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327. 区间和的个数

English Version

题目描述

给你一个整数数组 nums 以及两个整数 lower 和 upper 。求数组中，值位于范围 [lower, upper] （包含 lower 和 upper）之内的 区间和的个数 。

区间和 S(i, j) 表示在 nums 中，位置从 i 到 j 的元素之和，包含 i 和 j (i ≤ j)。

 

示例 1： 
输入：nums = [-2,5,-1], lower = -2, upper = 2
输出：3
解释：存在三个区间：[0,0]、[2,2] 和 [0,2] ，对应的区间和分别是：-2 、-1 、2 。

示例 2：

输入：nums = [0], lower = 0, upper = 0
输出：1

 

提示：

  • 1 <= nums.length <= 105
  • -231 <= nums[i] <= 231 - 1
  • -105 <= lower <= upper <= 105
  • 题目数据保证答案是一个 32 位 的整数

解法

方法一：树状数组

树状数组，也称作“二叉索引树”（Binary Indexed Tree）或 Fenwick 树。 它可以高效地实现如下两个操作：

  1. 单点更新 update(x, delta)： 把序列 x 位置的数加上一个值 delta；
  2. 前缀和查询 query(x)：查询序列 [1,...x] 区间的区间和，即位置 x 的前缀和。

这两个操作的时间复杂度均为 $O(\log n)$。

本题中，对于每个下标 j，以 j 为右端点的下标对的数量，就等于 preSum[1..j] 中的所有整数，出现在区间 [preSum[j] - upper, preSum[j] - lower] 的次数。

lower <= preSum[j] - preSum[i - 1] <= upper，变形得 preSum[j] - upper <= preSum[i - 1] <= preSum[j] - lower。

我们可以用树状数组，从左到右扫描前缀和数组，每遇到一个前缀和 s，就在树状数组中查询区间 [preSum[j] - upper, preSum[j] - lower] 内的整数的数量，随后将 s 更新至树状数组。

方法二：线段树

线段树将整个区间分割为多个不连续的子区间，子区间的数量不超过 log(width)。更新某个元素的值，只需要更新 log(width) 个区间，并且这些区间都包含在一个包含该元素的大区间内。

  • 线段树的每个节点代表一个区间；
  • 线段树具有唯一的根节点，代表的区间是整个统计范围，如 [1, N]；
  • 线段树的每个叶子节点代表一个长度为 1 的元区间 [x, x]；
  • 对于每个内部节点 [l, r]，它的左儿子是 [l, mid]，右儿子是 [mid + 1, r], 其中 mid = ⌊(l + r) / 2⌋ (即向下取整)。

Python3 代码 TLE，Java 代码 AC。

Python3

树状数组：

class BinaryIndexedTree:
    def __init__(self, n):
        self.n = n
        self.c = [0] * (n + 1)

    @staticmethod
    def lowbit(x):
        return x & -x

    def update(self, x, delta):
        while x <= self.n:
            self.c[x] += delta
            x += BinaryIndexedTree.lowbit(x)

    def query(self, x):
        s = 0
        while x > 0:
            s += self.c[x]
            x -= BinaryIndexedTree.lowbit(x)
        return s


class Solution:
    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        presum = [0]
        for v in nums:
            presum.append(presum[-1] + v)
        alls = set()
        for s in presum:
            alls.add(s)
            alls.add(s - lower)
            alls.add(s - upper)
        alls = sorted(alls)
        m = {v: i for i, v in enumerate(alls, 1)}
        tree = BinaryIndexedTree(len(m))
        ans = 0
        for s in presum:
            i, j = m[s - upper], m[s - lower]
            ans += tree.query(j) - tree.query(i - 1)
            tree.update(m[s], 1)
        return ans

