3.Y - line1.Y) * (__int128)(line1.X - line2.X) <= (__int128)(line2.Y - line1.Y) * (__int128)(line1.X - line3.X); + return (__int128)(line3.second - line1.second) * (__int128)(line1.first - line2.first) <= (__int128)(line2.second - line1.second) * (__int128)(line1.first - line3.first); } int main(){ - io + ios_base::sync_with_stdio(0); cin.tie(0); ll n, s; cin >> n >> s; for(int i = 1; i <= n; i++){ @@ -449,6 +447,7 @@ int main(){ pret[i] += pret[i - 1]; prec[i] += prec[i - 1]; } + v[sz++] = pll(0, 0); for(int i = 1; i <= n; i++){ pll fl = find_line(pret[i]); @@ -507,28 +506,25 @@ CDQ 分治的過程就是對於一個序列 \\([l, r]\\),我們先遞迴解決 #pragma GCC optimize("O2") using namespace std; -typedef pair pii; -typedef long long ll; -typedef pair pll; -#define X first -#define Y second -#define eb emplace_back -#define pb pop_back -#define N 200005 +using ll = long long; +using pii = pair; +using pll = pair; +const ll N = 200005; const ll INF = 1e14; -ll dp[N], s[N], f[N]; +int s[N], f[N]; +ll dp[N]; ll cal(pll line, ll x){ - return x * line.X + line.Y; + return x * line.first + line.second; } ll cmp2(pll p1, pll p2, ll x){ - return (p2.Y - p1.Y) > (p1.X - p2.X) * x; + return (p2.second - p1.second) > (p1.first - p2.first) * x; } bool cmp(pll line1, pll line2, pll line3){ - return (line3.Y - line1.Y) * (line1.X - line2.X) <= (line2.Y - line1.Y) * (line1.X - line3.X); + return (line3.second - line1.second) * (line1.first - line2.first) <= (line2.second - line1.second) * (line1.first - line3.first); } pll find_line(vector v, ll x){ @@ -547,7 +543,7 @@ pll find_line(vector v, ll x){ vector solve(int l, int r){ vector v; if(l == r){ - v.eb(f[l], dp[l]); + v.emplace_back(f[l], dp[l]); return v; } int mid = (l + r) >> 1; @@ -558,13 +554,14 @@ vector solve(int l, int r){ dp[i] = min(dp[i], cal(fl, s[i])); } vector v2 = solve(mid + 1, r); + int idx1 = 0, idx2 = 0; while(idx1 < v1.size() || idx2 < v2.size()){ pll pl; if(idx2 >= v2.size() || (idx1 < v1.size() && v1[idx1] >= v2[idx2])) pl = v1[idx1++]; else pl = v2[idx2++]; - while(v.size() >= 2 && cmp(v[v.size() - 2], v.back(), pl)) v.pb(); - v.eb(pl); + while(v.size() >= 2 && cmp(v[v.size() - 2], v.back(), pl)) v.pop_back(); + v.emplace_back(pl); } return v; } @@ -665,45 +662,48 @@ int main(){ #pragma GCC optimize("O2") using namespace std; -typedef pair pii; -typedef long long ll; -typedef pair pll; -#define X first -#define Y second -#define N 1000005 +using ll = long long; +using pii = pair; +using pll = pair; +const ll N = 1000005; const ll INF = 1e18; -ll s[N], f[N]; +int s[N], f[N]; ll dp[N]; -pll seg[N * 4]; -ll cal(pll line, ll x){ - return line.X * x + line.Y; -} +struct LiChao{ + int n; + vector seg; + LiChao(int _n): n(_n){ + seg.assign(4 * n + 5, pll(0, INF)); + } -void insert(int l, int r, int id, pll line){ - if(l == r){ - if(cal(line, l) < cal(seg[id], l)) seg[id] = line; - return; + ll cal(pll line, ll x){ + return line.first * x + line.second; } - int mid = (l + r) >> 1; - if(line.X > seg[id].X) swap(line, seg[id]); - if(cal(line, mid) <= cal(seg[id], mid)){ - insert(l, mid, id * 2, seg[id]); - seg[id] = line; + void insert(int l, int r, int id, pll line){ + if(l == r){ + if(cal(line, l) < cal(seg[id], l)) seg[id] = line; + return; + } + int mid = (l + r) >> 1; + if(line.first > seg[id].first) swap(line, seg[id]); + if(cal(line, mid) <= cal(seg[id], mid)){ + insert(l, mid, id * 2, seg[id]); + seg[id] = line; + } + else{ + insert(mid + 1, r, id * 2 + 1, line); + } } - else{ - insert(mid + 1, r, id * 2 + 1, line); + ll query(int l, int r, int id, ll x){ + if(x < l || x > r) return INF; + if(l == r) return cal(seg[id], x); + int mid = (l + r) >> 1; + ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); + return min(val, cal(seg[id], x)); } -} - -ll query(int l, int r, int id, ll x){ - if(x < l || x > r) return INF; - if(l == r) return cal(seg[id], x); - int mid = (l + r) >> 1; - ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); - return min(val, cal(seg[id], x)); -} +}; int main(){ int n, x; @@ -714,13 +714,11 @@ int main(){ for(int i = 1; i <= n; i++){ cin >> f[i]; } - for(int i = 1; i <= 4e6; i++){ - seg[i] = pll(0, INF); - } - insert(1, 1e6, 1, pll(x, 0)); + LiChao lt(N); + lt.insert(1, 1e6, 1, pll(x, 0)); for(int i = 1; i <= n; i++){ - dp[i] = query(1, 1e6, 1, s[i]); - insert(1, 1e6, 1, pll(f[i], dp[i])); + dp[i] = lt.query(1, 1e6, 1, s[i]); + lt.insert(1, 1e6, 1, pll(f[i], dp[i])); } cout << dp[n] << "\n"; return 0; From f791bbb3e8960809ee47533f772f9eead507130a Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Fri, 18 Aug 2023 11:02:14 +0800 Subject: [PATCH 44/53] add template code for lichao tree --- src/dp/convex_hull_optimization.md | 43 ++++++++++++++++++++++++++++++ 1 file changed, 43 insertions(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 459290ad..085f1e6a 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -651,6 +651,49 @@ int main(){ } ``` +我們可以將以上兩種操作包成 template code 以方便使用: + +

