HackerEarth Maximize the distance in graph problem solution

In this HackerEarth Maximize the distance in graph problem solution Given an undirected complete graph G with n vertices. The weight of the edge between (i,j) is equal to C[i][j] Let D(i,j) be the shortest distance between vertices i,j and W(G) be the sum of all shortest distances between any two vertices. You can remove at most k edges from graph G in order to obtain a graph G’. You have to maximize the ratio of the sum of all shortest distances between any two vertices in new graph G’ (after removing the edges) and sum of all shortest distances between any two vertices in old graph G (before reomoving the edges): W(G’)/W(G). G must remain connected.

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ld = long double;
const int inf = 1e9 + 1e5;

mt19937 rr;

using Edge = pair<int, int>;

struct DSU {
  vector<int> col, rk;

  DSU(int n): col(n), rk(n) {
    iota(col.begin(), col.end(), 0);
  }

  int get(int u) {
    if (u == col[u]) {
      return u;
    }
    return col[u] = get(col[u]);
  }

  bool join(int u, int v) {
    u = get(u), v = get(v);
    if (u == v) {
      return false;
    }
    if (rk[u] > rk[v]) {
      swap(u, v);
    }
    if (rk[u] == rk[v]) {
      ++rk[v];
    }
    col[u] = v;
    return true;
  }
};

const int maxn = 128;
const int C = 1 << 20;
int a[maxn][maxn];
int n, k;
int clique = 1, rem_in_clique = 0;

int b[maxn][maxn];

ll score(int g[maxn][maxn]) {
  int t[maxn][maxn];
  memcpy(t, g, sizeof(t));
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      for (int k = 0; k < n; ++k) {
        t[j][k] = min(t[j][k], t[j][i] + t[i][k]);
      }
    }
  }
  ll res = 0;
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      assert(t[i][j] <= (1 << 20) * (n - 1));
      res += t[i][j];
    }
  }
  return res;
}

vector<int> find_path_1(int u) {
  vector<int> path{u};
  vector<char> used(n);
  used[u] = true;
  for (int i = 0; i < n - 1; ++i) {
    int v = -1;
    for (int j = 0; j < n; ++j) {
      if (!used[j] && (v == -1 || a[u][j] > a[u][v])) {
        v = j;
      }
    }
    assert(v != -1);
    u = v;
    path.push_back(u);
    used[u] = true;
  }
  reverse(path.begin(), path.end());
  return path;
}

vector<int> find_path_2(int u, bool with_two) {
  vector<int> path{u};
  vector<char> used(n);
  used[u] = true;
  int L = n / 2 - 1, R = n / 2 + 1;
  while (L >= 0 || R < n) {
    bool in_front = true;
    if (R < n && (L <= clique || L + 1 <= n - R)) {
      in_front = false;
    }
    bool two_vertices = false;
    if (in_front) {
      if (--L >= 0 && with_two) {
        two_vertices = true;
        --L;
      }
    } else {
      if (++R < n && with_two) {
        two_vertices = true;
        ++R;
      }
    }
    int u = in_front ? path.front() : path.back();
    int v1 = -1, v2 = -1;
    if (!two_vertices) {
      for (int j = 0; j < n; ++j) {
        if (used[j]) {
          continue;
        }
        if (v1 == -1 || (a[u][j] > a[u][v1] && rr())) {
          v1 = j;
        }
      }
      assert(v1 != -1);
    } else {
      for (int j1 = 0; j1 < n; ++j1) {
        if (used[j1]) {
          continue;
        }
        for (int j2 = 0; j2 < n; ++j2) {
          if (used[j2] || j2 == j1) {
            continue;
          }
          if (v1 == -1 || a[u][j1] + a[j1][j2] > a[u][v1] + a[v1][v2]) {
            v1 = j1, v2 = j2;
          }
        }
      }
      assert(v1 != -1 && v2 != -1);
    }
    used[v1] = true;
    if (two_vertices) {
      used[v2] = true;
    }
    if (!in_front) {
      path.push_back(v1);
      if (two_vertices) {
        path.push_back(v2);
      }
    } else {
      path.insert(path.begin(), v1);
      if (two_vertices) {
        path.insert(path.begin(), v2);
      }
    }
  }
  assert(L == -1 && R == n);
  return path;
}

vector<int> find_path_3() {
  vector<vector<int>> g(n);
  struct E {
    int u, v, w;
  };
  vector<E> es;
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < i; ++j) {
      es.push_back(E{i, j, a[i][j]});
    }
  }
  sort(es.begin(), es.end(), [](E a, E b) {
    return a.w > b.w;
  });
  int iters = n - 1;
  DSU dsu(n);
  for (auto e : es) {
    if ((int) g[e.u].size() <= 1 &&
        (int) g[e.v].size() <= 1 &&
        dsu.get(e.u) != dsu.get(e.v)) {
      --iters;
      dsu.join(e.u, e.v);
      g[e.u].push_back(e.v);
      g[e.v].push_back(e.u);
    }
  }
  int u = 0, prev = -1;
  while ((int) g[u].size() == 2) {
    ++u;
  }
  vector<int> path{u};
  for (int iter = 0; iter < n - 1; ++iter) {
    for (auto v : g[u]) {
      if (v == prev) {
        continue;
      }
      prev = u;
      u = v;
      path.push_back(u);
      break;
    }
  }
  return path;
}

const int M = 10;
vector<int> rebuild(vector<int> path, vector<int> ws) {
  pair<ll, int>  d[1 << M][M];
  memset(d, -1, sizeof(d));

  int n = (int) path.size();
  assert(n <= M);

  auto w = [&](int i, int j) -> ll {
    return a[path[i]][path[j]];
  };

