[ZJU 2112] Dynamic Rankings

[题目链接]

https://www.lydsy.com/JudgeOnline/problem.php?id=1901

[算法]

首先 ， 考虑没有修改操作

不妨建立可持久化线段树 ， 第i棵树维护区间[1 , i]中的数的出现个数 ， 则可以通过在线段树上二分的方式求出答案

那么 ， 若有修改操作 ， 我们不妨使用树状数组套可持久化线段树

树状数组中的第i个元素为一棵可持久化线段树 ， 代表区间[i , i - lowbit(i) + 1]每个数的出现次数

询问时可以同样二分 ， 只需用最多log(N)棵线段树作差即可

时间复杂度 : O(NlogN ^ 2)

[代码]

#include<bits/stdc++.h>
using namespace std;
#define MAXN 200010
#define MAXP 5000010
#define MAXLOG 20
typedef long long ll;
typedef unsigned long long ull;
typedef long double ld;

int n , m , len , L , R;
int a[MAXN] , root[MAXN] , val[MAXN] , x[MAXN] , y[MAXN] , l[MAXN] , r[MAXN] , k[MAXN] , ql[MAXLOG] , qr[MAXLOG];
char type[MAXN][];

struct Presitent_Segment_Tree
{
        int sz;
        int sum[MAXP] , lson[MAXP] , rson[MAXP];
        Presitent_Segment_Tree()
        {
                sz = ;
                memset(root ,  , sizeof(root));
                memset(lson ,  , sizeof(lson));
                memset(rson ,  , sizeof(rson));
        }
        inline void modify(int &k , int old , int l , int r , int pos , int value)
        {
                k = ++sz;
                lson[k] = lson[old] , rson[k] = rson[old];
                sum[k] = sum[old] + value;
                if (l == r) return;
                int mid = (l + r) >> ;
                if (mid >= pos) modify(lson[k] , lson[k] , l , mid , pos , value);
                else modify(rson[k] , rson[k] , mid +  , r , pos , value);
        }
        inline int query(int l , int r , int k)
        {
                int cnt = ;
                if (l == r)
                        return l;
                for (int i = ; i <= L; i++) cnt -= sum[lson[ql[i]]];
                for (int i = ; i <= R; i++) cnt += sum[lson[qr[i]]];
                int mid = (l + r) >> ;
                if (cnt >= k)
                {
                        for (int i = ; i <= L; i++) ql[i] = lson[ql[i]];
                        for (int i = ; i <= R; i++) qr[i] = lson[qr[i]];
                        return query(l , mid , k);
                } else
                {
                        for (int i = ; i <= L; i++) ql[i] = rson[ql[i]];
                        for (int i = ; i <= R; i++) qr[i] = rson[qr[i]];
                        return query(mid +  , r , k - cnt);
                }
        }
} PST;
struct Binary_Indexed_Tree
{
        inline int lowbit(int x)
        {
                return x & (-x);
        }
        inline void modify(int pos , int x , int val)
        {
                for (int i = pos; i <= n; i += lowbit(i))
                        PST.modify(root[i] , root[i] ,  , len , x , val);
        }
        inline int query(int l , int r , int k)
        {
                L =  , R = ;
                for (int i = l - ; i; i -= lowbit(i))
                        ql[++L] = root[i];
                for (int i = r; i; i -= lowbit(i))
                        qr[++R] = root[i];
                return PST.query( , len , k);
        }
} BIT;

template <typename T> inline void chkmax(T &x , T y) { x = max(x , y); }
template <typename T> inline void chkmin(T &x , T y) { x = min(x , y); }
template <typename T> inline void read(T &x)
{
        T f = ; x = ;
        char c = getchar();
        for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;
        for (; isdigit(c); c = getchar()) x = (x << ) + (x << ) + c - '';
        x *= f;
}

int main()
{

        scanf("%d%d" , &n , &m);
        len = n;
        for (int i = ; i <= n; i++)
        {
                scanf("%d" , &a[i]);
                val[i] = a[i];
        }
        for (int i = ; i <= m; i++)
        {
                scanf("%s" , type[i]);
                if (type[i][] == 'C')
                {
                        scanf("%d%d" , &x[i] , &y[i]);
                        val[++len] = y[i];
                } else
                        scanf("%d%d%d" , &l[i] , &r[i] , &k[i]);
        }
        sort(val +  , val + len + );
        len = unique(val +  , val + len + ) - val - ;
        for (int i = ; i <= n; i++) a[i] = lower_bound(val +  , val + len +  , a[i]) - val;
        for (int i = ; i <= m; i++)
                if (type[i][] == 'C') y[i] = lower_bound(val +  , val + len +  , y[i]) - val;
        for (int i = ; i <= n; i++) BIT.modify(i , a[i] , );
        for (int i = ; i <= m; i++)
        {
                if (type[i][] == 'C')
                {
                        BIT.modify(x[i] , a[x[i]] , -);
                        BIT.modify(x[i] , y[i] , );
                        a[x[i]] = y[i];
                } else printf("%d\n" , val[BIT.query(l[i] , r[i] , k[i])]);
        }

        return ;
}