算法学习：主席树（可持久化线段树）

【前置知识】

 线段树

 

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【定义】

【可持久化】能够保存历史版本，方便操作区间等，减少复杂度

 

 

 

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【主席树】

  可解决的经典问题区间第k大/小

  时空复杂度为O(nlogn)

 

 

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【模板题】

【luogu 3834】 

#include<cstdio>
#include<iostream>
#include<algorithm>
using namespace std;
const int MAXN = 200010;
const int MAXM = 200010;
int l[MAXN], a[MAXN],T[MAXN];
int sum[21*MAXN], ls[21 * MAXN], rs[21 * MAXN];
int n, m;
int cnt;
int build(int l, int r)
{
    int root=++cnt;
    sum[root] = 0;
    int mid = (l + r) >> 1;
    if (l < r)
    {
        ls[root] = build(l, mid);
        rs[root] = build(mid + 1, r);
    }
    return root;
}
int update(int pre, int l, int r,int x)
{
    int root=++cnt;
    sum[root] = sum[pre] + 1;
    ls[root] = ls[pre];
    rs[root] = rs[pre];
    int mid = (l + r) >> 1;
    if (l < r)
    {
        if (x <= mid)
            ls[root] = update(ls[pre], l, mid, x);
        else
            rs[root] = update(rs[pre], mid + 1, r,x);
    }
    return root;
}
int query(int l,int r,int x, int y, int num)
{
    if (l >= r)
        return l;
    int t = sum[ls[y]] - sum[ls[x]];
    int mid = (l + r) >> 1;
    if (num > t)
        return query(mid + 1, r, rs[x], rs[y], num - t);
    else
        return query(l, mid, ls[x], ls[y], num);

}
int main()
{
    scanf("%d%d", &n, &m);
    for (int i = 1; i <= n; i++)
    {
        scanf("%d", &a[i]);
        l[i] = a[i];
    }
    sort(l + 1, l + 1 + n);
    int maxn = unique(l + 1, l + 1 + n) - l - 1;
    T[0] = build(1, maxn);
    for (int i = 1; i <= n; i++)
    {
        int t = lower_bound(l + 1, l + 1 + maxn,a[i]) - l;
        T[i] = update(T[i - 1], 1, maxn,t);
    }
    for (int i = 1; i <= m; i++)
    {
        int x, y,k;
        scanf("%d%d%d", &x, &y, &k);
        printf("%d\n", l[query(1,maxn,T[x-1],T[y],k)]);
    }
    return 0;
}