树状数组模板

1.单点修改 区间查询

const int MAXN=1e5+8;
typedef long long ll;
int n;
ll a[MAXN];
inline int lowbit(int x){return x&(-x);}
inline void add(int x,ll val){//单点修改
   while(x){
  	 	a[x]+=val;
	   	x+=lowbit(x);
   	}
}
inline ll sum(int x){//返回a[x]前缀和
    ll res=0;
    while(x){
    	res+=d[x];
    	x-=lowbit(x);
    }
    return res;
}
inline ll range_sum(int x,int y){//返回区间和
	return sum[y]-sum[x-1];
}

2.区间修改 单点查询

差分数组\(d[i]= a[i]- a[i-1]\)，故\(a[i]=d[1]+d[2]+...+d[i]\)
维护 \(d[i]\) 前缀和即可维护 \(a[i]\)
若\(d[i]+=k\) 则：\(a[j]'=d[1]+d[2]+...+d[j] = a[j]+k\) 其中 \((i\le j)\)
故区间 \([l , r ]\)上加上k 等价于 \(d[l]+k，d[r+1]-k\)

const int MAXN=1e5+8;
typedef long long ll;
int n;
ll d[MAXN];//差分数组
inline int lowbit(int x){return x&(-x);}
inline void add(int x,ll val){
   while(x){
  	 	d[x]+=val;
	   	x+=lowbit(x);
   	}
}
inline void range_add(int x,int y,ll val){//区间修改
    add(x,val);
    add(y+1,-val);
}
inline ll ask(int x){//返回a[x]
    ll res=0;
    while(x){
    	res+=d[x];
    	x-=lowbit(x);
    }
    return res;
}

3.区间修改 区间查询

\(d[i]= a[i]- a[i-1]\)
\(dd[i]=i*d[i]\)
数组a前缀和\(\sum_{i=0}^k a[i]=a[1]+a[2]+...+a[k]=id[1]+(i-1)d[2]+...+d[k]\)
\(=(k+1)*(d[1]+d[2]+...+d[k])-(d[1]+2d[2]+...+kd[k] )\)
所以，维护\(d[i]\) 和 \(dd[i]\) 即可。

const int MAXN=1e5+8;
typedef long long ll;
int n;
ll d[MAXN],dd[MAXN];
inline int lowbit(int x){return x&(-x);}
inline void add(int x,ll val){//d和dd单点修改
    for(int i=x;i<=n;i+=lowbit(i)){
        d[i]+=val,dd[i]+=val*x;
    }
}
inline void range_add(int x,int y,ll val){
	add(x,val);
	add(y+1.-val);
}
inline ll sum(int x){//前缀和
    ll res=0;
    for(int i=x;i;i-=lowbit(i))
        res+=(x+1)*d[i]-dd[i];
    return res;
}
inline ll range_sum(int x,int y){
	return sum(y)-sum(x-1);
}