    Template Code + +```cpp +struct LiChao{ + int n; + vector seg; + LiChao(int _n): n(_n){ + seg.assign(4 * n + 5, pll(0, INF)); + } + + ll cal(pll line, ll x){ + return line.first * x + line.second; + } + void insert(int l, int r, int id, pll line){ + if(l == r){ + if(cal(line, l) < cal(seg[id], l)) seg[id] = line; + return; + } + int mid = (l + r) >> 1; + if(line.first > seg[id].first) swap(line, seg[id]); + if(cal(line, mid) <= cal(seg[id], mid)){ + insert(l, mid, id * 2, seg[id]); + seg[id] = line; + } + else{ + insert(mid + 1, r, id * 2 + 1, line); + } + } + ll query(int l, int r, int id, ll x){ + if(x < l || x > r) return INF; + if(l == r) return cal(seg[id], x); + int mid = (l + r) >> 1; + ll val = x <= mid ? query(l, mid, id * 2, x) : query(mid + 1, r, id * 2 + 1, x); + return min(val, cal(seg[id], x)); + } +}; +``` + +
    +
    + --- 現在我們來看題目,一樣是 [CSES - Monster Game II](https://cses.fi/problemset/task/2085),但這次我們考慮李超線段樹的作法。想法很直觀,我們只要先在李超樹上查詢最小值以得到當前 dp 值,再將新的線插入樹中就好。時間複雜度 \\(O(n\log n)\\)。 From e7320929d234706e8f56096ce7481c615b5a4786 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 10:48:21 +0800 Subject: [PATCH 45/53] sigma fixed --- src/dp/convex_hull_optimization.md | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 085f1e6a..ee99a985 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -27,7 +27,7 @@ \\[a=f_j\\] \\[b=f(j)\\] -因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。在求 \\(f(i)\\) 時,我們的題目便轉變成:給定一個 \\(x\\),也就是給定 \\(s_i\\) ,求他帶入一個直線集合中,哪一條能有最小的 \\(y\\),也就是哪個 \\(j\\) 能有最小的 \\(f(j) + f_j * s_i\\) +因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。在求 \\(f(i)\\) 時,我們的題目便轉變成:給定一個 \\(x\\),也就是給定 \\(s_i\\) ,求他帶入一個直線集合中,哪一條能有最小的 \\(y\\),也就是哪個 \\(j\\) 能有最小的 \\(f(j) + f_j \times s_i\\) 接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集,應該會長類似這個樣子: @@ -143,7 +143,7 @@ int main(){ > [TIOJ 1676 - 烏龜疊疊樂](https://tioj.ck.tp.edu.tw/problems/1676) > > 給定一個長度為 \\(n\\) 的陣列與一個整數 \\(k\\),你可以將陣列切成數段連續的區間,區間大小不能超過 \\(k\\),求切完之後的最大價值。 -> 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \Sigma^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 +> 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \sum\limits^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 > > - \\(1 \leq k \leq n \leq 500000\\) @@ -153,15 +153,15 @@ int main(){ 其中 \\(suf(i+1)\\) 是後綴和,範圍從 \\(i+1\\) 到 \\(n\\)。 -除了 \\(suf(i+1)\\) 這一項外,其他部分還蠻直觀的。而 \\(suf(i+1)\\) 這一項對應到題目的 \\((m-1)\times \Sigma^r_{i=l}a_i\\),我們看一張示意圖以更好理解 \\(suf(i+1)\\) 的意義: +除了 \\(suf(i+1)\\) 這一項外,其他部分還蠻直觀的。而 \\(suf(i+1)\\) 這一項對應到題目的 \\((m-1)\times \sum\limits^r_{i=l}a_i\\),我們看一張示意圖以更好理解 \\(suf(i+1)\\) 的意義: -上面的示意圖對每個區間畫出了其 \\((m-1)\times \Sigma^r_{i=l}a_i\\) 的值。我們可以將它們重新分割變成下面這樣: +上面的示意圖對每個區間畫出了其 \\((m-1)\times \sum\limits^r_{i=l}a_i\\) 的值。我們可以將它們重新分割變成下面這樣: -可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \Sigma^r_{i=l}a_i\\),這個部分的總和。 +可以發現這每一塊其實就是一段後綴,而且每個後綴的起點對應到每個區間的起點(除了第一個區間不考慮)。我們轉移式中的 \\(i\\) 是區間終點,因此 \\(i+1\\) 便對應到區間起點。而所有的 \\(suf(i+1)\\) 加起來,就對應到所有區間的 \\((m-1)\times \sum\limits^r_{i=l}a_i\\),這個部分的總和。 