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답안 #947364

# 제출 시각 아이디 문제 언어 결과 실행 시간 메모리
947364 2024-03-16T03:13:26 Z GrindMachine Star Trek (CEOI20_startrek) C++17
100 / 100
 106 ms 31768 KB 
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>

using namespace std;
using namespace __gnu_pbds;

template<typename T> using Tree = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
typedef long long int ll;
typedef long double ld;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;

#define fastio ios_base::sync_with_stdio(false); cin.tie(NULL)
#define pb push_back
#define endl '\n'
#define sz(a) (int)a.size()
#define setbits(x) __builtin_popcountll(x)
#define ff first
#define ss second
#define conts continue
#define ceil2(x,y) ((x+y-1)/(y))
#define all(a) a.begin(), a.end()
#define rall(a) a.rbegin(), a.rend()
#define yes cout << "Yes" << endl
#define no cout << "No" << endl

#define rep(i,n) for(int i = 0; i < n; ++i)
#define rep1(i,n) for(int i = 1; i <= n; ++i)
#define rev(i,s,e) for(int i = s; i >= e; --i)
#define trav(i,a) for(auto &i : a)

template<typename T>
void amin(T &a, T b) {
    a = min(a,b);
}

template<typename T>
void amax(T &a, T b) {
    a = max(a,b);
}

#ifdef LOCAL
#include "debug.h"
#else
#define debug(x) 42
#endif

/*

refs:
edi
https://youtu.be/Tjv78ZThV5c
(solution inspired by these sources)

build backwards from the dth to the 0th dimension
when we are at the ith dimension (i < d), do the following:
connect to the root of the (i+1)th tree
pick a root for the ith tree
(for tree 0, root = 1)

we only care about the root's state (W/L), not the exact node
maintain dp[i][W/L] = #of ways to make the root of ith tree W/L after fixing all teleporters in trees >= i
for each root, calculate the #of nodes s.t when their state changes from L --> W, the state of the root is W
naive calculation = O(n^2) (n dfs calls)
optimized = O(n) (rerooting dp)

these values can be used to transition from dp[i][W/L] to dp[i-1][W/L]
transitions can be sped up using matrix expo
need some care when handling 0th tree (because root = 1 is fixed, can't pick arbitrary root)

*/

const int MOD = 1e9 + 7;
const int N = 1e5 + 5;
const int inf1 = int(1e9) + 5;
const ll inf2 = ll(1e18) + 5;

struct Matrix {
    vector<vector<ll>> a;
    int n, m;

    Matrix() {

    }

    Matrix(int row, int col) {
        n = row, m = col;
        a = vector<vector<ll>>(row, vector<ll>(col));
    }

    Matrix operator*(const Matrix &mat2) {
        int n2 = mat2.n, m2 = mat2.m;
        Matrix res(n, m2);

        rep(i, n) {
            rep(j, m2) {
                rep(k, m) {
                    ll temp = (a[i][k] * mat2.a[k][j]) % MOD;
                    res.a[i][j] = (res.a[i][j] + temp) % MOD;
                }
            }
        }

        return res;
    }

    void exp(ll b) {
        Matrix res(n, m);
        Matrix curr = *this;
        rep(i, n) res.a[i][i] = 1;

        while (b) {
            if (b & 1) res = res * curr;
            curr = curr * curr;
            b /= 2;
        }

        a = res.a;
    }
};

vector<ll> adj[N];
vector<ll> dp1(N), dp2(N);

void dfs1(ll u, ll p){
    dp1[u] = 0;
    trav(v,adj[u]){
        if(v == p) conts;
        dfs1(v,u);
        dp1[u] += (dp1[v] == 0);
    }
}

void dfs2(ll u, ll p){
    trav(v,adj[u]){
        if(v == p) conts;
        ll val = dp2[u]-(dp1[v] == 0);
        dp2[v] += (val == 0);
        dfs2(v,u);
    }
}

ll dp3[N][2], dp4[N][2];
vector<ll> dp5(N);
vector<ll> win_cnt(N), lose_cnt(N);
ll win_sum[N][2], lose_sum[N][2];

void dfs3(ll u, ll p){
    dp3[u][0] = 1;

    trav(v,adj[u]){
        if(v == p) conts;
        dfs3(v,u);
    }

    vector<ll> win,lose;
    trav(v,adj[u]){
        if(v == p) conts;
        if(dp1[v]){
            win.pb(v);
            win_cnt[u]++;
            rep(j,2){
                win_sum[u][j] += dp3[v][j];
            }
        }
        else{
            lose.pb(v);
            lose_cnt[u]++;
            rep(j,2){
                lose_sum[u][j] += dp3[v][j];
            }
        }
    }

