ds/segtree/beats_kinetic.hpp

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• Last update: 2024-08-26 04:14:45+09:00
• Include: #include "ds/segtree/beats_kinetic.hpp"
• Link: https://atcoder.jp/contests/jsc2024-final/tasks/jsc2024_final_d
• Link: https://codeforces.com/blog/entry/82094#comment-688448

Depends on

• ds/segtree/segtree_beats.hpp

Verified with

• test/1_mytest/kinetic.test.cpp

Code

#include "ds/segtree/segtree_beats.hpp"

// (x[i],y[i]) からなる列. a>=0 であるときに y[i] :- y[i] + ax[i] + b という作用ができる
// x には単調性は要らない. x,sum(a):T1, y,sum(b):T2, T1*T1<=T2.
// https://codeforces.com/blog/entry/82094#comment-688448
// https://atcoder.jp/contests/jsc2024-final/tasks/jsc2024_final_d
template <typename T1, typename T2>
struct Beats_Kinetic_Max {
  struct Mono_X {
    struct X {
      T1 x;
      T2 y;
      T1 nxt_change;
      bool fail;
    };
    using value_type = X;
    static X op(X L, X R) {
      X M;
      if (L.y < R.y) { swap(L, R); }
      M.x = L.x, M.y = L.y;
      M.nxt_change = min(L.nxt_change, R.nxt_change);
      if (L.x < R.x) {
        T2 t = floor<T2>(L.y - R.y, R.x - L.x);
        chmin(M.nxt_change, t + 1);
      }
      M.fail = 0;
      return M;
    }
    static constexpr X unit() { return {0, -infty<T2>, infty<T1>, 0}; }
    bool commute = true;
  };
  struct Mono_A {
    using X = pair<T1, T2>;
    using value_type = X;
    static constexpr X op(const X& x, const X& y) { return {x.fi + y.fi, x.se + y.se}; }
    static constexpr X unit() { return {0, 0}; }
    bool commute = true;
  };
  struct Beats {
    using Monoid_X = Mono_X;
    using Monoid_A = Mono_A;
    using X = typename Monoid_X::value_type;
    using A = typename Monoid_A::value_type;
    static X act(X& M, const A& a, int cnt) {
      assert(!M.fail && a.fi >= 0);
      if (M.nxt_change <= a.fi) {
        M.fail = 1;
        return M;
      }
      M.y += T2(a.fi) * M.x + a.se;
      M.nxt_change -= a.fi;
      return M;
    }
  };
  using S = typename Mono_X::X;
  SegTree_Beats<Beats> seg;
  Beats_Kinetic_Max(vc<T1>& X, vc<T2>& Y) {
    seg.build(len(X), [&](int i) -> S { return {X[i], Y[i], infty<T1>, 0}; });
  }
  template <typename F>
  Beats_Kinetic_Max(int n, F f) {
    seg.build(n, [&](int i) -> S {
      auto [x, y] = f(i);
      return {x, y, infty<T1>, 0};
    });
  }

  void set(int i, T1 x, T2 y) { seg.set(i, {x, y, infty<T1>, 0}); }

  // (x,y)
  pair<T1, T2> prod(int l, int r) {
    auto e = seg.prod(l, r);
    return {e.x, e.y};
  }
  pair<T1, T2> prod_all() {
    auto e = seg.prod_all();
    return {e.x, e.y};
  }
  void apply(int l, int r, T1 a, T2 b) { seg.apply(l, r, {a, b}); }
};

// (x[i],y[i]) からなる列. a>=0 であるときに y[i] :- y[i] + ax[i] + b という作用ができる
// x には単調性は要らない. x,sum(a):T1, y,sum(b):T2, T1*T1<=T2.
// https://codeforces.com/blog/entry/82094#comment-688448
// https://atcoder.jp/contests/jsc2024-final/tasks/jsc2024_final_d
template <typename T1, typename T2>
struct Beats_Kinetic_Min {
  struct Mono_X {
    struct X {
      T1 x;
      T2 y;
      T1 nxt_change;
      bool fail;
    };
    using value_type = X;
    static X op(X L, X R) {
      X M;
      if (L.y > R.y) { swap(L, R); }
      M.x = L.x, M.y = L.y;
      M.nxt_change = min(L.nxt_change, R.nxt_change);
      if (L.x > R.x) {
        T2 t = floor<T2>(R.y - L.y, L.x - R.x);
        chmin(M.nxt_change, t + 1);
      }
      M.fail = 0;
      return M;
    }
    static constexpr X unit() { return {0, infty<T2>, infty<T1>, 0}; }
    bool commute = true;
  };
  struct Mono_A {
    using X = pair<T1, T2>;
    using value_type = X;
    static constexpr X op(const X& x, const X& y) { return {x.fi + y.fi, x.se + y.se}; }
    static constexpr X unit() { return {0, 0}; }
    bool commute = true;
  };
  struct Beats {
    using Monoid_X = Mono_X;
    using Monoid_A = Mono_A;
    using X = typename Monoid_X::value_type;
    using A = typename Monoid_A::value_type;
    static X act(X& M, const A& a, int cnt) {
      assert(!M.fail && a.fi >= 0);
      if (M.nxt_change <= a.fi) {
        M.fail = 1;
        return M;
      }
      M.y += T2(a.fi) * M.x + a.se;
      M.nxt_change -= a.fi;
      return M;
    }
  };
  using S = typename Mono_X::X;
  SegTree_Beats<Beats> seg;
  Beats_Kinetic_Min(vc<T1>& X, vc<T2>& Y) {
    seg.build(len(X), [&](int i) -> S { return {X[i], Y[i], infty<T1>, 0}; });
  }
  template <typename F>
  Beats_Kinetic_Min(int n, F f) {
    seg.build(n, [&](int i) -> S {
      auto [x, y] = f(i);
      return {x, y, infty<T1>, 0};
    });
  }