瞭解轉移式的推導後,我們回來觀察這個轉移式。它看起來與 \\(y=ax+b\\) 的形式不太像,但我們試著將它展開: @@ -368,17 +368,17 @@ int main(){ 先列轉移式。令 \\(f(i)\\) 為完成前 \\(i\\) 個任務的最小 cost。 -轉移式為:\\(f(i)=\underset{jk}{\min}(f(i-1, k)-\Sigma_{p=k+1}^j\space t_p+t_j \times (j-k))\\) +則轉移式為:\\(f(i, j)=\underset{j>k}{\min}(f(i-1, k)-\sum\limits_{p=k+1}^j\space t_p+t_j \times (j-k))\\) \\(=\underset{j>k}{\min}(f(i-1, k)-suf(k+1)+suf(j+1)+t_{j} \times (j-k))\\) @@ -981,7 +981,7 @@ int main(){ 令 \\(f(i, j)\\) 為將前 \\(j\\) 個 elements 分成 \\(i\\) 個 subarrays 的最小 cost。 -\\(f(i, j)=\underset{j>k}{\min}(f(i-1, k)+(\Sigma_{t=k+1}^{j}x_t)^2)\\) +\\(f(i, j)=\underset{j>k}{\min}(f(i-1, k)+(\sum\limits_{t=k+1}^{j}x_t)^2)\\) \\(=\underset{j>k}{\min}(f(i-1, k)+(pre_{j}-pre_k)^2)\\) From 78017c40eb821101de85ac25eefbabcf4744a7d9 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 11:24:19 +0800 Subject: [PATCH 46/53] add explaination for comparison between lines --- src/dp/convex_hull_optimization.md | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index ee99a985..fbfb7c0e 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -75,6 +75,26 @@ + 假設: + + \\(line1: y=a_1 x+b_1\\) + + \\(line2: y=a_2 x+b_2\\) + + \\(line3: y=a_3 x+b_3\\) + + 則 \\(line1\\) 與 \\(line3\\) 交點的 \\(x\\) 座標為: + \\(\frac{b_3-b_1}{a_1-a_3}\\) + + \\(line1\\) 與 \\(line2\\) 交點的 \\(x\\) 座標為: + \\(\frac{b_2-b_1}{a_1-a_2}\\) + + 因此,\\(line2\\) 要被刪除的條件為: + \\(\frac{b_3-b_1}{a_1-a_3} \le \frac{b_2-b_1}{a_1-a_2}\\) + + 為了避免浮點數誤差,我們將分母做移項,得到 \\(line2\\) 要被刪除的條件為: + \\((b_3-b_1)(a_1-a_2) \le (b_2-b_1)(a_1-a_3)\\) + 綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(f(i)\\),便將 pair \\((f_i, f(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們檢查是否有直線在這次插入後,再也不會被當成答案,有的話就從後端 pop 掉。 在查詢最小值時,我們不斷比較隊列中第一個與第二個直線,如果將當前的 \\(x\\) 代入兩條直線後,發現代入第二條直線有更好的解,則將第一條直線 pop 掉。 From 5a176cf7515cf395d60863503a96b13c3b9ed938 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 11:31:11 +0800 Subject: [PATCH 47/53] add intro for dynamic hull --- src/dp/convex_hull_optimization.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index fbfb7c0e..24fa47af 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -792,7 +792,9 @@ int main(){ #### 動態凸包 -動態凸包,顧名思義就是動態維護的凸包,支援的操作有插入直線與詢問在 \\(x\\) 處的最大值。這個東西有模板可以用,應該蠻好找的,因此這邊不討論如何實作,只討論如何使用它。不過非常建議將動態凸包的模板放到 codebook 內,當遇到斜率優化的題目且斜率不單調時可以直接套。 +動態凸包,顧名思義就是動態維護的凸包,支援的操作有插入直線與詢問在 \\(x\\) 處的最大值。概念上是用 std::set 維護在凸包上的線段,每次查詢時二分搜答案在哪條線段上、每次插入判斷目前凸包中是否有需要被砍掉的線段。 + +這個東西有模板可以用,應該蠻好找的,因此這邊不討論如何實作,只討論如何使用它。不過非常建議將動態凸包的模板放到 codebook 內,當遇到斜率優化的題目且斜率不單調時可以直接套。 插入直線與查詢最大值恰好就是我們需要的兩個操作,因此就直接使用即可。如果轉移式是要求最小值,我們就只要把所有東西都加負號並改取最大值就好。 