线段树：

class Node:
    def __init__(self):
        self.l = 0
        self.r = 0
        self.v = 0

class SegmentTree:
    def __init__(self, n):
        self.tr = [Node() for _ in range(4 * n)]
        self.build(1, 1, n)

    def build(self, u, l, r):
        self.tr[u].l = l
        self.tr[u].r = r
        if l == r:
            return
        mid = (l + r) >> 1
        self.build(u << 1, l, mid)
        self.build(u << 1 | 1, mid + 1, r)

    def modify(self, u, x, v):
        if self.tr[u].l == x and self.tr[u].r == x:
            self.tr[u].v += v
            return
        mid = (self.tr[u].l + self.tr[u].r) >> 1
        if x <= mid:
            self.modify(u << 1, x, v)
        else:
            self.modify(u << 1 | 1, x, v)
        self.pushup(u)

    def pushup(self, u):
        self.tr[u].v = self.tr[u << 1].v + self.tr[u << 1 | 1].v

    def query(self, u, l, r):
        if self.tr[u].l >= l and self.tr[u].r <= r:
            return self.tr[u].v
        mid = (self.tr[u].l + self.tr[u].r) >> 1
        v = 0
        if l <= mid:
            v += self.query(u << 1, l, r)
        if r > mid:
            v += self.query(u << 1 | 1, l, r)
        return v

class Solution:
    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        s = [0]
        for x in nums:
            s.append(s[-1] + x)
        alls = set()
        for v in s:
            alls.add(v)
            alls.add(v - lower)
            alls.add(v - upper)
        m = {v: i for i, v in enumerate(sorted(alls), 1)}
        tree = SegmentTree(len(m))
        ans = 0
        for v in s:
            l, r = m[v - upper], m[v - lower]
            ans += tree.query(1, l, r)
            tree.modify(1, m[v], 1)
        return ans

Java

树状数组：

class Solution {
    public int countRangeSum(int[] nums, int lower, int upper) {
        int n = nums.length;
        long[] preSum = new long[n + 1];
        for (int i = 0; i < n; ++i) {
            preSum[i + 1] = preSum[i] + nums[i];
        }
        TreeSet<Long> ts = new TreeSet<>();
        for (long s : preSum) {
            ts.add(s);
            ts.add(s - upper);
            ts.add(s - lower);
        }
        Map<Long, Integer> m = new HashMap<>();
        int idx = 1;
        for (long s : ts) {
            m.put(s, idx++);
        }
        int ans = 0;
        BinaryIndexedTree tree = new BinaryIndexedTree(m.size());
        for (long s : preSum) {
            int i = m.get(s - upper);
            int j = m.get(s - lower);
            ans += tree.query(j) - tree.query(i - 1);
            tree.update(m.get(s), 1);
        }
        return ans;
    }
}

class BinaryIndexedTree {
    private int n;
    private int[] c;

    public BinaryIndexedTree(int n) {
        this.n = n;
        c = new int[n + 1];
    }

    public void update(int x, int delta) {
        while (x <= n) {
            c[x] += delta;
            x += lowbit(x);
        }
    }

    public int query(int x) {
        int s = 0;
        while (x > 0) {
            s += c[x];
            x -= lowbit(x);
        }
        return s;
    }

    public static int lowbit(int x) {
        return x & -x;
    }
}

线段树：

class Solution {
    public int countRangeSum(int[] nums, int lower, int upper) {
        int n = nums.length;
        long[] preSum = new long[n + 1];
        for (int i = 0; i < n; ++i) {
            preSum[i + 1] = preSum[i] + nums[i];
        }
        TreeSet<Long> ts = new TreeSet<>();
        for (long s : preSum) {
            ts.add(s);
            ts.add(s - upper);
            ts.add(s - lower);
        }
        Map<Long, Integer> m = new HashMap<>();
        int idx = 1;
        for (long s : ts) {
            m.put(s, idx++);
        }
        int ans = 0;
        SegmentTree tree = new SegmentTree(m.size());
        for (long s : preSum) {
            int l = m.get(s - upper);
            int r = m.get(s - lower);
            ans += tree.query(1, l, r);
            tree.modify(1, m.get(s), 1);
        }
        return ans;
    }
}

class Node {
    int l;
    int r;
    int v;
}

class SegmentTree {
    private Node[] tr;

