【数据结构】树状数组

单点加值，区间求和

这个是最基础的树状数组的用法了。初始值是Init之后一个一个Add上去的。

struct BinaryIndexTree {

    static const int MAXN = 500000 + 5;

    int n;
    int sm[MAXN];

    void Add(int x, int v) {
        for(int i = x; i <= n; i += i & (-i))
            sm[i] += v;
    }

    ll Sum(int x) {
        ll res = 0;
        for(int i = x; i; i -= i & (-i))
            res += sm[i];
        return res;
    }

    void Init(int _n) {
        n = _n;
        memset(sm, 0, sizeof(sm[0]) * (n + 1));
    }

} bit;

已通过：
https://www.luogu.com.cn/problem/P3374

单点改值，区间求和

额外记录一个va数组，然后每次比对这次要“加”的值是多少，套用单点加值。

struct BinaryIndexTree {

    static const int MAXN = 500000 + 5;

    int n;
    int va[MAXN];
    int sm[MAXN];

    void Add(int x, int v) {
        for(int i = x; i <= n; i += i & (-i))
            sm[i] += v;
    }

    ll Sum(int x) {
        ll res = 0;
        for(int i = x; i; i -= i & (-i))
            res += sm[i];
        return res;
    }

    void Init(int _n) {
        n = _n;
        memset(va, 0, sizeof(va[0]) * (n + 1));
        memset(sm, 0, sizeof(sm[0]) * (n + 1));
    }

    void Modify(int x, int v) {
        int d = v - va[x];
        Add(x, d);
        va[x] = v;
    }

} bit;

二维偏序

namespace BinaryIndexTree {

    const int MAXN = 200000 + 5;

    int n;
    int nn;
    int a[MAXN];
    int aa[MAXN];

    int bit[MAXN];

    void Add(int x, int v) {
        for(int i = x; i <= nn; i += i & (-i))
            bit[i] += v;
    }

    ll Sum(int x) {
        ll res = 0;
        for(int i = x; i; i -= i & (-i))
            res += bit[i];
        return res;
    }

    void Init1() {
        n = 0;
        nn = 0;
    }

    void Insert(int v) {
        a[++n] = v;
        aa[++nn] = v;
    }

    void Init2() {
        sort(aa + 1, aa + 1 + nn);
        nn = unique(aa + 1, aa + 1 + nn) - (aa + 1);
        memset(bit, 0, sizeof(bit[0]) * (nn + 1));
        for(int i = 1; i <= n; ++i) {
            a[i] = lower_bound(aa + 1, aa + 1 + nn, a[i]) - aa;
            Add(a[i], 1);
        }
    }

}

区间修改区间求和

设差分数组 (d_i=a_i-a_{i-1}) ，显然有 (a_i=sumlimits_{j=1}^i d_j) ，那么 (Sum(x)=sumlimits_{i=1}^x a_i=sumlimits_{i=1}^x sumlimits_{j=1}^i d_j = sumlimits_{i=1}^x (x-i+1)*d_i=(x+1)sumlimits_{i=1}^xd_i - sumlimits_{i=1}^x i*d_i) 。

struct BinaryIndexTree {
//    const int MAXN = 3e5 + 5;
    int n;
    ll d1[MAXN], d2[MAXN];
    void Add(int x, int v) {
        for(int i = x; i <= n; i += i & (-i)) d1[i] += v, d2[i] += 1LL * x * v;
    }
    ll Sum(int x) {
        ll res = 0;
        for(int i = x; i; i -= i & (-i)) res += 1LL * (x + 1) * d1[i] - d2[i];
        return res;
    }
    int RangeAdd(int l, int r, int v) {
        Add(l, v), Add(r + 1, -v);
    }
    ll RangeSum(int l, int r) {
        return Sum(r) - Sum(l - 1);
    }
    void Init(int _n) {
        n = _n;
        memset(d1, 0, sizeof(d1[0]) * (n + 1));
        memset(d2, 0, sizeof(d2[0]) * (n + 1));
    }
} bit;