From 8f99f46e9053919280721bab0c3c0b6d982dc380 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 11:37:27 +0800 Subject: [PATCH 48/53] fixed hard tab --- src/dp/convex_hull_optimization.md | 50 +++++++++++++++--------------- 1 file changed, 25 insertions(+), 25 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 24fa47af..d9009d94 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -811,34 +811,34 @@ using ll = long long; const ll N = 200005; struct Line { - mutable ll k, m, p; - bool operator<(const Line& o) const { return k < o.k; } - bool operator<(ll x) const { return p < x; } + mutable ll k, m, p; + bool operator<(const Line& o) const { return k < o.k; } + bool operator<(ll x) const { return p < x; } }; struct LineContainer : multiset> { - // (for doubles, use inf = 1/.0, div(a,b) = a/b) - static const ll inf = LLONG_MAX; - ll div(ll a, ll b) { // floored division - return a / b - ((a ^ b) < 0 && a % b); } - bool isect(iterator x, iterator y) { - if (y == end()) return x->p = inf, 0; - if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; - else x->p = div(y->m - x->m, x->k - y->k); - return x->p >= y->p; - } - void add(ll k, ll m) { - auto z = insert({k, m, 0}), y = z++, x = y; - while (isect(y, z)) z = erase(z); - if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); - while ((y = x) != begin() && (--x)->p >= y->p) - isect(x, erase(y)); - } - ll query(ll x) { - assert(!empty()); - auto l = *lower_bound(x); - return l.k * x + l.m; - } + // (for doubles, use inf = 1/.0, div(a,b) = a/b) + static const ll inf = LLONG_MAX; + ll div(ll a, ll b) { // floored division + return a / b - ((a ^ b) < 0 && a % b); } + bool isect(iterator x, iterator y) { + if (y == end()) return x->p = inf, 0; + if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; + else x->p = div(y->m - x->m, x->k - y->k); + return x->p >= y->p; + } + void add(ll k, ll m) { + auto z = insert({k, m, 0}), y = z++, x = y; + while (isect(y, z)) z = erase(z); + if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); + while ((y = x) != begin() && (--x)->p >= y->p) + isect(x, erase(y)); + } + ll query(ll x) { + assert(!empty()); + auto l = *lower_bound(x); + return l.k * x + l.m; + } }; int s[N], f[N]; From da7cac08db22a66b9302903a54acb28087f122c0 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 11:42:09 +0800 Subject: [PATCH 49/53] part of textlint fixed --- src/dp/convex_hull_optimization.md | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index d9009d94..3cdc58ef 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -27,7 +27,7 @@ \\[a=f_j\\] \\[b=f(j)\\] -因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。