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00000000..3d053fed --- /dev/null +++ b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays @@ -0,0 +1,150 @@ +# LeetCode 第 4 号问题:寻找两个正序数组的中位数 + +> 本文首发于公众号「图解面试算法」,是 [图解 LeetCode ]() 系列文章之一。 +> +> 同步博客:https://www.algomooc.com + +题目来源于 LeetCode 上第 4 号问题:寻找两个正序数组的中位数。题目难度为 Hard,目前通过率为 29.0% 。 + +#### 题目描述 + +> 给定两个大小为 m 和 n 的正序(从小到大)数组 nums1 和 nums2。 +请你找出这两个正序数组的中位数,并且要求算法的时间复杂度为 O(log(m + n))。 +你可以假设 nums1 和 nums2 不会同时为空。 + +```java +示例1: +nums1 = [1, 3] +nums2 = [2] + +则中位数是 2.0 + +示例2: +nums1 = [1, 2] +nums2 = [3, 4] + +则中位数是 (2 + 3)/2 = 2.5 +``` + +#### 题目解析 +这道题网络上的解析都非常“高深”,很难理解。私以为它们都将简单的问题复杂化了。本题在一些处理上确实会有些麻烦,比如数组边界的处理,和偶数个数的中位数的处理。但其核心思想并不复杂。 + +首先,我们可以只考虑数字总个数为奇数的情况。让我们看下下图: + +![](../Animation/image1.png) + +蓝框是中位数左边的数(包括中位数),而橘框则为中位数右边的数。 + +3个显然的规则: +1.两个数组的蓝框总个数=(数字总个数+1)/2; +2.所有蓝框内的数都小于橘框内的数 +3.中位数为蓝框中最大的那一位(即数组1蓝框最后一位,或数组2蓝框最后一位) +![](../Animation/image2.png) +如图,我们要找到一组A,B,满足上面3条规则。 +对于规则1,我们在数组1中找任意A,然后根据规则1就能推算出对应的B的位置。 +对于规则2,由于数组1和2都是有序数组,即X1=和<=。 +下面提供了作者的一份代码,leetcode上的结果为:执行用时:2 ms;内存消耗:40.3 MB,都超过了100%的用户。读者可以参考一下。 + + +#### 代码实现 + +Java语言 + +```java +public class Solution { + public double findMedianSortedArrays(int[] nums1, int[] nums2) { + // 使nums1成为较短数组,不仅可以提高检索速度,同时可以避免一些边界问题 + if (nums1.length > nums2.length) { + int[] temp = nums1; + nums1 = nums2; + nums2 = temp; + } + + int len1 = nums1.length; + int len2 = nums2.length; + int leftLen = (len1 + len2 + 1) / 2; //两数组合并&排序后,左半边的长度 + + // 对数组1进行二分检索 + int start = 0; + int end = len1; + while (start <= end) { + // 两个数组的被测数A,B的位置(从1开始计算) + // count1 = 2 表示 num1 数组的第2个数字 + // 比index大1 + int count1 = start + ((end - start) / 2); + int count2 = leftLen - count1; + + if (count1 > 0 && nums1[count1 - 1] > nums2[count2]) { + // A比B的next还要大 + end = count1 - 1; + } else if (count1 < len1 && nums2[count2 - 1] > nums1[count1]) { + // B比A的next还要大 + start = count1 + 1; + } else { + // 获取中位数 + int result = (count1 == 0)? nums2[count2 - 1]: // 当num1数组的数都在总数组右边 + (count2 == 0)? nums1[count1 - 1]: // 当num2数组的数都在总数组右边 + Math.max(nums1[count1 - 1], nums2[count2 - 1]); // 比较A,B + if (isOdd(len1 + len2)) { + return result; + } + + // 处理偶数个数的情况 + int nextValue = (count1 == len1) ? nums2[count2]: + (count2 == len2) ? nums1[count1]: + Math.min(nums1[count1], nums2[count2]); + return (result + nextValue) / 2.0; + } + } + + return Integer.MIN_VALUE; // 绝对到不了这里 + } + + // 奇数返回true,偶数返回false + private boolean isOdd(int x) { + return (x & 1) == 1; + } +} +``` + +#### 动画理解 + +![](../Animation/Animation.gif) + +#### 复杂度分析 + ++ 时间复杂度:对数组进行二分查找,因此为O(logN) ++ 空间复杂度:O(1) + + + + + +``` +![](../../Pictures/qrcode.jpg) +``` \ No newline at end of file diff --git a/0004-median-of-two-sorted-arrays/Code/1.java b/0004-median-of-two-sorted-arrays/Code/1.java new file mode 100644 index 00000000..cfb36cfd --- /dev/null +++ b/0004-median-of-two-sorted-arrays/Code/1.java @@ -0,0 +1,54 @@ +public class Solution { + public double findMedianSortedArrays(int[] nums1, int[] nums2) { + // ʹnums1Ϊ϶,߼ٶ,ͬʱԱһЩ߽ + if (nums1.length > nums2.length) { + int[] temp = nums1; + nums1 = nums2; + nums2 = temp; + } + + int len1 = nums1.length; + int len2 = nums2.length; + int leftLen = (len1 + len2 + 1) / 2; //ϲ&,ߵij + + // 1жּ + int start = 0; + int end = len1; + while (start <= end) { + // ıA,Bλ(1ʼ) + // count1 = 2 ʾ num1 ĵ2 + // index1 + int count1 = start + ((end - start) / 2); + int count2 = leftLen - count1; + + if (count1 > 0 && nums1[count1 - 1] > nums2[count2]) { + // ABnextҪ + end = count1 - 1; + } else if (count1 < len1 && nums2[count2 - 1] > nums1[count1]) { + // BAnextҪ + start = count1 + 1; + } else { + // ȡλ + int result = (count1 == 0)? nums2[count2 - 1]: // num1ұ + (count2 == 0)? nums1[count1 - 1]: // num2ұ + Math.max(nums1[count1 - 1], nums2[count2 - 1]); // ȽA,B + if (isOdd(len1 + len2)) { + return result; + } + + // ż + int nextValue = (count1 == len1) ? nums2[count2]: + (count2 == len2) ? nums1[count1]: + Math.min(nums1[count1], nums2[count2]); + return (result + nextValue) / 2.0; + } + } + + return Integer.MIN_VALUE; // Ե + } + + // true,żfalse + private boolean isOdd(int x) { + return (x & 1) == 1; + } +} From 1d540ec394a9681951f9d2db7252b7dd9b5025c1 Mon Sep 17 00:00:00 2001 From: Ziyuan Zhang Date: Fri, 8 May 2020 19:47:07 -0500 Subject: [PATCH 2/5] Rename 0004-median-of-two-sorted-arrays to 0004-median-of-two-sorted-arrays.md --- ...of-two-sorted-arrays => 0004-median-of-two-sorted-arrays.md} | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) rename 0004-median-of-two-sorted-arrays/Article/{0004-median-of-two-sorted-arrays => 0004-median-of-two-sorted-arrays.md} (97%) diff --git a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md similarity index 97% rename from 0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays rename to 0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md index 3d053fed..d09915b9 100644 --- a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays +++ b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md @@ -147,4 +147,4 @@ public class Solution { ``` ![](../../Pictures/qrcode.jpg) -``` \ No newline at end of file +``` From 0a528f26fee1e7f86d04dfe441ab31de75cf6de0 Mon Sep 17 00:00:00 2001 From: Ziyuan Zhang Date: Fri, 8 May 2020 19:49:15 -0500 Subject: [PATCH 3/5] 0004_Solved 0004_Solved --- .../Article/0004-median-of-two-sorted-arrays.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md index d09915b9..945a3ebf 100644 --- a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md +++ b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md @@ -51,6 +51,7 @@ nums2 = [3, 4] **情况1:** ![](../Animation/case1.png) + 首先,我们选择数组1进行操作。取其中间值9 。(因此 A=9) 根据规则1,我们在数组2中找到对应值(B = 4)。(一共有11个数,(11+1) / 2 = 6,因此蓝色框总数为6) 紧接着,我们根据规则2判断A(9)是否小于B.next(5),以及B(4)是否小于A.next(11)。 显然,A比B.next,也就是一个橘框还要大。这是不允许的。可见A只能取比9更小的数字了。如果取更大的数字,那B就会更小,更不可能满足规则2。所以这种情况下我们要向左进行二分。 @@ -58,11 +59,13 @@ nums2 = [3, 4] **情况2:** ![](../Animation/case2.png) + 这种情况下B比A.next,也就是一个橘框还要大。这是不允许的。可见A只能取比9更大的数字了。如果取更小的数字,那B就会更大,更不可能满足规则2。所以这种情况下我们要向右进行二分。 **情况3:** ![](../Animation/case3.png) + 随着我们不断地二分,中位数显然必然会出现。 如图上这种情况,A小于B.next,且B小于A.next。 那么,显然,A,B中较大的那一项就是中位数(规则3)。 From bcacf82697c61af3c6a28a8767137c740e6d51ed Mon Sep 17 00:00:00 2001 From: Ziyuan Zhang Date: Fri, 8 May 2020 19:51:56 -0500 Subject: [PATCH 4/5] 0004_Solved 0004_Solved --- .../Article/0004-median-of-two-sorted-arrays.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md index 945a3ebf..d4f15f8a 100644 --- a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md +++ b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md @@ -148,6 +148,4 @@ public class Solution { -``` ![](../../Pictures/qrcode.jpg) -``` From 74f153720e5f2478444462ff74b7573cd87a9e87 Mon Sep 17 00:00:00 2001 From: Ziyuan Zhang Date: Fri, 8 May 2020 19:58:08 -0500 Subject: [PATCH 5/5] 0004_Solved 0004_Solved --- .../Article/0004-median-of-two-sorted-arrays.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md index d4f15f8a..889ec882 100644 --- a/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md +++ b/0004-median-of-two-sorted-arrays/Article/0004-median-of-two-sorted-arrays.md @@ -31,7 +31,7 @@ nums2 = [3, 4] 首先,我们可以只考虑数字总个数为奇数的情况。让我们看下下图: -![](../Animation/image1.png) +![](../Animation/image1.PNG) 蓝框是中位数左边的数(包括中位数),而橘框则为中位数右边的数。 @@ -39,7 +39,7 @@ nums2 = [3, 4] 1.两个数组的蓝框总个数=(数字总个数+1)/2; 2.所有蓝框内的数都小于橘框内的数 3.中位数为蓝框中最大的那一位(即数组1蓝框最后一位,或数组2蓝框最后一位) -![](../Animation/image2.png) +![](../Animation/image2.PNG) 如图,我们要找到一组A,B,满足上面3条规则。 对于规则1,我们在数组1中找任意A,然后根据规则1就能推算出对应的B的位置。 对于规则2,由于数组1和2都是有序数组,即X1

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