Comet OJ - Contest #5 简要题解

好久没更博了，还是象征性地更一次。

依然延续了简要题解的风格。

题目链接

https://cometoj.com/contest/46

题解

A. 迫真字符串

记 \(s_i\) 表示数字 \(i\) 出现的次数，答案为 \(\min\{\lfloor\frac{s_1}{3}\rfloor, \lfloor\frac{s_4}{2}\rfloor, s_5\}\)。

#include<bits/stdc++.h>

using namespace std;

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  string s;
  cin >> s;
  int a = 0, b = 0, c = 0, n = s.length();
  for (int i = 0; i < n; ++i) {
    if (s[i] == '1') {
      ++a;
    } else if (s[i] == '4') {
      ++b;
    } else if (s[i] == '5') {
      ++c;
    }
  }
  cout << min(min(a / 3, b / 2), c) << '\n';
  return 0;
}

B. 迫真数论

暴力。

#include<bits/stdc++.h>

using namespace std;

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  int tt;
  long long n;
  cin >> tt;
  while (tt--) {
    cin >> n;
    int answer = 0;
    auto f = [&] (int x) {
      int result = 0;
      while (x) {
        result += x % 10;
        x /= 10;
      }
      return result;
    };
    for (int i = 1; i <= 200; ++i) {
      if (n % i == 0 && f(i) == (i >> 1)) {
        ++answer;
      }
    }
    cout << answer << '\n';
  }
  return 0;
}

C. 迫真小游戏

贪心。

#include<bits/stdc++.h>

using namespace std;

const int N = 567890;

int n, a[N], depth[N];
vector<int> adj[N], nodes[N];
bool visit[N];

void dfs(int x, int f) {
  nodes[depth[x] = depth[f] + 1].push_back(x);
  for (auto y : adj[x]) {
    if (y != f) {
      dfs(y, x);
    }
  }
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  cin >> n;
  for (int i = 1, x, y; i < n; ++i) {
    cin >> x >> y;
    adj[x].push_back(y);
    adj[y].push_back(x);
  }
  dfs(1, 0);
  for (int i = 1; i <= n; ++i) {
    cin >> a[i];
  }
  multiset<int> s;
  priority_queue<int, vector<int>, greater<int>> q;
  for (int i = 2; i <= n; ++i) {
    s.insert(a[i]);
  }
  cout << 1 << " \n"[n == 1];
  int j = 1, tt = 1;
  while (tt != n) {
    while (s.size() && j < *s.begin()) {
      ++j;
      for (auto x : nodes[j]) {
        q.push(x);
      }
    }
    int x = q.top();
    s.erase(s.find(a[x]));
    q.pop();
    cout << x << " \n"[++tt == n];
  }
  return 0;
}

D. 迫真图论

假设 \(n, m\) 同阶。将点按度数大小是否超过 \(\sqrt n\) 分类，记为大点和小点，那么小点的度数不超过 \(\sqrt n\)，大点的总数量不超过 \(O(\sqrt n)\)，这样就可以暴力做了。对于每个大点，用一棵 trie 维护与其相邻的小点间的边的信息；对于所有小点与小点间的边、大点与大点间的边的信息，可以用一棵全局的树状数组维护。大点的权值对 trie 的影响可以用一个tag标记记录，剩下的修改与查询操作就比较显然了。

在代码实现中，点按度数大小分类的阈值可以设得比 \(\sqrt n\) 稍大一些。

#include<bits/stdc++.h>

using namespace std;

const int N = 1 << 18, sq = 2000, mod = 998244353;

int n, m, q, tt, degree[N], a[N], x[N], y[N], z[N], ch[N * 100][2], root[N], tag[N], now_tag;
vector<pair<int, int>> adj[N], sadj[N];
long long sum[N * 100];
bool type[N];

class bit {
  long long a[N];

 public:
  bit() {
    memset(a, 0, sizeof a);
  }

  void add(int x, int y) {
    if (!x) {
      a[0] += y;
      return;
    }
    while (x < N) {
      a[x] += y;
      x += x & -x;
    }
  }

  long long sum(int x) {
    long long result = a[0];
    while (x) {
      result += a[x];
      x -= x & -x;
    }
    return result;
  }
} tree;

void insert(int& x, int d, int y, int z, int now_tag) {
  if (!x) {
    x = ++tt;
  }
  sum[x] += z;
  if (!~d) {
    return;
  }
  insert(ch[x][(y >> d & 1) ^ (now_tag >> d & 1)], d - 1, y, z, now_tag);
}