    public SegmentTree(int n) {
        tr = new Node[4 * n];
        for (int i = 0; i < tr.length; ++i) {
            tr[i] = new Node();
        }
        build(1, 1, n);
    }

    public void build(int u, int l, int r) {
        tr[u].l = l;
        tr[u].r = r;
        if (l == r) {
            return;
        }
        int mid = (l + r) >> 1;
        build(u << 1, l, mid);
        build(u << 1 | 1, mid + 1, r);
    }

    public void modify(int u, int x, int v) {
        if (tr[u].l == x && tr[u].r == x) {
            tr[u].v += v;
            return;
        }
        int mid = (tr[u].l + tr[u].r) >> 1;
        if (x <= mid) {
            modify(u << 1, x, v);
        } else {
            modify(u << 1 | 1, x, v);
        }
        pushup(u);
    }

    public void pushup(int u) {
        tr[u].v = tr[u << 1].v + tr[u << 1 | 1].v;
    }

    public int query(int u, int l, int r) {
        if (tr[u].l >= l && tr[u].r <= r) {
            return tr[u].v;
        }
        int mid = (tr[u].l + tr[u].r) >> 1;
        int v = 0;
        if (l <= mid) {
            v += query(u << 1, l, r);
        }
        if (r > mid) {
            v += query(u << 1 | 1, l, r);
        }
        return v;
    }
}

C++

class BinaryIndexedTree {
public:
    int n;
    vector<int> c;

    BinaryIndexedTree(int _n)
        : n(_n)
        , c(_n + 1) { }

    void update(int x, int delta) {
        while (x <= n) {
            c[x] += delta;
            x += lowbit(x);
        }
    }

    int query(int x) {
        int s = 0;
        while (x > 0) {
            s += c[x];
            x -= lowbit(x);
        }
        return s;
    }

    int lowbit(int x) {
        return x & -x;
    }
};

class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        vector<long long> preSum(n + 1);
        for (int i = 0; i < n; ++i) preSum[i + 1] = preSum[i] + nums[i];
        set<long long> alls;
        for (auto& s : preSum) {
            alls.insert(s);
            alls.insert(s - upper);
            alls.insert(s - lower);
        }
        unordered_map<long long, int> m;
        int idx = 1;
        for (auto& v : alls) m[v] = idx++;
        BinaryIndexedTree* tree = new BinaryIndexedTree(m.size());
        int ans = 0;
        for (auto& s : preSum) {
            int i = m[s - upper], j = m[s - lower];
            ans += tree->query(j) - tree->query(i - 1);
            tree->update(m[s], 1);
        }
        return ans;
    }
};

Go

type BinaryIndexedTree struct {
	n int
	c []int
}

func newBinaryIndexedTree(n int) *BinaryIndexedTree {
	c := make([]int, n+1)
	return &BinaryIndexedTree{n, c}
}

func (this *BinaryIndexedTree) lowbit(x int) int {
	return x & -x
}

func (this *BinaryIndexedTree) update(x, delta int) {
	for x <= this.n {
		this.c[x] += delta
		x += this.lowbit(x)
	}
}

func (this *BinaryIndexedTree) query(x int) int {
	s := 0
	for x > 0 {
		s += this.c[x]
		x -= this.lowbit(x)
	}
	return s
}

func countRangeSum(nums []int, lower int, upper int) int {
	n := len(nums)
	presum := make([]int, n+1)
	for i, v := range nums {
		presum[i+1] = presum[i] + v
	}
	alls := make(map[int]bool)
	for _, s := range presum {
		alls[s] = true
		alls[s-upper] = true
		alls[s-lower] = true
	}
	var t []int
	for s, _ := range alls {
		t = append(t, s)
	}
	sort.Ints(t)
	m := make(map[int]int)
	for i, v := range t {
		m[v] = i + 1
	}
	ans := 0
	tree := newBinaryIndexedTree(len(alls))
	for _, s := range presum {
		i, j := m[s-upper], m[s-lower]
		ans += tree.query(j) - tree.query(i-1)
		tree.update(m[s], 1)
	}
	return ans
}

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