树状数组

lowbit操作：

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

更新操作：

void update(int x, int num){
    for(int i = x; i <= N; i += lowbit(i)){
        tree[i] += num;
    }
}

求和操作：

int getSum(int x){
    int res = 0;
    for(int i = x; i > 0; i -= lowbit(i)){
        res += tree[i];
    }
    return res;
}

单点更新，区间查询
https://cn.vjudge.net/problem/HDU-1166

#include<bits/stdc++.h>
using namespace std;
const int maxn = 50000+5;
int d[maxn];
int t,n;
string c;

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


void update(int x, int num){
    for(int i = x; i <= n; i += lowbit(i)){
        d[i] += num;
    }
}


 int getSum(int x){
    int res = 0;
    for(int i = x; i > 0; i -= lowbit(i)){
        res += d[i];
    }
    return res;
}

int main(){
  int a,x,y;
  cin>>t;
  for(int i=1;i<=t;i++){
    cin>>n;
    printf("Case %d:\n",i);
    memset(d,0,sizeof d);
    for(int l=1;l<=n;l++){
      cin>>a;
      update(l,a);
    }
    while(cin>>c){
      if(c=="Query"){
        cin>>x>>y;
        cout<<getSum(y)-getSum(x-1)<<endl;//单点求和，差分树状数组里面的值不是差分数组的值
      }else if(c=="Add"){
        cin>>x>>y;
        update(x,y);//单点更新，树状数组存前缀和
      }else if(c=="Sub"){
        cin>>x>>y;
        update(x,-y);
      }else{
        break;
      }
    }
  }
  return 0;
}

区间更新，单点查询
https://www.luogu.org/problemnew/show/P3368

#include<bits/stdc++.h>
using namespace std;
const int maxn = 500000+5;
int a[maxn],d[maxn];
int n,m;

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


void update(int x, int num){
    for(int i = x; i <= n; i += lowbit(i)){
        d[i] += num;
    }
}


int getSum(int x){
    int res = 0;
    for(int i = x; i > 0; i -= lowbit(i)){
        res += d[i];
    }
    return res;
}


int main(){
  int A,B,C,D;
  cin>>n>>m;
  a[0]=0;
  for(int i=1;i<=n;i++){
    cin>>a[i];
    update(i,a[i]-a[i-1]);//区间更新，树状数组存差分
  }

  for(int i=0;i<m;i++){
    cin>>A;
    if(A==1){
      cin>>B>>C>>D;
      update(B,D);
      update(C+1,-D);
    }
    if(A==2){
      cin>>B;
      cout<<getSum(B)<<endl;
    }
  }
  return 0;
}

区间更新，区间查询
https://cn.vjudge.net/problem/POJ-3468

#include <iostream>
#include <cstdio>
using namespace std;
typedef long long LL;
const int maxn = 1e5 + 7;
LL tree1[maxn], tree2[maxn], a[maxn];
int n, m;
int lowbit(int x){
    return x & (-x);
}

void update(int i, LL x){
    for(int j = i; j <= n; j += lowbit(j)){
        tree1[j] += x;
        tree2[j] += x * (i-1);
    }
}

LL getSum(int i){
    LL res = 0;
    for(int j = i; j >= 1; j -= lowbit(j)){
        res += i * tree1[j] - tree2[j];
    }
    return res;
}

int main()
{
    scanf("%d %d", &n, &m);
    for(int i = 1; i <= n; i++){
        scanf("%lld", &a[i]);
        update(i, a[i] - a[i-1]);
    }
    char c;
    int x, y, k;
    while(m--){
        scanf(" %c", &c);
        if(c == 'C'){
            scanf("%d %d %d", &x, &y, &k);
            update(x, k);
            update(y+1, -k);
        }
        else {
            scanf("%d %d", &x, &y);
            printf("%lld\n", getSum(y) - getSum(x-1));
        }
    }
    return 0;
}