Codeforces 1091E

题意：给定n个点的度数，请你添加第n+1个点，输出所有可能的第n+1个点的度数

做法：1. 查看链接知道了下面这个定理

A sequence of non-negative integers $ d_1\geq\cdots\geq d_n$ can be represented as the degree sequence of a finite simple graph on \(n\) vertices if and only if $ d_1+\cdots+d_n$ is even and\(\sum _{i=1}^{k}d_{i}\leq k(k-1)+\sum _{i=k+1}^{n}\min(d_{i},k)\)
holds for every k in \({\displaystyle 1\leq k\leq n}\).

2.排序之后，\([low(p), up(p)]\) 表示在\([p, p+1]\) 之间插入新的点度数\(d_t\)的取值范围。

(1) \(d_{p+1} \leq d_t \leq d_p\)

(2) 当 \(d_t\) 在不等式左侧时
\[ d_t \leq k(k+1)+\sum _{i=k+1}^{n}\min(d_{i},k+1) - \sum _{i=1}^{k}d_{i}, p \leq k \leq n \\ \Downarrow \\ up(p) = min(up(p + 1) , p(p+1) + \sum_{i=p+1}^n min(d_i,p+1) - \sum_{i=1}^p d_i) \]

(3) 当 \(d_t\) 在不等式右侧时
\[ \sum_{i=1}^k d_i - k(k-1) - \sum_{i=k+1}^n min(d_i,k) <= d_t , 1 \leq k \leq p \\ \Downarrow\\ low(p) = max(\sum_{i=1}^p d_i - p(p-1) - \sum_{i=p+1}^n min(d_i,p), low(p-1)) \\ \]

3.猜测答案为连续的一段值，将上一步求出的区间并起来

4.又因为无向简单图，每条边对度数贡献为2，由此可以确定答案的奇偶性

5.确定上下界时，需要维护\(\sum_{i = k+1}^{n} min(d_i,k)\) 和\(\sum_{i = k}^{n} min(d_i,k)\) 我们倒着枚举\(k\)，用优先队列维护小于等于当前\(k\)的集合，以及这个集合内所有元素的和，就可以预处理出这个式子，因为k是递减的因此保证了正确性

6.一定要分析清楚再写。。。代码细节很多

#include <bits/stdc++.h>
typedef long long ll;
constexpr int N = 600005;
constexpr ll inf = 1e18;
using namespace std;
int n;
priority_queue<ll> q;
ll f1[N], f2[N], sum[N], low[N], up[N], d[N];
int main() {scanf("%d",&n);for(int i = 1; i <= n; ++i) scanf("%lld",&d[i]);sort(d+1,d+1+n); reverse(d+1,d+1+n);for(int i = 1; i <= n; ++i) sum[i] = sum[i-1] + d[i];d[0] = n; d[n+1] = 0;for(int i = 0; i <= n; ++i) up[i] = inf, low[i] = -inf;ll tmp = 0;for(int i = n; i >= 0; --i) {while(!q.empty() && q.top() > i) tmp -= q.top(), q.pop();f2[i] = tmp + 1ll*(n - i - q.size())*i;if(d[i] <= i) q.push(d[i]), tmp += d[i];f1[i] = tmp + 1ll*(n - i + 1 - q.size())*i;}for(int i = n; i >= 0; --i) {if(i!=n) up[i] = min(up[i+1], 1ll*i*(i+1) + f1[i+1] - sum[i]);else up[i] = 1ll*i*(i+1) - sum[i];}for(int i = 1; i <= n; ++i) {low[i] = max(low[i-1], sum[i] - 1ll*i*(i-1) - f2[i]);}for(int i = 0; i <= n; ++i) low[i] = max(low[i], d[i+1]), up[i] = min(up[i], d[i]);ll L = inf, R = -inf;for(ll i = 0; i <= n; ++i) if(low[i] <= up[i]) L = min(L, low[i]), R = max(R, up[i]);if(L > R) printf("-1");for(ll i = L; i <= R; ++i) if(!((sum[n]-i)&1)) printf("%lld ",i);puts("");return 0;
}