  void set(int i, T1 x, T2 y) { seg.set(i, {x, y, infty<T1>, 0}); }

  // (x,y)
  pair<T1, T2> prod(int l, int r) {
    auto e = seg.prod(l, r);
    return {e.x, e.y};
  }
  pair<T1, T2> prod_all() {
    auto e = seg.prod_all();
    return {e.x, e.y};
  }
  void apply(int l, int r, T1 a, T2 b) { seg.apply(l, r, {a, b}); }
};

#line 2 "ds/segtree/segtree_beats.hpp"

template <typename ActedMonoid>
struct SegTree_Beats {
  using AM = ActedMonoid;
  using MX = typename AM::Monoid_X;
  using MA = typename AM::Monoid_A;
  using X = typename MX::value_type;
  using A = typename MA::value_type;
  int n, log, size;
  vc<X> dat;
  vc<A> laz;

  SegTree_Beats() {}
  SegTree_Beats(int n) { build(n); }
  template <typename F>
  SegTree_Beats(int n, F f) {
    build(n, f);
  }
  SegTree_Beats(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::unit(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    dat.assign(size << 1, MX::unit());
    laz.assign(size, MA::unit());
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  void update(int k) { dat[k] = MX::op(dat[2 * k], dat[2 * k + 1]); }
  void set(int p, X x) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    dat[p] = x;
    for (int i = 1; i <= log; i++) update(p >> i);
  }

  X get(int p) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    return dat[p];
  }

  /*
  void all_apply(int k, A a) {
    dat[k] = ActedMonoid::act(dat[k], a);
    if (k < size) {
      laz[k] = MA::op(laz[k], a);
      if (dat[k].fail) push(k), update(k);
    }
  }
  */

  vc<X> get_all() {
    FOR(k, 1, size) { push(k); }
    return {dat.begin() + size, dat.begin() + size + n};
  }

  X prod(int l, int r) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return MX::unit();
    l += size, r += size;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) push(l >> i);
      if (((r >> i) << i) != r) push((r - 1) >> i);
    }
    X xl = MX::unit(), xr = MX::unit();
    while (l < r) {
      if (l & 1) xl = MX::op(xl, dat[l++]);
      if (r & 1) xr = MX::op(dat[--r], xr);
      l >>= 1, r >>= 1;
    }
    return MX::op(xl, xr);
  }

  X prod_all() { return dat[1]; }

  void apply(int l, int r, A a) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return;
    l += size, r += size;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) push(l >> i);
      if (((r >> i) << i) != r) push((r - 1) >> i);
    }
    int l2 = l, r2 = r;
    while (l < r) {
      if (l & 1) apply_at(l++, a);
      if (r & 1) apply_at(--r, a);
      l >>= 1, r >>= 1;
    }
    l = l2, r = r2;
    for (int i = 1; i <= log; i++) {
      if (((l >> i) << i) != l) update(l >> i);
      if (((r >> i) << i) != r) update((r - 1) >> i);
    }
  }

private:
  void apply_at(int k, A a) {
    int sz = 1 << (log - topbit(k));
    dat[k] = AM::act(dat[k], a, sz);
    if (k < size) {
      laz[k] = MA::op(laz[k], a);
      if (dat[k].fail) push(k), update(k);
    }
  }

  void push(int k) {
    if (laz[k] == MA::unit()) return;
    apply_at(2 * k, laz[k]), apply_at(2 * k + 1, laz[k]);
    laz[k] = MA::unit();
  }
};
#line 2 "ds/segtree/beats_kinetic.hpp"

