【Codeforces Round #853 (Div. 2)】C. Serval and Toxel‘s Arrays【题解】

题目

Toxel likes arrays. Before traveling to the Paldea region, Serval gave him an array $a$ as a gift. This array has $n$ pairwise distinct elements.

In order to get more arrays, Toxel performed $m$ operations with the initial array. In the $i$-th operation, he modified the pi-th element of the ($i-1$)-th array to $v_i$, resulting in the $i$-th array (the initial array a is numbered as $0$). During modifications, Toxel guaranteed that the elements of each array are still pairwise distinct after each operation.

Finally, Toxel got $m+1$ arrays and denoted them as $A_0=a,A_1,\dots ,A_m$. For each pair $(i,j)(0\le i<j\le m)$, Toxel defines its value as the number of distinct elements of the concatenation of $A_i$ and $A_j$. Now Toxel wonders, what is the sum of the values of all pairs? Please help him to calculate the answer.

It is guaranteed that the sum of $n$ and the sum of $m$ over all test cases do not exceed $2\cdot 10^5$.

思路

①初始化所有在原始数组中的数的数量为$m+1$。因为一共有$m+1$个数组，那么我一开始假定所有数组都与原始数组保持相同。
②对于第$i(0\le i＜m)$个数组，也就是给出一个位置的修改，那么这个位置上原来的数对应的数量就要相应地减少$m-i$，因为在当前包括后面一共$m-i$个数组中这个位置上的数都要修改，并将其变成$v$，然后$v$的数量相应地增加$m-i$。这样，我们就统计出了这$m+1$个数组中所有不同的数分别出现的次数。
③现在统计这$m+1$个数组中不同的数分别对答案产生的贡献之和。设某个数的数量是$cnt$，那么含有这个数的$cnt$个数组与其它$m+1-cnt$个数组组合时这个数产生的贡献是$cnt\ast (m+1-cnt)$；此外，含有这个数的$cnt$个数组两两组合时产生的$C_{cnt}^2=cnt\ast (cnt-1)/2$个组合分别产生了$1$的贡献。因此，答案就是：
$ans={\sum }_1^{n+m}(cnt[i]\ast (m+1-cnt[i])+cnt[i]\ast (cnt[i]-1)/2)$。

代码

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll maxn = 2e5+5;
ll n,m,a[maxn],cnt[maxn<<1];
void solve(){cin>>n>>m;memset(cnt,0,sizeof(cnt));for(ll i=1;i<=n;i++)cin>>a[i], cnt[a[i]] = m+1;for(ll i=0;i<m;i++){ll p,v;cin>>p>>v;cnt[a[p]] -= (m-i);a[p] = v;cnt[v] += (m-i);}ll ans = 0;for(ll i=1;i<=n+m;i++){if(cnt[i]==0)continue;ans += cnt[i]*(m+1-cnt[i]);ans += cnt[i]*(cnt[i]-1)/2;}cout<<ans<<endl;
}
int main() {ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);ll tests;cin>>tests;while(tests--){solve();}return 0;
}