树状数组初步

引入

树状数组用于求区间和，其修改和查询的复杂度都是\(O(logn)\)，非常好写，比较小巧。

几种基础用法，关于权值树状数组在另一篇博客。

单点修改，区间查询

区间和

HDU-1166 敌兵布阵

模版：

#include <bits/stdc++.h>
#define N 50005
using namespace std;
typedef long long ll;
int max(int a, int b) {
	return a > b ? a : b;
}
int min(int a, int b) {
	return a < b ? a : b;
}
int d[N];
int n;
void update(int x, int v) {
	for (int i = x; i <= n; i += i & (-i))
		d[i] += v;
}
ll query(int x) {
	ll ans = 0;
	for (int i = x; i; i -= i & (-i))
		ans += d[i];
	return ans;
}
int main() {
	int t;
	scanf("%d", &t);
	int cnt = 0;
	while (t--) {
		memset(d, 0, sizeof(d));
		scanf("%d", &n);
		for (int i = 1; i <= n; i++) {
			int v;
			scanf("%d", &v);
			update(i, v);
		}
		cnt++;
		printf("Case %d:\n", cnt);
		while (1) {
			char ch[10];
			scanf("%s", ch);
			int x, y;
			if (ch[0] == 'Q') {
				scanf("%d%d", &x, &y);
				printf("%lld\n", query(y) - query(x - 1));
			}
			else if (ch[0] == 'A') {
				scanf("%d%d", &x, &y);
				update(x, y);
			}
			else if (ch[0] == 'S') {
				scanf("%d%d", &x, &y);
				update(x, -y);
			}
			else break;
		}
	}
	return 0;
}

区间最小值

不建议使用树状数组求区间最小值，比较难以理解，而且写起来并不简单

HDU-1754 I Hate It

模版

#include <cstdio>
#include <cstring>
#define N 200050
using namespace std;
int h[N], a[N];
int lowbit(int x) {
	return x & (-x);
}
int n, m;
int max(int a, int b) {return a > b ? a : b;}
void update(int x) {
	int lx;
	while (x <= n) {
		h[x] = a[x];
		lx = lowbit(x);
		for (int i = 1; i < lx; i <<= 1)
			h[x] = max(h[x], h[x - i]);
		x += lowbit(x);
	}
}
int query(int l, int r) {
	int ans = 0;
	while (r >= l) {
		ans = max(a[r], ans);
		r--;
		for (; r - lowbit(r) >= l; r -= lowbit(r))
			ans = max(h[r], ans);
	}
	return ans;
}
int main() {
	while (scanf("%d%d", &n, &m) != EOF) {
		memset(h, 0, sizeof(h));
		for (int i = 1; i <= n; i++) {
			scanf("%d", &a[i]);
			update(i);
		}
		while (m--) {
			char ch[2];
			int x, y;
			scanf("%s%d%d", ch, &x, &y);
			if (ch[0] == 'Q') {
				printf("%d\n", query(x, y));
			}
			else {
				a[x] = y;
				update(x);
			}
		}
	}
	return 0;
}

单点修改，矩阵求和

直接使用二维树状数组即可。

附：矩阵前缀和求一部分和的公式：

\(sumv(x2, y2) - sumv(x1 - 1, y2) - sumv(x2, y11 - 1) + sumv(x1 - 1, y11 - 1)\)

\(sumv(x,y)\)为从(1,1)到(x,y)的前缀和

题目：HihoCoder-1336 Matrix Sum

模版：

#include <cstdio>
#include <cstring>
#define p 1000000007
#define N 1050
using namespace std;
typedef long long ll;
ll d[N][N];
int n, m;
int lowbit(int x) {
	return x & (-x);
}
void update(int x, int y, ll v) {
	for (int i = x; i <= n; i += lowbit(i)) {
		for (int j = y; j <= n; j += lowbit(j)) {
			d[i][j] += v;
		}
	}
}
ll sumv(int x, int y) {
	ll ans = 0;
	for (int i = x; i; i -= lowbit(i)) {
		for (int j = y; j; j -= lowbit(j)) {
			ans += d[i][j];
		}
	}
	return ans;
}
ll query(int x1, int y11, int x2, int y2) {
	return sumv(x2, y2) - sumv(x1 - 1, y2) - sumv(x2, y11 - 1) + sumv(x1 - 1, y11 - 1);
}

int main() {
	scanf("%d%d", &n, &m);
	for (int i = 1; i <= m; i++) {
		char ch[10];
		scanf("%s", ch);
		if (ch[0] == 'A') {
			int x, y; ll v;
			scanf("%d%d%lld", &x, &y, &v);
			x++; y++;
			update(x, y, v);
		}
		else {
			int x1, y11, x2, y2;
			scanf("%d%d%d%d", &x1, &y11, &x2, &y2);
			x1++; y11++; x2++; y2++;
			printf("%lld\n", (query(x1, y11, x2, y2) + p) % p);
		}
	}
	return 0;
}

树状数组求逆序对

使用树状数组维护一个位置之前一共有了多少数，当第i个数a[i]加进来时，先update更新，在询问这个数之前一共有了多少数，使用i-get(a[i])即为对逆序对的贡献，累计即可

题目：洛谷-P1966 火柴排队

此题题意是让a数组和b数组之间每个位置均对应为相同的大小顺序，问最少需要移动几次，即把a的位置移动到b的位置需要移动多少次，排序之后将a映射到b求逆序对即可

模版

#include<bits/stdc++.h>
#define maxn 100060
#define p 99999997
using namespace std;
int c[maxn], d[maxn], n;
inline int getnum(){
	char c; int ans = 0; bool flag = false;
	while (!isdigit(c = getchar()) && c != '-');
	if (c == '-') flag = true; else ans = c - '0';
	while (isdigit(c = getchar())) ans = ans * 10 + c - '0';
	return ans * (flag ? -1 : 1);
}
struct node{
	long long v; int pos;
}a[maxn], b[maxn];
int cmp(node x, node y){
	return x.v < y.v;
}
inline int lowbit(int x){
	return x & (-x);
}
inline void updata(int x){
	while (x <= n){
		d[x]++;
		x += lowbit(x);
	}
}
inline int getsum(int x){
	int ans = 0;
	while (x > 0){
		ans += d[x] % p;
		x -= lowbit(x);
	}
	return ans;
}
int main(){
	n = getnum();
	for (int i = 1; i <= n; i++){
		a[i].v = getnum();
		a[i].pos = i;
	}
	for (int i = 1; i <= n; i++){
		b[i].v = getnum();
		b[i].pos = i;
	}
	sort(a + 1, a + n + 1, cmp);
	sort(b + 1, b + n + 1, cmp);
	for (int i = 1; i <= n; i++){
		c[b[i].pos] = a[i].pos;
	}
	int ans = 0;
	for (int i = 1; i <= n; i++){
		updata(c[i]);
		ans += i - getsum(c[i]);
		ans %= p;
	}
	printf("%d", ans);
	return 0;
}