// (x[i],y[i]) からなる列. a>=0 であるときに y[i] :- y[i] + ax[i] + b という作用ができる
// x には単調性は要らない. x,sum(a):T1, y,sum(b):T2, T1*T1<=T2.
// https://codeforces.com/blog/entry/82094#comment-688448
// https://atcoder.jp/contests/jsc2024-final/tasks/jsc2024_final_d
template <typename T1, typename T2>
struct Beats_Kinetic_Max {
  struct Mono_X {
    struct X {
      T1 x;
      T2 y;
      T1 nxt_change;
      bool fail;
    };
    using value_type = X;
    static X op(X L, X R) {
      X M;
      if (L.y < R.y) { swap(L, R); }
      M.x = L.x, M.y = L.y;
      M.nxt_change = min(L.nxt_change, R.nxt_change);
      if (L.x < R.x) {
        T2 t = floor<T2>(L.y - R.y, R.x - L.x);
        chmin(M.nxt_change, t + 1);
      }
      M.fail = 0;
      return M;
    }
    static constexpr X unit() { return {0, -infty<T2>, infty<T1>, 0}; }
    bool commute = true;
  };
  struct Mono_A {
    using X = pair<T1, T2>;
    using value_type = X;
    static constexpr X op(const X& x, const X& y) { return {x.fi + y.fi, x.se + y.se}; }
    static constexpr X unit() { return {0, 0}; }
    bool commute = true;
  };
  struct Beats {
    using Monoid_X = Mono_X;
    using Monoid_A = Mono_A;
    using X = typename Monoid_X::value_type;
    using A = typename Monoid_A::value_type;
    static X act(X& M, const A& a, int cnt) {
      assert(!M.fail && a.fi >= 0);
      if (M.nxt_change <= a.fi) {
        M.fail = 1;
        return M;
      }
      M.y += T2(a.fi) * M.x + a.se;
      M.nxt_change -= a.fi;
      return M;
    }
  };
  using S = typename Mono_X::X;
  SegTree_Beats<Beats> seg;
  Beats_Kinetic_Max(vc<T1>& X, vc<T2>& Y) {
    seg.build(len(X), [&](int i) -> S { return {X[i], Y[i], infty<T1>, 0}; });
  }
  template <typename F>
  Beats_Kinetic_Max(int n, F f) {
    seg.build(n, [&](int i) -> S {
      auto [x, y] = f(i);
      return {x, y, infty<T1>, 0};
    });
  }

  void set(int i, T1 x, T2 y) { seg.set(i, {x, y, infty<T1>, 0}); }

  // (x,y)
  pair<T1, T2> prod(int l, int r) {
    auto e = seg.prod(l, r);
    return {e.x, e.y};
  }
  pair<T1, T2> prod_all() {
    auto e = seg.prod_all();
    return {e.x, e.y};
  }
  void apply(int l, int r, T1 a, T2 b) { seg.apply(l, r, {a, b}); }
};

// (x[i],y[i]) からなる列. a>=0 であるときに y[i] :- y[i] + ax[i] + b という作用ができる
// x には単調性は要らない. x,sum(a):T1, y,sum(b):T2, T1*T1<=T2.
// https://codeforces.com/blog/entry/82094#comment-688448
// https://atcoder.jp/contests/jsc2024-final/tasks/jsc2024_final_d
template <typename T1, typename T2>
struct Beats_Kinetic_Min {
  struct Mono_X {
    struct X {
      T1 x;
      T2 y;
      T1 nxt_change;
      bool fail;
    };
    using value_type = X;
    static X op(X L, X R) {
      X M;
      if (L.y > R.y) { swap(L, R); }
      M.x = L.x, M.y = L.y;
      M.nxt_change = min(L.nxt_change, R.nxt_change);
      if (L.x > R.x) {
        T2 t = floor<T2>(R.y - L.y, L.x - R.x);
        chmin(M.nxt_change, t + 1);
      }
      M.fail = 0;
      return M;
    }
    static constexpr X unit() { return {0, infty<T2>, infty<T1>, 0}; }
    bool commute = true;
  };
  struct Mono_A {
    using X = pair<T1, T2>;
    using value_type = X;
    static constexpr X op(const X& x, const X& y) { return {x.fi + y.fi, x.se + y.se}; }
    static constexpr X unit() { return {0, 0}; }
    bool commute = true;
  };
  struct Beats {
    using Monoid_X = Mono_X;
    using Monoid_A = Mono_A;
    using X = typename Monoid_X::value_type;
    using A = typename Monoid_A::value_type;
    static X act(X& M, const A& a, int cnt) {
      assert(!M.fail && a.fi >= 0);
      if (M.nxt_change <= a.fi) {
        M.fail = 1;
        return M;
      }
      M.y += T2(a.fi) * M.x + a.se;
      M.nxt_change -= a.fi;
      return M;
    }
  };
  using S = typename Mono_X::X;
  SegTree_Beats<Beats> seg;
  Beats_Kinetic_Min(vc<T1>& X, vc<T2>& Y) {
    seg.build(len(X), [&](int i) -> S { return {X[i], Y[i], infty<T1>, 0}; });
  }
  template <typename F>
  Beats_Kinetic_Min(int n, F f) {
    seg.build(n, [&](int i) -> S {
      auto [x, y] = f(i);
      return {x, y, infty<T1>, 0};
    });
  }

  void set(int i, T1 x, T2 y) { seg.set(i, {x, y, infty<T1>, 0}); }

  // (x,y)
  pair<T1, T2> prod(int l, int r) {
    auto e = seg.prod(l, r);
    return {e.x, e.y};
  }
  pair<T1, T2> prod_all() {
    auto e = seg.prod_all();
    return {e.x, e.y};
  }
  void apply(int l, int r, T1 a, T2 b) { seg.apply(l, r, {a, b}); }
};

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