    if(sz(lose) == 0){
        rep(j,2){
            dp3[u][j] += win_sum[u][j^1];
        }
    }
    else if(sz(lose) == 1){
        rep(j,2){
            dp3[u][j] += lose_sum[u][j^1];
        }
    }

    rep(j,2){
        dp4[u][j] = dp3[u][j];
    }
}

void dfs4(ll u, ll p){
    trav(v,adj[u]){
        if(v == p) conts;

        ll win = win_cnt[u], lose = lose_cnt[u];
        array<ll,2> wsum,lsum;
        wsum.fill(0), lsum.fill(0);

        rep(j,2){
            wsum[j] = win_sum[u][j];
            lsum[j] = lose_sum[u][j];
        }

        if(dp1[v]){
            win--;
            rep(j,2){
                wsum[j] -= dp3[v][j];
            }
        }
        else{
            lose--;
            rep(j,2){
                lsum[j] -= dp3[v][j];
            }
        }

        array<ll,2> dpu;
        dpu.fill(0);
        dpu[0] = 1;

        if(lose == 0){
            rep(j,2){
                dpu[j] += wsum[j^1];
            }
        }
        else if(lose == 1){
            rep(j,2){
                dpu[j] += lsum[j^1];
            }
        }

        ll val = dp2[u];
        val -= (dp1[v] == 0);

        if(val){
            win_cnt[v]++;
            rep(j,2){
                win_sum[v][j] += dpu[j];
            }
        }
        else{
            lose_cnt[v]++;
            rep(j,2){
                lose_sum[v][j] += dpu[j]; 
            }
        }

        dp4[v][0] = 1, dp4[v][1] = 0;

        if(lose_cnt[v] == 0){
            rep(j,2){
                dp4[v][j] += win_sum[v][j^1];
            }
        }
        else if(lose_cnt[v] == 1){
            rep(j,2){
                dp4[v][j] += lose_sum[v][j^1];
            }
        }

        dfs4(v,u);
    }
}

void dfs5(ll u, ll p, ll depth, ll r){
    if(!dp1[u]){
        ll val = 1;
        if(depth&1) val = 0;
        dp5[r] -= dp2[r];
        dp5[r] += val;
    }

    vector<ll> win,lose;
    trav(v,adj[u]){
        if(v == p) conts;
        if(dp1[v]) win.pb(v);
        else lose.pb(v);
    }

    if(sz(lose) == 0){
        trav(v,win){
            dfs5(v,u,depth+1,r);
        }
    }
    else if(sz(lose) == 1){
        dfs5(lose[0],u,depth+1,r);
    }
}

void solve(int test_case)
{
    ll n,d; cin >> n >> d;
    rep1(i,n-1){
        ll u,v; cin >> u >> v;
        adj[u].pb(v), adj[v].pb(u);
    }

    dfs1(1,-1);
    rep1(i,n) dp2[i] = dp1[i];
    dfs2(1,-1);

    dfs3(1,-1);
    dfs4(1,-1);

    rep1(i,n){
        amin(dp1[i],1ll);
        amin(dp2[i],1ll);
    }

    rep1(i,n){
        dp5[i] = n*dp2[i];
        dp5[i] -= (dp4[i][0]+dp4[i][1])*dp2[i];
        dp5[i] += dp4[i][0];
    }

    ll win_ways_w = 0, lose_ways_w = 0;
    rep1(r,n){
        if(dp2[r]){
            win_ways_w += n;
        }
        else{
            lose_ways_w += n;
        }
    }

    ll win_ways_l = 0, lose_ways_l = 0;
    rep1(r,n){
        win_ways_l += dp5[r];
        lose_ways_l += n-dp5[r];
    }

    Matrix base(1,2);
    rep1(i,n){
        base.a[0][dp2[i]]++;
    }

    Matrix mat(2,2);
    mat.a = {
        {lose_ways_l%MOD, win_ways_l%MOD},
        {lose_ways_w%MOD, win_ways_w%MOD}
    };

    mat.exp(d-1);
    base = base*mat;

    ll ans = 0;
    if(dp2[1]){
        ans += n*base.a[0][1];
    }
    
    ans += dp5[1]*base.a[0][0];
    ans %= MOD;

    cout << ans << endl;
}

int main()
{
    fastio;

    int t = 1;
    // cin >> t;

    rep1(i, t) {
        solve(i);
    }

    return 0;
}

Compilation message

startrek.cpp: In member function 'Matrix Matrix::operator*(const Matrix&)':
startrek.cpp:93:13: warning: unused variable 'n2' [-Wunused-variable]
   93 |         int n2 = mat2.n, m2 = mat2.m;
      |             ^~