在求 \\(f(i)\\) 時,我們的題目便轉變成:給定一個 \\(x\\),也就是給定 \\(s_i\\) ,求他帶入一個直線集合中,哪一條能有最小的 \\(y\\),也就是哪個 \\(j\\) 能有最小的 \\(f(j) + f_j \times s_i\\) +因此對於所有的 \\(j\\),我們可以將所有的 \\(y=ax+b\\) 在座標平面上畫出來。在求 \\(f(i)\\) 時,我們的題目便轉變成:給定一個 \\(x\\),也就是給定 \\(s_i\\),求他帶入一個直線集合中,哪一條能有最小的 \\(y\\),也就是哪個 \\(j\\) 能有最小的 \\(f(j) + f_j \times s_i\\) 接下來便要思考如何找最小值。如果要代入每一條直線再取極值才能找到答案,並不會優化複雜度,但我們可以來觀察一下這個線集,應該會長類似這個樣子: @@ -83,16 +83,14 @@ \\(line3: y=a_3 x+b_3\\) - 則 \\(line1\\) 與 \\(line3\\) 交點的 \\(x\\) 座標為: - \\(\frac{b_3-b_1}{a_1-a_3}\\) + 則 \\(line1\\) 與 \\(line3\\) 交點的 \\(x\\) 座標為:\\(\frac{b_3-b_1}{a_1-a_3}\\) - \\(line1\\) 與 \\(line2\\) 交點的 \\(x\\) 座標為: - \\(\frac{b_2-b_1}{a_1-a_2}\\) + \\(line1\\) 與 \\(line2\\) 交點的 \\(x\\) 座標為:\\(\frac{b_2-b_1}{a_1-a_2}\\) - 因此,\\(line2\\) 要被刪除的條件為: - \\(\frac{b_3-b_1}{a_1-a_3} \le \frac{b_2-b_1}{a_1-a_2}\\) + 因此,\\(line2\\) 要被刪除的條件為:\\(\frac{b_3-b_1}{a_1-a_3} \le \frac{b_2-b_1}{a_1-a_2}\\) 為了避免浮點數誤差,我們將分母做移項,得到 \\(line2\\) 要被刪除的條件為: + \\((b_3-b_1)(a_1-a_2) \le (b_2-b_1)(a_1-a_3)\\) 綜合上述,我們可以使用單調隊列,開一個 deque 完成這兩項操作。具體而言,每算完一個 \\(f(i)\\),便將 pair \\((f_i, f(i))\\) 插入隊列尾端,分別代表直線的 \\(a\\) 與 \\(b\\)。插入之前,我們檢查是否有直線在這次插入後,再也不會被當成答案,有的話就從後端 pop 掉。 From 02390d24482d36fd28253299ece1a4e083913663 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Sat, 19 Aug 2023 11:44:36 +0800 Subject: [PATCH 50/53] textlint fixed --- src/dp/convex_hull_optimization.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 3cdc58ef..c85da700 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -502,7 +502,7 @@ int main(){ CDQ 分治的過程就是對於一個序列 \\([l, r]\\),我們先遞迴解決 \\([l, mid]\\) 的子問題,再處理 \\([l, mid]\\) 對 \\([mid + 1, r]\\) 的貢獻,最後再遞迴處理 \\([mid + 1, r]\\)。 -以這題來說,對於當前遞迴到的區間 \\([l, r]\\),我們會先遞迴左半區間 \\([l, mid]\\) ,接著用計算好的左半區間,去更新右半區間 \\([mid + 1, r]\\) 的 dp 值,最後再往下遞迴右半區間。 +以這題來說,對於當前遞迴到的區間 \\([l, r]\\),我們會先遞迴左半區間 \\([l, mid]\\),接著用計算好的左半區間,去更新右半區間 \\([mid + 1, r]\\) 的 dp 值,最後再往下遞迴右半區間。 前面提到我們想創造出斜率的單調性,因此在遞迴過程中我們要維護一個斜率單調的序列,且裡面沒有不會被取到的直線。先不論直線如何計算,討論維護的部分。維護的方式類似 merge sort 的過程,只是要注意新增直線到序列尾端時,一樣要做 pop 的動作,將不可能會被取到的直線刪除。 From 82c14b82699c86f4a1c9c1eca7104814744bd306 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 30 Aug 2023 21:46:21 +0800 Subject: [PATCH 51/53] minor errors fixed --- src/dp/convex_hull_optimization.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index c85da700..03aaaba8 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -545,7 +545,7 @@ bool cmp(pll line1, pll line2, pll line3){ return (line3.second - line1.second) * (line1.first - line2.first) <= (line2.second - line1.second) * (line1.first - line3.first); } -pll find_line(vector v, ll x){ +pll find_line(vector &v, ll x){ int l = 0, r = v.size() - 2, mid, ans = max(0, r + 1); while(l <= r){ mid = (l + r) >> 1; @@ -621,7 +621,7 @@ int main(){ 1. 