  ll score_old = 0;
  for (int i = 0; i < n - 1; ++i) {
    score_old += ws[i] * w(i, i + 1);
  }

  d[1][0] = {0, -1};
  for (int mask = 0; mask < (1 << n); ++mask) {
    int bits = __builtin_popcount(mask);
    for (int u = 0; u < n; ++u) {
      if (d[mask][u].first == -1) {
        continue;
      }
      for (int v = 0; v < n; ++v) {
        if (mask & (1 << v)) {
          continue;
        }
        int nmask = mask | (1 << v);
        ll ncost = d[mask][u].first + ws[bits - 1] * w(u, v);
        if (d[nmask][v].first < ncost) {
          d[nmask][v] = {ncost, u};
        }
      }
    }
  }
  int mask = (1 << n) - 1;
  int u = n - 1;
  ll score_new = d[mask][u].first;
  assert(score_new != -1);
  vector<int> res;
  for (int i = 0; i < n - 1; ++i) {
    res.push_back(path[u]);
    int v = d[mask][u].second;
    mask ^= 1 << u;
    u = v;
  }
  res.push_back(path[u]);
  reverse(res.begin(), res.end());
  return res;
}

int rnd(int l, int r) {
  return rr() % (r - l + 1) + l;
}

void optimize_path(vector<int>& path, int iters) {
  if (clique == n - 1) {
    return;
  }
  for (int iter = 0; iter < iters; ++iter) {
    int l = rnd(clique + 1, max(clique + 1, n - M));
    int r = min(l + M, n);
    vector<int> ws;
    for (int i = l + 1; i < r; ++i) {
      ws.push_back(i * (n - i));
    }
    vector<int> subpath;
    for (int i = l; i < r; ++i) {
      subpath.push_back(path[i]);
    }
    rebuild(subpath, ws);
    for (int i = l; i < r; ++i) {
      path[i] = subpath[i - l];
    }
  }
}

vector<int> random_path() {
  vector<int> path(n);
  iota(path.begin(), path.end(), 0);
  shuffle(path.begin(), path.end(), rr);
  return path;
}

int attempt = 0;
vector<int> find_path() {
  if (attempt++ == 0) {
    return find_path_3();
  }
  if (attempt <= n) {
    int u = attempt++ - 1;
    return find_path_2(u, true);
  }
  if (attempt <= 2 * n) {
    int u = attempt++ - n - 1;
    return find_path_2(u, false);
  }
  cerr << "donen";
  return random_path();
}

vector<Edge> solve(vector<int> path) {
  optimize_path(path, 10);
  vector<Edge> es;
  for (int i = 0; i < n - 1; ++i) {
    es.emplace_back(path[i], path[i + 1]);
  }
  for (int i = 0; i < n && (int) es.size() < k; ++i) {
    vector<pair<int, int>> to;
    for (int j = 0; j < i - 1; ++j) {
      to.emplace_back(a[path[i]][path[j]], j);
    }
    sort(to.rbegin(), to.rend());
    for (auto e : to) {
      es.emplace_back(path[i], path[e.second]);
      if ((int) es.size() == k) {
        break;
      }
    }
  }
  return es;
}

void print(vector<Edge> es) {
  assert((int) es.size() == k);
  int q = n * (n - 1) / 2 - (int) es.size();
  bool keep[maxn][maxn];
  memset(keep, 0, sizeof(keep));
  for (auto e : es) {
    int u = e.first, v = e.second;
    keep[u][v] = keep[v][u] = true;
  }
  cout << q << 'n';
  for (int i = 0; i < n; ++i) {
    for (int j = i + 1; j < n; ++j) {
      if (!keep[i][j]) {
        cout << i + 1 << ' ' << j + 1 << 'n';
        --q;
      }
    }
  }
  assert(q == 0);
}

void build_graph(vector<Edge> es) {
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      b[i][j] = i == j ? 0 : inf;
    }
  }
  for (auto e : es) {
    int u, v;
    tie(u, v) = e;
    b[u][v] = b[v][u] = a[u][v];
  }
}

signed main() {
#ifdef LOCAL
  assert(freopen("test.in", "r", stdin));
#endif
  assert(cin >> n >> k);
#ifndef LOCAL
  assert(n == 128);
#endif
  assert(0 <= k && k <= (n - 1) * (n - 1));
  k = max(n - 1, n * (n - 1) / 2 - k);

  int cc = n - 1;
  if (cc < k) {
    for (int i = 2; i < n; ++i) {
      cc += i - 1;
      if (cc >= k) {
        rem_in_clique = cc - k;
        clique = i;
        break;
      }
    }
  }
  //cerr << clique << ' ' << rem_in_clique << 'n';

  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      assert(cin >> a[i][j]);
      if (i == j) {
        assert(a[i][j] == 0);
      } else {
        assert(1 <= a[i][j] && a[i][j] <= (1 << 20));
      }
      if (i > j) {
        assert(a[i][j] == a[j][i]);
      }
    }
  }
  string tmp;
  assert(!(cin >> tmp));
  ll sum_dist = score(a);
  cerr << "sum_dist: " << sum_dist << 'n';
  ll best = 0;
  vector<Edge> ans;
  while (clock() < 1.8 * CLOCKS_PER_SEC) {
    auto path = find_path();
    auto es = solve(path);
    build_graph(es);
    ll cur = score(b);
    if (cur > best) {
      optimize_path(path, 1000);
      auto es = solve(path);
      build_graph(es);
      ll cur2 = score(b);
      assert(cur2 >= cur);
      cur = cur2;

      cerr << "new best: " << best << " -> " << cur << 'n';
      ans = es;
      best = cur;
    }
  }
  print(ans);
  cerr << "score: " << best * 1. / sum_dist << 'n';
}