CF911E Stack Sorting

洛谷题目链接:CF911E Stack Sorting

Codeforces题目链接:Stack Sorting

题意翻译

给你一排列的一部分,让你补全整个排列使其字典序最大并且经过一个栈调整顺序之后能够顺序输出

题目描述

Let's suppose you have an array aa , a stack ss (initially empty) and an array bb (also initially empty).

You may perform the following operations until both aa and ss are empty:

• Take the first element of aa , push it into ss and remove it from aa (if aa is not empty);
• Take the top element from ss , append it to the end of array bb and remove it from ss (if ss is not empty).

You can perform these operations in arbitrary order.

If there exists a way to perform the operations such that array bb is sorted in non-descending order in the end, then array aa is called stack-sortable.

For example, [3,1,2][3,1,2] is stack-sortable, because bb will be sorted if we perform the following operations:

1. Remove 33 from aa and push it into ss ;
2. Remove 11 from aa and push it into ss ;
3. Remove 11 from ss and append it to the end of bb ;
4. Remove 22 from aa and push it into ss ;
5. Remove 22 from ss and append it to the end of bb ;
6. Remove 33 from ss and append it to the end of bb .

After all these operations b=[1,2,3]b=[1,2,3] , so [3,1,2][3,1,2] is stack-sortable. [2,3,1][2,3,1] is not stack-sortable.

You are given kk first elements of some permutation pp of size nn (recall that a permutation of size nn is an array of size nn where each integer from 11 to nn occurs exactly once). You have to restore the remaining n-kn−k elements of this permutation so it is stack-sortable. If there are multiple answers, choose the answer such that pp is lexicographically maximal (an array qq is lexicographically greater than an array pp iff there exists some integer kksuch that for every i<ki<k q_{i}=p_{i}qi​=pi​ , and q_{k}>p_{k}qk​>pk​ ). You may not swap or change any of first kk elements of the permutation.

Print the lexicographically maximal permutation pp you can obtain.

If there exists no answer then output -1.

输入输出格式

输入格式：

The first line contains two integers nn and kk ( 2<=n<=2000002<=n<=200000 , 1<=k<n1<=k<n ) — the size of a desired permutation, and the number of elements you are given, respectively.

The second line contains kk integers p_{1}p1​ , p_{2}p2​ , ..., p_{k}pk​ ( 1<=p_{i}<=n1<=pi​<=n ) — the first kk elements of pp . These integers are pairwise distinct.

输出格式：

If it is possible to restore a stack-sortable permutation pp of size nn such that the first kk elements of pp are equal to elements given in the input, print lexicographically maximal such permutation.

Otherwise print -1.

输入输出样例

输入样例#1：

5 3

3 2 1

输出样例#1：

3 2 1 5 4

输入样例#2：

5 3

2 3 1

输出样例#2：

-1

输入样例#3：

5 1

3

输出样例#3：

3 2 1 5 4

输入样例#4：

5 2

3 4

输出样例#4：

-1

一句话题意: 给出一个长度为\(n\)的排列的前\(k\)个,你需要构造第\(k+1\)到第\(n\)个使得这些数字按顺序入栈并在入栈后任意时候出栈可以使出栈顺序为\(1\)~\(n\).如果无法构造出这样的排列则输出\(-1\).如果存在多组解则输出字典序最大的那一个.

话说这道题是我在CF上第一道非比赛时A掉的题目,也是第一个超过C题的题目...

题解: 首先考虑判断\(-1\)的情况.其实前\(k\)个已经确定了,那么我们就可以直接从小到大的插入数字并判断.因为我们要求出栈的顺序是从小到大的,所以如果我们从小的开始插入,那么插入这个小数字之后马上就可以弹出栈,这样模拟过一遍之后就可以判断前\(k\)个是否能组成一组合法解.

然后再来考虑这个字典序的问题.既然要求字典序最大,那么显然能放一个大的数字就不能放比它小的.然后我们会发现,如果\(k==0\),那么后面的\(n\)个就可以直接从\(n\)到\(1\)构造. 也就是说,只要我们把前\(k\)个已经固定了的数字全部都能弹出栈之后,那么后面直接降序输出就可以了.

这里我们记录前\(k\)个元素中未被弹出栈的元素的最小值,然后将后来从小到大枚举变成从最小值-1开始枚举直到出现被选用的数字停下就可以了.

如果不懂看一下代码理解一下吧.

#include<bits/stdc++.h>
using namespace std;
const int N=200000+5;
const int inf=2147483647;

int n, k, a[N], vis[N], stk[N], ins[N], top = 0, cnt = 0, mn[N], topmn = 0, len = 0;
//vis记录一个数字是否已经被选用
//mn记录未被选用的数字的最小值(每一个可能成为最小值的数字都要记录进来)
//topmn为mn栈的栈顶指针
//ins记录前k个数字是否已经被弹出栈
//stk模拟放入数字的栈,top是它的栈顶

bool check(){
    int i = 1, pos = 1;
    while(i <= k){
        stk[++top] = a[i], vis[a[i]] = ins[a[i]] = 1, i++;
        while(top && stk[top] == pos) pos++, ins[stk[top--]] = 0;
    }
    for(int j=1;j<=n;j++){//直接从小到大放入数字,能取出就把数字都取出
        if(!vis[j]) stk[++top] = j;
        while(top && stk[top] == pos) ins[stk[top--]] = 0, pos++;
    }
    for(int i=1;i<=n;i++)
        if(ins[i]) return false;//如果仍有数字未被取出,则无法构成合法排列
    return true;
}

void work(){
    int i = 1, pos = 1, st = 0; top = 0, topmn = 0, mn[0] = inf, len = k;
    //cnt记录第1~k个数字中有几个没被弹出栈
    memset(vis, 0, sizeof(vis)); memset(ins, 0, sizeof(ins));
    vis[0] = 1;//重复使用数组需要清空
    //vis[0]=1是为了防止数组判断越界
    for(int j=1;j<=k;j++) st = max(st, a[j]);//st记录最后一个数字被弹出栈的位置,之后的就直接倒序输出就可以了
    while(i <= k){//模拟
        stk[++top] = a[i], vis[a[i]] = ins[a[i]] = 1, cnt++;
        if(mn[topmn] > a[i]) mn[++topmn] = a[i];
        while(top && stk[top] == pos){
            if(mn[topmn] == stk[top]) topmn--;
            pos++, ins[stk[top--]] = 0, cnt--;
        }
        i++;
    }
    while(cnt){
        for(int j=mn[topmn]-1;vis[j] == 0;j--){//注意这里的枚举顺序
            stk[++top] = j, a[++len] = j, vis[j] = 1;
            while(top && stk[top] == pos){
                if(mn[topmn] ==  stk[top]) topmn--;
                if(ins[stk[top]]) ins[stk[top--]] = 0, pos++, cnt--;
                else top--, pos++;
            }
        }
    }
    for(int i=st+1;i<=n;i++) a[i] = n-i+1+st;
    for(int i=1;i<=n;i++) cout << a[i] << ' '; cout << endl;
}

int main(){
    // freopen("data.in", "r", stdin);
    ios::sync_with_stdio(false);
    cin >> n >> k;
    for(int i=1;i<=k;i++) cin >> a[i];
    if(check()) work();
    else cout << -1 << endl;
    return 0;
}