long long query(int x, int d, int y, int now_tag) {
  if (!x) {
    return 0;
  }
  if (!~d) {
    return sum[x];
  }
  if (y >> d & 1) {
    if (now_tag >> d & 1) {
      return sum[ch[x][1]] + query(ch[x][0], d - 1, y, now_tag);
    } else {
      return sum[ch[x][0]] + query(ch[x][1], d - 1, y, now_tag);
    }
  } else {
    if (now_tag >> d & 1) {
      return query(ch[x][1], d - 1, y, now_tag);
    } else {
      return query(ch[x][0], d - 1, y, now_tag);
    }
  }
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  cin >> n >> m >> q;
  for (int i = 1; i <= n; ++i) {
    cin >> a[i];
  }
  for (int i = 1; i <= m; ++i) {
    cin >> x[i] >> y[i] >> z[i];
    adj[x[i]].emplace_back(y[i], i);
    adj[y[i]].emplace_back(x[i], i);
    ++degree[x[i]];
    ++degree[y[i]];
  }
  for (int i = 1; i <= n; ++i) {
    if (degree[i] <= sq) {
      type[i] = false;
    } else {
      type[i] = true;
    }
  }
  vector<int> snodes;
  for (int i = 1; i <= n; ++i) {
    if (type[i]) {
      tag[i] = a[i];
      snodes.push_back(i);
      for (auto p : adj[i]) {
        if (type[p.first]) {
          sadj[i].push_back(p);
        }
      }
    }
  }
  for (int i = 1; i <= n; ++i) {
    for (auto p : adj[i]) {
      int j = p.first;
      if (type[i] == type[j] && i < j) {
        tree.add(a[i] ^ a[j], z[p.second]);
      } else if (type[i] && !type[j]) {
        insert(root[i], 16, a[i] ^ a[j], z[p.second], tag[i]);
      }
    }
  }
  for (int i = 1, op, u, v; i <= q; ++i) {
    cin >> op >> u >> v;
    if (op == 1) {
      if (!type[u]) {
        for (auto p : adj[u]) {
          if (!type[p.first]) {
            tree.add(a[u] ^ a[p.first], -z[p.second]);
            tree.add(v ^ a[p.first], z[p.second]);
          } else {
            insert(root[p.first], 16, a[u] ^ a[p.first], -z[p.second], tag[p.first]);
            insert(root[p.first], 16, v ^ a[p.first], z[p.second], tag[p.first]);
          }
        }
      } else {
        tag[u] ^= a[u] ^ v;
        for (auto p : sadj[u]) {
          tree.add(a[u] ^ a[p.first], -z[p.second]);
          tree.add(v ^ a[p.first], z[p.second]);
        }
      }
      a[u] = v;
    } else if (op == 2) {
      int s = x[u], t = y[u];
      if (type[s] == type[t]) {
        tree.add(a[s] ^ a[t], -z[u]);
        tree.add(a[s] ^ a[t], v);
      } else {
        if (type[t]) {
          swap(s, t);
        }
        insert(root[s], 16, a[s] ^ a[t], -z[u], tag[s]);
        insert(root[s], 16, a[s] ^ a[t], v, tag[s]);
      }
      z[u] = v;
    } else {
      --u;
      long long answer = tree.sum(v) - (~u ? tree.sum(u) : 0);
      for (auto x : snodes) {
        answer += query(root[x], 16, v, tag[x]) - (~u ? query(root[x], 16, u, tag[x]) : 0);
      }
      cout << (answer % mod) << '\n';
    }
  }
  return 0;
}

E. 迫真大游戏

先只考虑 \(1\) 号分身。定义 \(f_i\) 表示当前还剩 \(i\) 个分身（必然包含 \(1\) 号分身），\(1\) 号分身最后消失的概率。那么有 \(f_i = \sum_\limits{j = 1}^i \binom{i - 1}{j - 1} (1 - p)^{j}p^{i- j}f_j\)，\(f_1 = 1\)，可以用分治 NTT 在 \(O(n \log^2 n)\) 的时间内求出所有 \(f_i\)，那么 \(1\) 号分身的答案即为 \(f_n\)。

现在考虑求其他分身的答案。为了让 \(x\) 号分身的答案也能用 \(f_i\) 求出，我们可以直接枚举前 \(x - 1\) 个人的消失情况，然后乘以对应的方案数和概率，发现又是一个卷积的形式，于是再做一次 NTT 即可。

