bzoj4695 最假女选手（势能线段树/吉司机线段树）题解

题意：

已知\(n\)个数字，进行以下操作：

• \(1.\)给一个区间\([L,R]\) 加上一个数\(x\)
• \(2.\)把一个区间\([L,R]\) 里小于\(x\) 的数变成\(x\)
• \(3.\)把一个区间\([L,R]\) 里大于\(x\) 的数变成\(x\)
• \(4.\)求区间\([L,R]\)的和
• \(5.\)求区间\([L,R]\)的最大值
• \(6.\)求区间\([L,R]\) 的最小值

思路：

吉司机线段树。

假如我们要进行把一个区间\([L,R]\) 里小于\(x\) 的数变成\(x\)。那么我们可以维护一个最小值\(Min\)和次小值\(sMin\)和最小值数量\(Minlen\)。那么，当\(Min\geq x\)时，显然这个区间不需要操作；当\(Min<x\)且\(sMin>x\)时，这时只要更新\(Min=x\)；当\(sMin\leq x\)时，继续往下\(dfs\)。操作\(3\)也是同理。

因为这里是没设重置的标记的，所以在\(pushdown\)时如果子节点和父节点产生冲突，那么以父节点为准。

复杂度\(O(mlog^2n)\)，\(m\)为操作数。

有点卡常。

代码：

#include<map>
#include<set>
#include<queue>
#include<cmath>
#include<stack>
#include<ctime>
#include<vector>
#include<cstdio>
#include<string>
#include<cstring>
#include<sstream>
#include<iostream>
#include<algorithm>
typedef long long ll;
typedef unsigned long long ull;
using namespace std;
const int maxn = 5e5 + 5;
const int MAXM = 3e6;
const ll MOD = 998244353;
const ull seed = 131;
const int INF = 1 << 30;

#define lson (rt << 1)
#define rson (rt << 1 | 1)
int a[maxn];
int Max[maxn << 2], Min[maxn << 2], sMax[maxn << 2], sMin[maxn << 2];
int Maxlen[maxn << 2], Minlen[maxn << 2];
ll sum[maxn << 2];
int lazy[maxn << 2];
inline void pushup(int rt){ //这边建议加上inline
    sum[rt] = sum[lson] + sum[rson];

    if(Max[lson] < Max[rson]){
        Max[rt] = Max[rson];
        Maxlen[rt] = Maxlen[rson];
        sMax[rt] = max(sMax[rson], Max[lson]);
    }
    if(Max[lson] > Max[rson]){
        Max[rt] = Max[lson];
        Maxlen[rt] = Maxlen[lson];
        sMax[rt] = max(sMax[lson], Max[rson]);
    }
    if(Max[lson] == Max[rson]){
        Max[rt] = Max[lson];
        Maxlen[rt] = Maxlen[lson] + Maxlen[rson];
        sMax[rt] = max(sMax[lson], sMax[rson]);
    }

