2021/finals/and.cpp

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

const int LIM = 4000005;

struct UnionFind {
  int N;
  vector<int> root, diff_to_root, rank, sz, diffsum;

  UnionFind(int _N):
   N(_N), root(_N), diff_to_root(_N), rank(_N), sz(_N, 1), diffsum(_N) {
    for (int i = 0; i < N; i++) {
      root[i] = i;
    }
  }

  int find(int i) {
    int r = root[i];
    if (r != i) {
      root[i] = find(r);
      diff_to_root[i] ^= diff_to_root[r];
    }
    return root[i];
  }

  bool merge(int i, int j, int d) {
    int ri = find(i), rj = find(j);
    d ^= diff_to_root[i] ^ diff_to_root[j];
    if (ri == rj) {
      return !d;
    }
    if (rank[ri] > rank[rj]) {
      swap(ri, rj);
    }
    root[ri] = rj;
    diff_to_root[ri] = d;
    sz[rj] += sz[ri];
    diffsum[rj] += d ? sz[ri] - diffsum[ri] : diffsum[ri];
    if (rank[ri] == rank[rj]) {
      rank[rj]++;
    }
    return true;
  }
};

int N, K;

void solve() {
  vector<int> both(LIM), ans(LIM);
  vector<vector<vector<int>>> ind(LIM, vector<vector<int>>(2));
  int maxL = 0;
  // Input.
  cin >> N >> K;
  for (int i = 0; i < N; i++) {
    string s[2];
    cin >> s[0] >> s[1];
    int len = max(s[0].size(), s[1].size());
    maxL = max(maxL, len);
    for (int b = 0; b < len; b++) {
      int ch[2];
      for (int j : {0, 1}) {
        int slen = s[j].size();
        ch[j] = b < slen ? s[j][slen - b - 1] - '0' : 0;
      }
      if (ch[0] && ch[1]) {
        both[b]++;
      } else {
        for (int j : {0, 1}) {
          if (ch[j]) {
            ind[b][j].push_back(i);
          }
        }
      }
    }
  }
  // Consider each bit, starting from the most significant.
  UnionFind U(N);
  for (int b = maxL - 1; b >= 0; b--) {
    ans[b] = 0;
    // Bit included twice?
    if (both[b] == N) {
      ans[b] = 2;
      continue;
    }
    // Bit not included at all?
    if (both[b] + ind[b][0].size() + ind[b][1].size() < N) {
      continue;
    }
    // Include bit once if possible.
    int k = 0;
    bool imp = false;
    UnionFind U2 = U;
    for (int i = 0; !imp && i < 2; i++) {
      for (int j = 0; !imp && j < (int)ind[b][i].size() - 1; j++) {
        if (!U2.merge(ind[b][i][j], ind[b][i][j + 1], 0)) {
          imp = true;
        }
      }
    }
    if (imp) {
      continue;
    }
    if (ind[b][0].size() > 0 && ind[b][1].size() > 0) {
      if (!U2.merge(ind[b][0][0], ind[b][1][0], 1)) {
        continue;
      }
    }
    for (int i = 0; i < N; i++) {
      if (U2.find(i) == i) {
        k += min(U2.diffsum[i], U2.sz[i] - U2.diffsum[i]);
      }
    }
    if (k > K) {
      continue;
    }
    U = U2;
    ans[b] = 1;
  }
  // Add up answer in binary.
  for (int i = 0; i < maxL; i++) {
    if (i == maxL - 1 && ans[i] >= 2) {
      ans[maxL++] = 0;
    }
    ans[i + 1] += ans[i] / 2;
    ans[i] %= 2;
  }
  // Output.
  while (maxL > 1 && !ans[maxL - 1]) {
    maxL--;
  }
  for (int i = maxL - 1; i >= 0; i--) {
    cout << ans[i];
  }
  cout << endl;
}

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