Subtask #1
0.0 / 0.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10584 KB Output is correct
2 Correct 4 ms 10844 KB Output is correct

Subtask #2
7.0 / 7.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 4 ms 10584 KB Output is correct
3 Correct 3 ms 10584 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct

Subtask #3
8.0 / 8.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 3 ms 10588 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct

Subtask #4
15.0 / 15.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 3 ms 10588 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct
7 Correct 3 ms 10844 KB Output is correct
8 Correct 3 ms 10844 KB Output is correct
9 Correct 3 ms 10844 KB Output is correct
10 Correct 3 ms 10844 KB Output is correct
11 Correct 4 ms 10844 KB Output is correct

Subtask #5
15.0 / 15.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 3 ms 10588 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct
7 Correct 3 ms 10844 KB Output is correct
8 Correct 3 ms 10844 KB Output is correct
9 Correct 3 ms 10844 KB Output is correct
10 Correct 3 ms 10844 KB Output is correct
11 Correct 4 ms 10844 KB Output is correct
12 Correct 95 ms 23316 KB Output is correct
13 Correct 78 ms 31768 KB Output is correct
14 Correct 52 ms 16648 KB Output is correct
15 Correct 65 ms 16468 KB Output is correct
16 Correct 56 ms 16580 KB Output is correct

Subtask #6
20.0 / 20.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 3 ms 10588 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct
7 Correct 3 ms 10844 KB Output is correct
8 Correct 3 ms 10844 KB Output is correct
9 Correct 3 ms 10844 KB Output is correct
10 Correct 3 ms 10844 KB Output is correct
11 Correct 4 ms 10844 KB Output is correct
12 Correct 3 ms 10584 KB Output is correct
13 Correct 3 ms 10844 KB Output is correct
14 Correct 3 ms 10588 KB Output is correct
15 Correct 3 ms 10596 KB Output is correct
16 Correct 3 ms 10588 KB Output is correct
17 Correct 2 ms 10588 KB Output is correct
18 Correct 3 ms 10588 KB Output is correct
19 Correct 3 ms 10588 KB Output is correct
20 Correct 3 ms 10588 KB Output is correct
21 Correct 4 ms 10840 KB Output is correct
22 Correct 3 ms 10844 KB Output is correct
23 Correct 3 ms 10828 KB Output is correct
24 Correct 3 ms 10844 KB Output is correct
25 Correct 3 ms 10840 KB Output is correct
26 Correct 3 ms 10844 KB Output is correct
27 Correct 3 ms 10844 KB Output is correct
28 Correct 3 ms 10840 KB Output is correct
29 Correct 3 ms 10844 KB Output is correct
30 Correct 3 ms 10844 KB Output is correct

Subtask #7
20.0 / 20.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10588 KB Output is correct
2 Correct 3 ms 10588 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 3 ms 10588 KB Output is correct
5 Correct 3 ms 10588 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct
7 Correct 3 ms 10844 KB Output is correct
8 Correct 3 ms 10844 KB Output is correct
9 Correct 3 ms 10844 KB Output is correct
10 Correct 3 ms 10844 KB Output is correct
11 Correct 4 ms 10844 KB Output is correct
12 Correct 95 ms 23316 KB Output is correct
13 Correct 78 ms 31768 KB Output is correct
14 Correct 52 ms 16648 KB Output is correct
15 Correct 65 ms 16468 KB Output is correct
16 Correct 56 ms 16580 KB Output is correct
17 Correct 3 ms 10584 KB Output is correct
18 Correct 3 ms 10844 KB Output is correct
19 Correct 3 ms 10588 KB Output is correct
20 Correct 3 ms 10596 KB Output is correct
21 Correct 3 ms 10588 KB Output is correct
22 Correct 2 ms 10588 KB Output is correct
23 Correct 3 ms 10588 KB Output is correct
24 Correct 3 ms 10588 KB Output is correct
25 Correct 3 ms 10588 KB Output is correct
26 Correct 4 ms 10840 KB Output is correct
27 Correct 3 ms 10844 KB Output is correct
28 Correct 3 ms 10828 KB Output is correct
29 Correct 3 ms 10844 KB Output is correct
30 Correct 3 ms 10840 KB Output is correct
31 Correct 3 ms 10844 KB Output is correct
32 Correct 3 ms 10844 KB Output is correct
33 Correct 3 ms 10840 KB Output is correct
34 Correct 3 ms 10844 KB Output is correct
35 Correct 3 ms 10844 KB Output is correct
36 Correct 78 ms 23420 KB Output is correct
37 Correct 106 ms 31568 KB Output is correct
38 Correct 45 ms 16464 KB Output is correct
39 Correct 56 ms 16480 KB Output is correct
40 Correct 56 ms 16356 KB Output is correct
41 Correct 82 ms 27840 KB Output is correct
42 Correct 64 ms 30124 KB Output is correct
43 Correct 36 ms 17096 KB Output is correct
44 Correct 56 ms 16468 KB Output is correct
45 Correct 57 ms 16484 KB Output is correct