    if(Min[lson] > Min[rson]){
        Min[rt] = Min[rson];
        Minlen[rt] = Minlen[rson];
        sMin[rt] = min(sMin[rson], Min[lson]);
    }
    if(Min[lson] < Min[rson]){
        Min[rt] = Min[lson];
        Minlen[rt] = Minlen[lson];
        sMin[rt] = min(sMin[lson], Min[rson]);
    }
    if(Min[lson] == Min[rson]){
        Min[rt] = Min[lson];
        Minlen[rt] = Minlen[lson] + Minlen[rson];
        sMin[rt] = min(sMin[lson], sMin[rson]);
    }
}
inline void pushdown(int rt, int l, int r){
    int m = (l + r) >> 1;
    if(lazy[rt]){
        sum[lson] += 1LL * (m - l + 1) * lazy[rt];
        sum[rson] += 1LL * (r - m) * lazy[rt];
        Max[lson] += lazy[rt];
        Max[rson] += lazy[rt];
        sMax[lson] += lazy[rt];
        sMax[rson] += lazy[rt];
        Min[lson] += lazy[rt];
        Min[rson] += lazy[rt];
        sMin[lson] += lazy[rt];
        sMin[rson] += lazy[rt];
        lazy[lson] += lazy[rt];
        lazy[rson] += lazy[rt];
        lazy[rt] = 0;
    }
    if(Max[lson] > Max[rt]){	//要和父节点保持一致
        if(sMax[lson] == Max[lson]) sMax[lson] = Max[rt];
        if(Min[lson] == Max[lson]) Min[lson] = Max[rt];
        if(sMin[lson] == Max[lson]) sMin[lson] = Max[rt];
        sum[lson] += 1LL * (Max[rt] - Max[lson]) * Maxlen[lson];
        Max[lson] = Max[rt];
    }
    if(Max[rson] > Max[rt]){
        if(sMax[rson] == Max[rson]) sMax[rson] = Max[rt];
        if(Min[rson] == Max[rson]) Min[rson] = Max[rt];
        if(sMin[rson] == Max[rson]) sMin[rson] = Max[rt];
        sum[rson] += 1LL * (Max[rt] - Max[rson]) * Maxlen[rson];
        Max[rson] = Max[rt];
    }
    if(Min[lson] < Min[rt]){
        if(sMin[lson] == Min[lson]) sMin[lson] = Min[rt];
        if(Max[lson] == Min[lson]) Max[lson] = Min[rt];
        if(sMax[lson] == Min[lson]) sMax[lson] = Min[rt];
        sum[lson] += 1LL * (Min[rt] - Min[lson]) * Minlen[lson];
        Min[lson] = Min[rt];
    }
    if(Min[rson] < Min[rt]){
        if(sMin[rson] == Min[rson]) sMin[rson] = Min[rt];
        if(Max[rson] == Min[rson]) Max[rson] = Min[rt];
        if(sMax[rson] == Min[rson]) sMax[rson] = Min[rt];
        sum[rson] += 1LL * (Min[rt] - Min[rson]) * Minlen[rson];
        Min[rson] = Min[rt];
    }
}
void build(int l, int r, int rt){
    lazy[rt] = 0;
    if(l == r){
        sum[rt] = Max[rt] = Min[rt] = a[l];
        sMax[rt] = -INF;
        sMin[rt] = INF;
        Maxlen[rt] = Minlen[rt] = 1;
        return;
    }
    int m = (l + r) >> 1;
    build(l, m, lson);
    build(m + 1, r, rson);
    pushup(rt);
}
void add(int L, int R, int l, int r, int v, int rt){
    if(L <= l && R >= r){
        sum[rt] += 1LL * v * (r - l + 1);
        Max[rt] += v;
        Min[rt] += v;
        sMax[rt] += v;
        sMin[rt] += v;
        lazy[rt] += v;
        return;
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    if(L <= m)
        add(L, R, l, m, v, lson);
    if(R > m)
        add(L, R, m + 1, r, v, rson);
    pushup(rt);
}
void Less(int L, int R, int l, int r, int v, int rt){
    if(Min[rt] >= v) return;
    if(L <= l && R >= r && sMin[rt] > v){//>保证Minlen不变
        if(Max[rt] == Min[rt]) Max[rt] = v;
        if(sMax[rt] == Min[rt]) sMax[rt] = v;
        sum[rt] += 1LL * (v - Min[rt]) * Minlen[rt] ;
        Min[rt] = v;
        return;
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    if(L <= m)
        Less(L, R, l, m, v, lson);
    if(R > m)
        Less(L, R, m + 1, r, v, rson);
    pushup(rt);
}
void More(int L, int R, int l, int r, int v, int rt){
    if(Max[rt] <= v) return;
    if(L <= l && R >= r && sMax[rt] < v){   //<保证Maxlen不变
        if(Min[rt] == Max[rt]) Min[rt] = v;
        if(sMin[rt] == Max[rt]) sMin[rt] = v;
        sum[rt] += 1LL * (v - Max[rt]) * Maxlen[rt];
        Max[rt] = v;
        return;
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    if(L <= m)
        More(L, R, l, m, v, lson);
    if(R > m)
        More(L, R, m + 1, r, v, rson);
    pushup(rt);
}
ll querySum(int L, int R, int l, int r, int rt){
    if(L <= l && R >= r){
        return sum[rt];
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    ll ret = 0;
    if(L <= m)
        ret += querySum(L, R, l, m, lson);
    if(R > m)
        ret += querySum(L, R, m + 1, r, rson);
    return ret;
}
int queryMax(int L, int R, int l, int r, int rt){
    if(L <= l && R >= r){
        return Max[rt];
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    int MAX = -INF;
    if(L <= m)
        MAX = max(MAX, queryMax(L, R, l, m, lson));
    if(R > m)
        MAX = max(MAX, queryMax(L, R, m + 1, r, rson));
    return MAX;
}
int queryMin(int L, int R, int l, int r, int rt){
    if(L <= l && R >= r){
        return Min[rt];
    }
    pushdown(rt, l, r);
    int m = (l + r) >> 1;
    int MIN = INF;
    if(L <= m)
        MIN = min(MIN, queryMin(L, R, l, m, lson));
    if(R > m)
        MIN = min(MIN, queryMin(L, R, m + 1, r, rson));
    return MIN;
}
inline bool read(int &num){
        char in;
        bool IsN=false;
        in = getchar();
        if(in == EOF) return false;
        while(in != '-' && (in < '0' || in > '9')) in = getchar();
        if(in == '-'){ IsN = true; num = 0;}
        else num = in - '0';
        while(in = getchar(),in >= '0' && in <= '9'){
                num *= 10, num += in-'0';
        }
        if(IsN) num = -num;
        return true;
}

int main(){
    int n;
    read(n);
    for(int i = 1; i <= n; i++) read(a[i]);
    build(1, n, 1);
    int m;
    read(m);
    while(m--){
        int l, r, x, op;
        read(op), read(l), read(r);
        if(op <= 3) read(x);
        if(op == 1) add(l, r, 1, n, x, 1);
        else if(op == 2) Less(l, r, 1, n, x, 1);
        else if(op == 3) More(l, r, 1, n, x, 1);
        else if(op == 4) printf("%lld\n", querySum(l, r, 1, n, 1));
        else if(op == 5) printf("%d\n", queryMax(l, r, 1, n, 1));
        else printf("%d\n", queryMin(l, r, 1, n, 1));
    }
    return 0;
}