2021/quals/gold_mine_ch2.cpp

#include <algorithm>
#include <cstring>
#include <iostream>
#include <vector>
using namespace std;

const int LIM = 52;

inline int& setmax(int &l, int r) {
  return l = max(l, r);
}

int N, K, C[LIM];
vector<int> adj[LIM];

// dp[i][j][k] = max value in i's subtree,
// with j new paths present,
// and with a free path ongoing from i's parent if k=1.
int dp[LIM][LIM][2];

// dp2[i][j][k][c] = max value after first i children of current node,
// with j new paths present,
// and with a free path available for use if k=1,
// and with at least one child connected if c=1.
int dp2[LIM][LIM][2][2];

void rec(int i, int parent) {
  int nc = 0;
  for (int j : adj[i]) {
    if (j != parent) {
      rec(j, i);
      nc++;
    }
  }
  // Sub-DP.
  for (int j = 0; j <= nc; j++) {
    memset(dp2[j], -1, sizeof dp2[j]);
  }
  dp2[0][0][0][0] = 0;
  nc = 0;
  for (int j : adj[i]) {
    if (j == parent) {
      continue;
    }
    for (int a1 = 0; a1 <= K; a1++) {
      for (int x : {0, 1}) {
        for (int y : {0, 1}) {
          int d = dp2[nc][a1][x][y];
          if (d < 0) {
            continue;
          }
          for (int a2 = 0; a2 <= K - a1; a2++) {
            // Connect to child.
            int d2 = dp[j][a2][1];
            if (d2 >= 0) {
              setmax(dp2[nc + 1][a1 + a2 + (x ? 0 : 1)][1 - x][1], d + d2);
            }
            // Don't connect to child.
            d2 = dp[j][a2][0];
            if (d2 >= 0) {
              setmax(dp2[nc + 1][a1 + a2][x][y], d + d2);
            }
          }
        }
      }
    }
    nc++;
  }
  // Combine into main DP.
  memset(dp[i], -1, sizeof dp[i]);
  for (int o : {0, 1}) {
    for (int j = 0; j <= K; j++) {
      for (int x : {0, 1}) {
        for (int y : {0, 1}) {
          if (!i && !y) {
            continue;  // Root must connect to at least one child.
          }
          int d = dp2[nc][j][x][y];
          if (d < 0) {
            continue;
          }
          // Include current node's value?
          if (o || y) {
            d += C[i];
          }
          // Free path ongoing from parent?
          setmax(dp[i][j - (o && x ? 1 : 0)][o], d);
        }
      }
    }
  }
}

int solve() {
  cin >> N >> K;
  for (int i = 0; i < N; i++) {
    cin >> C[i];
    adj[i].clear();
  }
  for (int i = 0; i < N - 1; i++) {
    int a, b;
    cin >> a >> b;
    a--, b--;
    adj[a].push_back(b);
    adj[b].push_back(a);
  }
  // DP
  rec(0, -1);
  int ans = C[0];
  for (int i = 0; i <= K; i++) {
    setmax(ans, dp[0][i][0]);
  }
  return ans;
}

int main() {
  int T;
  cin >> T;
  for (int t = 1; t <= T; t++) {
    cout << "Case #" << t << ": " << solve() << endl;
  }
  return 0;
}