在 \\(\[L, R)\\) 中,\\(f\\) 大於 \\(g\\) 的部分較多。 - 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(f\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(G\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(f\\) 則保留下來存在當前節點。 + 我們可以將當前區間分為兩個區間:\\(f\\) 比較大的區間與 \\(g\\) 較大的區間,以下稱這兩個區間為 \\(F\\) 和 \\(G\\),如下圖。我們可以發現 \\(G\\) 一定能被 \\(\[L, mid\]\\) 或 \\(\[mid + 1, R)\\) 其中之一完全包含。我們將 \\(g\\) 向下推,像在線段樹上更新懶惰標記一樣,更新對應的子節點。\\(f\\) 則保留下來存在當前節點。 @@ -629,7 +629,7 @@ int main(){ 2. 在 \\(\[L, R)\\) 中,\\(f\\) 小於 \\(g\\) 的部分較多。 - 與上一個情況相似,這次我們將 \\(G\\) 存於當前節點,將 \\(f\\) 向下推。 + 與上一個情況相似,這次我們將 \\(g\\) 存於當前節點,將 \\(f\\) 向下推。 判斷方式:\\(f\\) 在 \\(mid\\) 的值小於 \\(g\\) 則成立。 @@ -888,7 +888,7 @@ int main(){ 先將所有矩形依照 \\(x\\) 座標由大至小 sort,這時 \\(y\\) 座標也會由小至大排好。 -令 \\f(i\\) 為考慮前 \\(i\\) 個矩形,且要取第 \\(i\\) 個矩形,所能得到的最大值。 +令 \\(f(i)\\) 為考慮前 \\(i\\) 個矩形,且要取第 \\(i\\) 個矩形,所能得到的最大值。 轉移式為:\\(f(i)=\underset{j [CF 311B - Cats Transport](https://codeforces.com/contest/311/problem/B) > -> 有 \\(p\\) 個 feeders 想把 \\(m\\) 隻貓接回農場。貓現在分布在各個山上,共有 \\(n\\) 座山,由左至右編號 \\(1\sim n\\),相鄰的兩座山 \\(i\\) 與 \\(i-1\\) 山隔著 \\(d_i\\) 公尺。每隻貓都會給定其在哪座山上,以及一個時間 \\(t_i\\),表示經過 \\(t_i\\) 單位時間後貓會開始等待被接走。現在 feeders 要從 \\(1\\) 號山出發往右走,以每單位時間一公尺的速度去接貓,當 feeder 抵達一座山,他會接走該座山中所有正在等待的貓。現在我們要安排 feeders 的出發時間,使得所有貓的等待時間總和最短。 +> 有 \\(p\\) 個 feeders 想把 \\(m\\) 隻貓接回農場。貓現在分布在各個山上,共有 \\(n\\) 座山,由左至右編號 \\(1\sim n\\),相鄰的兩座山 \\(i\\) 與 \\(i-1\\) 山隔著 \\(d_i\\) 公尺。每隻貓都會給定其在第 \\(h_i\\) 座山上,以及一個時間 \\(t_i\\),表示經過 \\(t_i\\) 單位時間後貓會開始等待被接走。現在 feeders 要從 \\(1\\) 號山出發往右走,以每單位時間一公尺的速度去接貓,當 feeder 抵達一座山,他會接走該座山中所有正在等待的貓。現在我們要安排 feeders 的出發時間,使得所有貓的等待時間總和最短。 > > - \\(2 \leq n \leq 10^5\\) > - \\(1 \leq m \leq 10^5\\) @@ -915,7 +915,7 @@ int main(){
    Solution -先將所有貓的 \\(t_i\\) 減掉 \\(h_1\\) 至 \\(h_i\\) 的距離,這樣就可以得到每隻貓最早能被接走時,feeder 的出發時間。 +先將所有貓的 \\(t_i\\) 減掉第一座山至第 \\(h_i\\) 座山的距離,這樣就可以得到每隻貓最早能被接走時,feeder 的出發時間。 接著將貓照處理過的 \\(t_i\\) 由小至大 sort,題目便轉成如何將序列切成 \\(p\\) 個部分,使得 cost 最小。 From 76fe65cce61bda3405eaec9c2b3e33eb10b9a8d1 Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 30 Aug 2023 21:47:09 +0800 Subject: [PATCH 52/53] minor errors fixed in SUMMARY.md --- src/SUMMARY.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 92295e67..60f4e73c 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -11,7 +11,7 @@ - [Greedy](greedy/intro.md) - [Basic Greedy](greedy/basic.md) - [Dynamic Programming]() - - [Convex hull optimization](dp/convex_hull_optimization.md) + - [Convex Hull Optimization](dp/convex_hull_optimization.md) - [Graph]() - [Tree]() - [Lowest Common Ancestor](graph/lca.md) From 9fc1fe2483b6a6e4535e1af73c3ee1a1c1dff6de Mon Sep 17 00:00:00 2001 From: