[20251122] BOJ / P2 / MST의 기댓값 / 권혁준

Author: oncsr

khj20006/202511/22 BOJ P2 MST의 기댓값.md

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+```cpp
+#include <bits/stdc++.h>
+using namespace std;
+using ll = long long;
+
+const ll MOD = 1e9 + 7;
+
+int N, M;
+vector<tuple<int, int, int>> edges;
+map<ll, ll> events;
+vector<int> root;
+vector<ll> cost;
+ll X = 0, P = 0;
+
+ll power(ll a, ll b) {
+    if(b == 0) return 1;
+    if(b == 1) return a % MOD;
+    ll h = power(a, b>>1) % MOD;
+    h = (h * h) % MOD;
+    return (b&1) ? h * a % MOD : h;
+}
+
+ll g(ll x) { return x*(x+1)%MOD*power(2,MOD-2)%MOD; }
+ll h(ll x) { return x*(x+1)%MOD*(2*x+1)%MOD*power(6,MOD-2)%MOD; }
+
+int find(int x) { return x == root[x] ? x : root[x] = find(root[x]); }
+
+int main(){
+    cin.tie(0)->sync_with_stdio(0);
+
+    cin>>N>>M;
+    P = power((1000000001LL * N % MOD * (N-1) % MOD * power(2, MOD-2) % MOD), MOD-2);
+    for(int a,b,c;M--;) {
+        cin>>a>>b>>c;
+        edges.emplace_back(c,a,b);
+    }
+
+    root.resize(N+1);
+    cost.resize(N+1);
+    iota(root.begin(), root.end(), 0);
+    fill(cost.begin(), cost.end(), 1);
+    cost[0] = 0;
+    
+    sort(edges.begin(), edges.end());
+    
+    for(auto [c,a,b] : edges) {
+        int x = find(a), y = find(b);
+        if(x == y) continue;
+        events[c] = (events[c] + cost[x]*cost[y]) % MOD;
+        events[0] = (events[0] + (cost[x]*cost[y] % MOD * (1000000001 - c) % MOD)) % MOD;
+        cost[y] += cost[x];
+        root[x] = y;
+        X += c;
+    }
+
+    ll C = 0, ans = 0, D = 0;
+    for(auto it = events.rbegin();it != events.rend();) {
+        // cur - i
+        // C + it->second
+        ll k = it->second;
+        D = (D + k) % MOD;
+        C = (C + D) % MOD;
+        if(it->first == 0) {
+            ans = (ans + (C * X % MOD)) % MOD;
+            break;
+        }
+        ll s = it->first;
+        ll e = (++it)->first;
+        ll cnt = s - e - 1;
+        
+        ll cur = (X - s) % MOD;
+        ll _1 = cur * C % MOD * (cnt + 1) % MOD;
+        ll _2 = cur * D % MOD * g(cnt) % MOD;
+        ll _3 = C * g(cnt) % MOD;
+        ll _4 = D * h(cnt) % MOD;
+        ans = (ans + (_1 + _2 + _3 + _4)) % MOD;
+        // cout<<"1 : "<<_1<<'\t';
+        // cout<<"2 : "<<_2<<'\t';
+        // cout<<"3 : "<<_3<<'\t';
+        // cout<<"4 : "<<_4<<'\n';
+        C = (C + D * cnt) % MOD;
+    }
+    cout<<ans * P % MOD;
+
+}
+```