POJ 3468 A Simple Problem with Integers (分块)

Description

You have \(N\) integers, \(A_1, A_2, ... , A_N\). You need to deal with two kinds of operations. One type of operation is to add some given number to each number in a given interval. The other is to ask for the sum of numbers in a given interval.

Input

The first line contains two numbers \(N\) and \(Q\). \(1 ≤ N,Q ≤ 100000\).

The second line contains \(N\) numbers, the initial values of \(A_1, A_2, ... , A_N\). \(-1000000000 \le A_i \le 1000000000\).

Each of the next \(Q\) lines represents an operation.

"\(C\ a\ b\ c\)" means adding c to each of \(A_a, A_{a+1}, \ ...\ , A_b\). \(-10000 \le c \le 10000\).

"\(Q\ a\ b\)" means querying the sum of \(A_a, A_{a+1}, \ ...\ , A_b\).

Output

You need to answer all \(Q\) commands in order. One answer in a line.

Sample Input

10 5
1 2 3 4 5 6 7 8 9 10
Q 4 4
Q 1 10
Q 2 4
C 3 6 3
Q 2 4

Sample Output

4
55
9
15

Hint

The sums may exceed the range of 32-bit integers.

Solution

题意

给定 \(n\) 个数和 \(q\) 个询问，询问包含两种：\(C\ a\ b\ c\) 代表区间 \([a, b]\) 的每个数加上 \(c\)，\(Q\ a\ b\) 输出区间 \([a, b]\) 的和。

题解

分块

区间更新模板题，本题可以使用树状数组、线段树和分块解决，这里使用的是分块。

#include <cstdio>
#include <iostream>
#include <cmath>
using namespace std;

typedef long long ll;

const int maxn = 1e5 + 10;

ll a[maxn], sum[maxn], add[maxn]; // add[] 是增量标记
int L[maxn], R[maxn]; // 存放每个块的左右边界
int block[maxn]; // 存放下标为 i 的元素的块号
int n, q;
int block_size; // 块的大小

// 分块 + 预处理
void init() {
    block_size = sqrt(n);
    for(int i = 1; i <= block_size; ++i) {
        L[i] = (i - 1) * block_size + 1;
        R[i] = i * block_size;
    }
    // 处理最后一块
    if(R[block_size] < n) {
        ++block_size;
        L[block_size] = R[block_size - 1] + 1;
        R[block_size] = n;
    }
    // 预处理每个块的区间和
    for(int i = 1; i <= block_size; ++i) {
        for(int j = L[i]; j <= R[i]; ++j) {
            block[j] = i;
            sum[i] += a[j];
        }
    }
}

// 将区间 [l, r] 内的所有元素加 c
void change(int l, int r, ll c) {
    int p = block[l], q = block[r]; // 取出左右区间所在的块号
    if(p == q) {
        // 在同一块直接块内暴力
        for(int i = l; i <= r; ++i) {
            a[i] += c;
        }
        sum[p] += c * (r - l + 1);
    } else {
        // 不在同一块，块内暴力，块间整块处理
        for(int i = p + 1; i <= q - 1; ++i) {
            add[i] += c;
        }
        // 块内暴力
        for(int i = l; i <= R[p]; ++i) {
            a[i] += c;
        }
        sum[p] += c * (R[p] - l + 1);
        for(int i = L[q]; i <= r; ++i) {
            a[i] += c;
        }
        sum[q] += c * (r - L[q] + 1);
    }
}

ll query(int l, int r) {
    int p = block[l], q = block[r]; // 取出左右区间所在的块号
    ll ans = 0;
    if(p == q) {
        for(int i = l; i <= r; ++i) {
            ans += a[i];
        }
        ans += add[p] * (r - l + 1);
    } else {
        // 块间暴力
        for(int i = p + 1; i <= q - 1; ++i) {
            ans += sum[i] + add[i] * (R[i] - L[i] + 1); // 注意不是乘以 block_size
        }
        // 块内暴力
        for(int i = l; i <= R[p]; ++i) {
            ans += a[i];
        }
        ans += add[p] * (R[p] - l + 1);
        for(int i = L[q]; i <= r; ++i) {
            ans += a[i];
        }
        ans += add[q] * (r - L[q] + 1);
    }
    return ans;
}

int main() {
    scanf("%d%d", &n, &q);
    for(int i = 1; i <= n; ++i) {
        scanf("%lld", &a[i]);
    }
    init();
    for(int i = 0; i < q; ++i) {
        char op;
        getchar(); scanf("%c", &op);
        int l, r;
        scanf("%d%d", &l, &r);
        if(op == 'C') {
            ll c;
            scanf("%lld", &c);
            change(l, r, c);
        } else {
            printf("%lld\n", query(l, r));
        }
    }
    return 0;
}

Reference

《算法竞赛进阶指南》 李煜东 著