Submission #1370310

#TimeUsernameProblemLanguageResultExecution timeMemory
1370310sarahspeedyBosses (BOI16_bosses)C++20
0 / 100
0 ms344 KiB
#include<bits/stdc++.h>
using namespace std;

#define int long long

signed main(){

    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int n;
    cin >> n;

    vector<vector<int>> boss(n + 1);

    for(int i = 1; i <= n; i++){

        int k;
        cin >> k;

        boss[i].resize(k);

        for(int j = 0; j < k; j++){

            cin >> boss[i][j];
        }
    }

    /*
        Correct observation:

        Minimum salary assignment for a rooted tree is:
        salary(u) = 1 + sum(salary(child))

        Hence total salary =
        sum of subtree sizes =
        n + sum(depth)

        So we want to maximize depths.

        Optimal structure:
        longest possible chain.

        We build DAG:
        u -> v means v accepts u as boss.

        Then answer = minimum possible sum
        over longest hierarchy chain decomposition.

        This becomes:
        answer = n + minimum path cover on DAG

        Since constraints/special guarantee allow valid construction,
        longest chain DP works.
    */

    vector<vector<int>> g(n + 1);
    vector<int> indeg(n + 1, 0);

    for(int v = 1; v <= n; v++){

        for(auto u : boss[v]){

            g[u].push_back(v);
            indeg[v]++;
        }
    }

    queue<int> q;

    vector<int> topo;

    for(int i = 1; i <= n; i++){

        if(indeg[i] == 0){

            q.push(i);
        }
    }

    while(!q.empty()){

        int u = q.front();
        q.pop();

        topo.push_back(u);

        for(auto v : g[u]){

            indeg[v]--;

            if(indeg[v] == 0){

                q.push(v);
            }
        }
    }

    vector<int> dp(n + 1, 1);

    for(auto u : topo){

        for(auto v : g[u]){

            dp[v] = max(dp[v], dp[u] + 1);
        }
    }

    int best = 0;

    for(int i = 1; i <= n; i++){

        best = max(best, dp[i]);
    }

    /*
        Minimum total salary for chain of length L:
        1 + 2 + ... + L
    */

    int ans = best * (best + 1) / 2;

    cout << ans << '\n';
}
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