aftermaaath Date: Wed, 30 Aug 2023 21:58:02 +0800 Subject: [PATCH 53/53] textlint fixed --- src/dp/convex_hull_optimization.md | 18 ++++++------------ 1 file changed, 6 insertions(+), 12 deletions(-) diff --git a/src/dp/convex_hull_optimization.md b/src/dp/convex_hull_optimization.md index 03aaaba8..87071519 100644 --- a/src/dp/convex_hull_optimization.md +++ b/src/dp/convex_hull_optimization.md @@ -160,8 +160,7 @@ int main(){ > [TIOJ 1676 - 烏龜疊疊樂](https://tioj.ck.tp.edu.tw/problems/1676) > -> 給定一個長度為 \\(n\\) 的陣列與一個整數 \\(k\\),你可以將陣列切成數段連續的區間,區間大小不能超過 \\(k\\),求切完之後的最大價值。 -> 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \sum\limits^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 +> 給定一個長度為 \\(n\\) 的陣列與一個整數 \\(k\\),你可以將陣列切成數段連續的區間,區間大小不能超過 \\(k\\),求切完之後的最大價值。> 一段長度為 \\(x\\)、為整個陣列由左至右數來第 \\(m\\) 個區間,其價值被定義為 \\((m-1)\times \sum\limits^r_{i=l}a_i-x^2\\)。其中 \\(l\\)、\\(r\\) 分別為區間左右界、\\(a_i\\) 為陣列中第 \\(i\\) 個值。 > > - \\(1 \leq k \leq n \leq 500000\\) @@ -189,8 +188,7 @@ int main(){ \\(=\underset{i>j>i-k}{\max}(f(j)-j^2+2ij)-i^2+suf(i+1)\\) -可以發現 \\(max\\) 裡面的部分變成 \\(y=ax+b\\) 的形式了: -\\[y=f(i)\\] +可以發現 \\(max\\) 裡面的部分變成 \\(y=ax+b\\) 的形式了:\\[y=f(i)\\] \\[x=2i\\] \\[a=j\\] \\[b=f(j)-j^2\\] @@ -398,8 +396,7 @@ int main(){ 其中 \\(pre_t(i)\\) 為 \\(T\\) 的前綴和,即 \\(\sum\limits_{k=1}^i T_k\\);\\(pre_c(i)\\) 為 \\(C\\) 的前綴和,即 \\(\sum\limits_{k=1}^i C_k\\)。 -我們可以得到: -\\[y=f(i)\\] +我們可以得到:\\[y=f(i)\\] \\[x=-pre_t(i)\\] \\[a=pre_c(j)\\] \\[b=f(j)-s \times pre_c(j)\\] @@ -966,8 +963,7 @@ int main(){ > [CF 1715E - Long Way Home](https://codeforces.com/problemset/problem/1715/E) > -> 有 \\(n\\) 個城市編號 \\(1 \sim n\\),從城市 \\(u\\) 移動到城市 \\(v\\) 的成本是 \\((u-v)^2\\)。 -> 現在 Stanley 想從城市 \\(1\\) 用至多 \\(k\\) 步走到城市 \\(n\\),請找出最小的成本。 +> 有 \\(n\\) 個城市編號 \\(1 \sim n\\),從城市 \\(u\\) 移動到城市 \\(v\\) 的成本是 \\((u-v)^2\\)。> 現在 Stanley 想從城市 \\(1\\) 用至多 \\(k\\) 步走到城市 \\(n\\),請找出最小的成本。 > > - \\(2 \le n \le 10^5, 1 \le k \le 20\\) @@ -992,8 +988,7 @@ int main(){ > [CSES - Subarray Squares](https://cses.fi/problemset/task/2086) > -> 將長度 \\(n\\) 的陣列分成 \\(k\\) 個子陣列,每個子陣列的成本是這個子列中數值總和的平方,此分法的總成本為每個子陣列的成本和。 -> 請輸出所有分法中的最小成本。 +> 將長度 \\(n\\) 的陣列分成 \\(k\\) 個子陣列,每個子陣列的成本是這個子列中數值總和的平方,此分法的總成本為每個子陣列的成本和。> 請輸出所有分法中的最小成本。 > > - \\(1 \le k \le n \le 3000\\) @@ -1020,8 +1015,7 @@ int main(){ > [TIOJ 1921 - 吐鈔機 2](https://tioj.ck.tp.edu.tw/problems/1921) > -> 有 \\(N\\) 台機器,每台機器都有 \\(D_i, P_i, G_i, R_i\\):售出的日期、購入的價格、每天產出的金額、賣出的價格。給定 \\(C, D\\):第 \\(0\\) 天持有的金額,停止交易的日期。 -> 問在停止交易當天所能持有的最大金額是多少。 +> 有 \\(N\\) 台機器,每台機器都有 \\(D_i, P_i, G_i, R_i\\):售出的日期、購入的價格、每天產出的金額、賣出的價格。給定 \\(C, D\\):第 \\(0\\) 天持有的金額,停止交易的日期。> 問在停止交易當天所能持有的最大金額是多少。 > > - \\(1 \le N \le 10^5\\) >