Segment tree beats (range minimize/maximize query, rmq, and rsq)

概要

以下のクエリを処理する

  • \(a_l, a_{l+1}, ..., a_{r-1}\) の各\(a_i\)について \(\min(a_i, x)\) に更新

  • \(a_l, a_{l+1}, ..., a_{r-1}\) の最大値を求める

  • \(a_l, a_{l+1}, ..., a_{r-1}\) の総和を求める

計算量

  • 区間chminクエリ: \(N\) 個の要素に対し \(Q\) 回のクエリで \(O((N+Q) \log N)\) (ならし計算量)

  • 最大値・総和計算: 各クエリ \(O(\log N)\)

実装

#include<algorithm>
using namespace std;
using ll = long long;

#define N 10003

class SegmentTree {
  const ll inf = 1e18;
  int n, n0;
  ll max_v[4*N], smax_v[4*N];
  ll sum[4*N], max_c[4*N];

  void update_node_max(int k, ll x) {
    sum[k] += (x - max_v[k]) * max_c[k];
    max_v[k] = x;
  }

  void push(int k) {
    if(max_v[k] < max_v[2*k+1]) {
      update_node_max(2*k+1, max_v[k]);
    }
    if(max_v[k] < max_v[2*k+2]) {
      update_node_max(2*k+2, max_v[k]);
    }
  }

  void update(int k) {
    sum[k] = sum[2*k+1] + sum[2*k+2];

    if(max_v[2*k+1] < max_v[2*k+2]) {
      max_v[k] = max_v[2*k+2];
      max_c[k] = max_c[2*k+2];
      smax_v[k] = max(max_v[2*k+1], smax_v[2*k+2]);
    } else if(max_v[2*k+1] > max_v[2*k+2]) {
      max_v[k] = max_v[2*k+1];
      max_c[k] = max_c[2*k+1];
      smax_v[k] = max(smax_v[2*k+1], max_v[2*k+2]);
    } else {
      max_v[k] = max_v[2*k+1];
      max_c[k] = max_c[2*k+1] + max_c[2*k+2];
      smax_v[k] = max(smax_v[2*k+1], smax_v[2*k+2]);
    }
  }

  void _update_min(ll x, int a, int b, int k, int l, int r) {
    if(b <= l || r <= a || max_v[k] <= x) {
      return;
    }
    if(a <= l && r <= b && smax_v[k] < x) {
      update_node_max(k, x);
      return;
    }

    push(k);
    _update_min(x, a, b, 2*k+1, l, (l+r)/2);
    _update_min(x, a, b, 2*k+2, (l+r)/2, r);
    update(k);
  }

  ll _query_max(int a, int b, int k, int l, int r) {
    if(b <= l || r <= a) {
      return 0;
    }
    if(a <= l && r <= b) {
      return max_v[k];
    }
    push(k);
    ll lv = _query_max(a, b, 2*k+1, l, (l+r)/2);
    ll rv = _query_max(a, b, 2*k+2, (l+r)/2, r);
    return max(lv, rv);
  }

  ll _query_sum(int a, int b, int k, int l, int r) {
    if(b <= l || r <= a) {
      return 0;
    }
    if(a <= l && r <= b) {
      return sum[k];
    }
    push(k);
    ll lv = _query_sum(a, b, 2*k+1, l, (l+r)/2);
    ll rv = _query_sum(a, b, 2*k+2, (l+r)/2, r);
    return lv + rv;
  }

public:
  SegmentTree(int n) {
    SegmentTree(n, nullptr);
  }

  SegmentTree(int n, ll *a) : n(n) {
    n0 = 1;
    while(n0 < n) n0 <<= 1;

    for(int i=0; i<n; ++i) {
      max_v[n0-1+i] = sum[n0-1+i] = (a != nullptr ? a[i] : 0);
      smax_v[n0-1+i] = -inf;
      max_c[n0-1+i] = 1;
    }

    for(int i=n; i<n0; ++i) {
      max_v[n0-1+i] = smax_v[n0-1+i] = -inf;
      sum[n0-1+i] = max_c[n0-1+i] = 0;
    }
    for(int i=n0-2; i>=0; i--) update(i);
  }

  // range chmin query
  void update_min(int a, int b, ll x) {
    return _update_min(x, a, b, 0, 0, n0);
  }

  // range maximum query
  ll query_max(int a, int b) {
    return _query_max(a, b, 0, 0, n0);
  }

  // range sum query
  ll query_sum(int a, int b) {
    return _query_sum(a, b, 0, 0, n0);
  }
};

Verified

参考