#include<bits/stdc++.h>

using namespace std;

const int N = 567890, mod = 998244353, root = 3;

void add(int& x, int y) {
  x += y;
  if (x >= mod) {
    x -= mod;
  }
}

int mul(int x, int y) {
  return (long long) x * y % mod;
}

int qpow(int x, int y) {
  int result = 1;
  for (; y; y >>= 1, x = mul(x, x)) {
    if (y & 1) {
      result = mul(result, x);
    }
  }
  return result;
}

int n, a, b, p, rev[N], f[N], fac[N], ifac[N], inv[N];

void dft(vector<int>& buffer, bool inv = false) {
  int n = buffer.size();
  for (int i = 0; i < n; ++i) {
    if (i < rev[i]) {
      swap(buffer[i], buffer[rev[i]]);
    }
  }
  for (int i = 1; i < n; i <<= 1) {
    int x = qpow(root, inv ? mod - 1 - (mod - 1) / (i << 1) : (mod - 1) / (i << 1));
    for (int j = 0; j < n; j += i << 1) {
      int y = 1;
      for (int k = 0; k < i; ++k, y = mul(y, x)) {
        int p = buffer[j + k], q = mul(y, buffer[i + j + k]);
        buffer[j + k] = (p + q) % mod;
        buffer[i + j + k] = (p - q + mod) % mod;
      }
    }
  }
  if (inv) {
    int x = qpow(n, mod - 2);
    for (int i = 0; i < n; ++i) {
      buffer[i] = mul(buffer[i], x);
    }
  }
}

vector<int> pmul(vector<int> x, vector<int> y) {
  int n = x.size() + y.size() - 1, len = 0;
  for (; (1 << len) < n; ++len);
  for (int i = 0; i < (1 << len); ++i) {
    rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << len - 1);
  }
  x.resize(1 << len);
  y.resize(1 << len);
  dft(x);
  dft(y);
  for (int i = 0; i < (1 << len); ++i) {
    x[i] = mul(x[i], y[i]);
  }
  dft(x, true);
  x.resize(n);
  return x;
}

void solve(int l, int r) {
  if (l == r) {
    if (l == 1) {
      f[1] = 1;
    } else {
      f[l] = mul(f[l], fac[l - 1]);
      f[l] = mul(f[l], qpow(1 - qpow(1 - p + mod, l) + mod, mod - 2));
    }
  } else {
    int mid = l + r >> 1;
    solve(l, mid);
    vector<int> foo(mid - l + 1), bar(r - l);
    for (int i = l; i <= mid; ++i) {
      foo[i - l] = mul(f[i], mul(qpow(1 - p + mod, i), ifac[i - 1]));
    }
    for (int i = 1; i <= r - l; ++i) {
      bar[i - 1] = mul(qpow(p, i), ifac[i]);
    }
    foo = pmul(foo, bar);
    for (int i = mid + 1; i <= r; ++i) {
      add(f[i], foo[i - l - 1]);
    }
    solve(mid + 1, r);
  }
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  cin >> n >> a >> b;
  p = mul(a, qpow(b, mod - 2));
  fac[0] = ifac[0] = inv[1] = fac[1] = ifac[1] = 1;
  for (int i = 2; i <= n; ++i) {
    inv[i] = mul(mod - mod / i, inv[mod % i]);
    fac[i] = mul(fac[i - 1], i);
    ifac[i] = mul(ifac[i - 1], inv[i]);
  }
  solve(1, n);
  vector<int> foo(n), bar(n);
  for (int i = 0; i < n; ++i) {
    foo[i] = mul(f[n - i], mul(qpow(p, i), ifac[i]));
    bar[i] = mul(qpow(1 - p + mod, i), ifac[i]);
  }
  foo = pmul(foo, bar);
  for (int i = 0; i < n; ++i) {
    cout << mul(foo[i], fac[i]) << '\n';
  }
  return 0;
}

F. 迫真树

二分答案 \(k\) 后通过做 dp 来判断是否存在合法方案。假设整棵树以 \(1\) 为根，设 \(f_{i, j}\) 表示 \(i\) 号点往子树内延伸的最长链长度为 \(j\)，且子树合法（即子树内最长链不超过 \(k\)）的最小代价（\(f_{i, 0}\) 则表示不选 \(i\) 点的最小代价），那么转移比较显然。注意到 dp 状态的第二维与点往下延伸的最长链长度有关，那么考虑用长链剖分，对 dp 转移分情况讨论后发现需要用到一段区间内的最优 dp 值，用线段树维护即可。