Subtask #8
15.0 / 15.0

# 결과 실행 시간 메모리 Grader output
1 Correct 3 ms 10584 KB Output is correct
2 Correct 4 ms 10844 KB Output is correct
3 Correct 3 ms 10588 KB Output is correct
4 Correct 4 ms 10584 KB Output is correct
5 Correct 3 ms 10584 KB Output is correct
6 Correct 3 ms 10588 KB Output is correct
7 Correct 3 ms 10588 KB Output is correct
8 Correct 3 ms 10588 KB Output is correct
9 Correct 3 ms 10588 KB Output is correct
10 Correct 3 ms 10588 KB Output is correct
11 Correct 3 ms 10588 KB Output is correct
12 Correct 3 ms 10588 KB Output is correct
13 Correct 3 ms 10588 KB Output is correct
14 Correct 3 ms 10844 KB Output is correct
15 Correct 3 ms 10844 KB Output is correct
16 Correct 3 ms 10844 KB Output is correct
17 Correct 3 ms 10844 KB Output is correct
18 Correct 4 ms 10844 KB Output is correct
19 Correct 95 ms 23316 KB Output is correct
20 Correct 78 ms 31768 KB Output is correct
21 Correct 52 ms 16648 KB Output is correct
22 Correct 65 ms 16468 KB Output is correct
23 Correct 56 ms 16580 KB Output is correct
24 Correct 3 ms 10584 KB Output is correct
25 Correct 3 ms 10844 KB Output is correct
26 Correct 3 ms 10588 KB Output is correct
27 Correct 3 ms 10596 KB Output is correct
28 Correct 3 ms 10588 KB Output is correct
29 Correct 2 ms 10588 KB Output is correct
30 Correct 3 ms 10588 KB Output is correct
31 Correct 3 ms 10588 KB Output is correct
32 Correct 3 ms 10588 KB Output is correct
33 Correct 4 ms 10840 KB Output is correct
34 Correct 3 ms 10844 KB Output is correct
35 Correct 3 ms 10828 KB Output is correct
36 Correct 3 ms 10844 KB Output is correct
37 Correct 3 ms 10840 KB Output is correct
38 Correct 3 ms 10844 KB Output is correct
39 Correct 3 ms 10844 KB Output is correct
40 Correct 3 ms 10840 KB Output is correct
41 Correct 3 ms 10844 KB Output is correct
42 Correct 3 ms 10844 KB Output is correct
43 Correct 78 ms 23420 KB Output is correct
44 Correct 106 ms 31568 KB Output is correct
45 Correct 45 ms 16464 KB Output is correct
46 Correct 56 ms 16480 KB Output is correct
47 Correct 56 ms 16356 KB Output is correct
48 Correct 82 ms 27840 KB Output is correct
49 Correct 64 ms 30124 KB Output is correct
50 Correct 36 ms 17096 KB Output is correct
51 Correct 56 ms 16468 KB Output is correct
52 Correct 57 ms 16484 KB Output is correct
53 Correct 72 ms 31756 KB Output is correct
54 Correct 73 ms 28404 KB Output is correct
55 Correct 30 ms 16580 KB Output is correct
56 Correct 66 ms 23384 KB Output is correct
57 Correct 70 ms 16756 KB Output is correct
58 Correct 54 ms 16732 KB Output is correct
59 Correct 54 ms 16464 KB Output is correct
60 Correct 56 ms 16324 KB Output is correct