#include<bits/stdc++.h>

using namespace std;

const int N = 123456;
const long long llinf = 1e18;

int n, m, tt, a[N], heavy[N], dfn[N], maxd[N];
vector<int> adj[N];
long long dp0[N];

class segment_t {
  long long a[N << 2], tag[N << 2];

 public:
  segment_t() {
    fill(a, a + (N << 2), llinf);
    memset(tag, 0, sizeof tag);
  }

  void mark(int x, long long y) {
    tag[x] += y;
    a[x] += y;
  }

  void push(int x) {
    if (tag[x]) {
      mark(x << 1, tag[x]);
      mark(x << 1 | 1, tag[x]);
      tag[x] = 0;
    }
  }

  void modify(int l, int r, int x, int ql, int qr, long long y) {
    if (ql <= l && r <= qr) {
      mark(x, y);
    } else {
      int mid = l + r >> 1;
      push(x);
      if (ql <= mid) {
        modify(l, mid, x << 1, ql, qr, y);
      }
      if (qr > mid) {
        modify(mid + 1, r, x << 1 | 1, ql, qr, y);
      }
      a[x] = min(a[x << 1], a[x << 1 | 1]);
    }
  }

  void modify(int l, int r, int x, int p, long long y) {
    if (l == r) {
      a[x] = min(a[x], y);
    } else {
      int mid = l + r >> 1;
      push(x);
      if (p <= mid) {
        modify(l, mid, x << 1, p, y);
      } else {
        modify(mid + 1, r, x << 1 | 1, p, y);
      }
      a[x] = min(a[x << 1], a[x << 1 | 1]);
    }
  }

  long long query(int l, int r, int x, int ql, int qr) {
    if (ql <= l && r <= qr) {
      return a[x];
    } else {
      int mid = l + r >> 1;
      long long result = llinf;
      push(x);
      if (ql <= mid) {
        result = min(result, query(l, mid, x << 1, ql, qr));
      }
      if (qr > mid) {
        result = min(result, query(mid + 1, r, x << 1 | 1, ql, qr));
      }
      return result;
    }
  }
};

void dfs(int x, int f) {
  maxd[x] = -1;
  for (auto y : adj[x]) {
    if (y != f) {
      dfs(y, x);
      if (maxd[y] > maxd[x]) {
        heavy[x] = y;
        maxd[x] = maxd[y];
      }
    }
  }
  ++maxd[x];
}

bool check(int k) {
  segment_t tree;
  auto dp = [&] (int x, int y) {
    return !y ? dp0[x] : tree.query(1, n, 1, dfn[x] + y - 1, dfn[x] + y - 1);
  };
  function<void (int, int)> dfs = [&] (int x, int f) {
    dp0[x] = a[x];
    dfn[x] = ++tt;
    if (maxd[x]) {
      dfs(heavy[x], x);
      dp0[x] += min(dp(heavy[x], 0), tree.query(1, n, 1, dfn[heavy[x]], dfn[heavy[x]] + min(maxd[heavy[x]], k - 1)));
      tree.modify(1, n, 1, dfn[x], dp(heavy[x], 0));
    } else {
      tree.modify(1, n, 1, dfn[x], 0);
    }
    for (auto y : adj[x]) {
      if (y != f && y != heavy[x]) {
        dfs(y, x);
        dp0[x] += min(dp(y, 0), tree.query(1, n, 1, dfn[y], dfn[y] + min(maxd[y], k - 1)));
        vector<long long> foo;
        vector<pair<int, long long>> bar;
        for (int j = 1; j <= min(maxd[y] + 2, k); ++j) {
          int l = 1, r = min(j, k - j + 1);
          if (l > r) {
            break;
          }
          long long t = dp(y, j - 1);
          foo.push_back(t + tree.query(1, n, 1, dfn[x] + l - 1, dfn[x] + r - 1));
          if (!bar.size() || (bar.size() && t < bar.back().second)) {
            bar.emplace_back(j - 1, t);
          }
        }
        long long last = 0;
        for (auto p : bar) {
          int l = p.first + 1, r = min(maxd[x] + 1, k - p.first);
          if (l <= r) {
            tree.modify(1, n, 1, dfn[x] + l - 1, dfn[x] + r - 1, p.second - last);
            last = p.second;
          }
        }
        for (int i = 0; i < foo.size(); ++i) {
          tree.modify(1, n, 1, dfn[x] + i, foo[i]);
        }
      }
    }
  };
  tt = 0;
  dfs(1, 0);
  return min(dp(1, 0), tree.query(1, n, 1, dfn[1], dfn[1] + min(maxd[1], k - 1))) <= m;
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout.tie(0);
  cin >> n >> m;
  long long all = 0;
  for (int i = 1; i <= n; ++i) {
    cin >> a[i];
    all += a[i];
  }
  if (all <= m) {
    cout << 0 << '\n';
    exit(0);
  }
  for (int i = 1, x, y; i < n; ++i) {
    cin >> x >> y;
    adj[x].push_back(y);
    adj[y].push_back(x);
  }
  dfs(1, 0);
  int l = 1, r = n;
  while (l != r) {
    int mid = l + r >> 1;
    if (check(mid)) {
      r = mid;
    } else {
      l = mid + 1;
    }
  }
  cout << l << '\n';
  return 0;
}