geo/closest_pair.hpp

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• Last update: 2024-09-14 09:20:23+09:00
• Include: #include "geo/closest_pair.hpp"

Depends on

• ds/hashmap.hpp
• geo/base.hpp
• random/base.hpp
• random/hash_pair.hpp
• random/shuffle.hpp

Verified with

• test/2_library_checker/geometry/closest_pair.test.cpp
• test/2_library_checker/geometry/closest_pair_dc.test.cpp
• test/4_aoj/CGL_5_A.test.cpp

Code

#include "geo/base.hpp"
#include "random/base.hpp"
#include "random/shuffle.hpp"
#include "ds/hashmap.hpp"
#include "random/hash_pair.hpp"

template <typename T>
pair<int, int> closest_pair(vc<Point<T>> points) {
  int N = len(points);
  assert(N >= 2);
  HashMap<int> MP(N);
  vc<int> I(N);
  iota(all(I), 0);
  shuffle(I);
  points = rearrange(points, I);

  auto calc = [&](int i, int j) -> T {
    return (points[j] - points[i]).dot(points[j] - points[i]);
  };

  T best = calc(0, 1);
  pair<int, int> res = {0, 1};
  T w = sqrtl(best);

  vc<int> nxt(N, -1);

  auto insert = [&](int i) -> void {
    u64 k = hash_pair<ll>({points[i].x / w, points[i].y / w});
    nxt[i] = MP.get(k, -1);
    MP[k] = i;
  };

  auto query = [&](int i) -> bool {
    ll a = points[i].x / w;
    ll b = points[i].y / w;
    bool upd = 0;
    FOR(dx, -1, 2) FOR(dy, -1, 2) {
      u64 k = hash_pair<ll>({a + dx, b + dy});
      int j = MP.get(k, -1);
      while (j != -1) {
        if (chmin(best, calc(i, j))) { upd = 1, res = {i, j}, w = sqrtl(best); }
        j = nxt[j];
      }
    }
    return upd;
  };

  if (best == T(0)) {
    res.fi = I[res.fi], res.se = I[res.se];
    return res;
  }

  insert(0), insert(1);
  FOR(i, 2, N) {
    if (query(i)) {
      if (best == T(0)) break;
      MP.build(N);
      FOR(j, i) insert(j);
    }
    insert(i);
  }
  res.fi = I[res.fi], res.se = I[res.se];
  return res;
}

pair<int, int> closest_pair_dc(vc<Point<ll>> point) {
  int N = len(point);
  assert(N >= 2);

  auto I = argsort(point);
  point = rearrange(point, I);

  ll best = -1;
  pair<int, int> best_pair = {-1, -1};

  auto upd = [&](int i, int j) -> void {
    Point<ll> p = point[i] - point[j];
    ll d = p.dot(p);
    if (best == -1 || best > d) { best = d, best_pair = {I[i], I[j]}; }
  };
  upd(0, 1);

  auto dfs = [&](auto &dfs, int L, int R) -> vc<int> {
    // return: [L,R) を y について sort したもの
    if (R == L + 1) return {L};
    int M = (L + R) / 2;
    vc<int> I0 = dfs(dfs, L, M);
    vc<int> I1 = dfs(dfs, M, R);
    vc<int> I;
    vc<int> near;
    int a = 0, b = 0;
    FOR(R - L) {
      int idx = [&]() -> int {
        if (a == len(I0)) return I1[b++];
        if (b == len(I1)) return I0[a++];
        int i = I0[a], j = I1[b];
        if (point[i].y < point[j].y) {
          ++a;
          return i;
        }
        ++b;
        return j;
      }();
      I.eb(idx);
      ll dx = point[M].x - point[idx].x;
      if (dx * dx > best) { continue; }
      FOR_R(k, len(near)) {
        int j = near[k];
        ll dy = point[idx].y - point[j].y;
        if (best == 0 || dy * dy > best) break;
        upd(idx, j);
      }
      near.eb(idx);
    }
    return I;
  };
  dfs(dfs, 0, N);
  return best_pair;
}

#line 2 "geo/base.hpp"
template <typename T>
struct Point {
  T x, y;

  Point() : x(0), y(0) {}

  template <typename A, typename B>
  Point(A x, B y) : x(x), y(y) {}

  template <typename A, typename B>
  Point(pair<A, B> p) : x(p.fi), y(p.se) {}

  Point operator+=(const Point p) {
    x += p.x, y += p.y;
    return *this;
  }
  Point operator-=(const Point p) {
    x -= p.x, y -= p.y;
    return *this;
  }
  Point operator+(Point p) const { return {x + p.x, y + p.y}; }
  Point operator-(Point p) const { return {x - p.x, y - p.y}; }
  bool operator==(Point p) const { return x == p.x && y == p.y; }
  bool operator!=(Point p) const { return x != p.x || y != p.y; }
  Point operator-() const { return {-x, -y}; }
  Point operator*(T t) const { return {x * t, y * t}; }
  Point operator/(T t) const { return {x / t, y / t}; }

  bool operator<(Point p) const {
    if (x != p.x) return x < p.x;
    return y < p.y;
  }
  T dot(Point other) { return x * other.x + y * other.y; }
  T det(Point other) { return x * other.y - y * other.x; }

  double norm() { return sqrtl(x * x + y * y); }
  double angle() { return atan2(y, x); }

  Point rotate(double theta) {
    static_assert(!is_integral<T>::value);
    double c = cos(theta), s = sin(theta);
    return Point{c * x - s * y, s * x + c * y};
  }
};

#ifdef FASTIO
template <typename T>
void rd(Point<T> &p) {
  fastio::rd(p.x), fastio::rd(p.y);
}
template <typename T>
void wt(Point<T> &p) {
  fastio::wt(p.x);
  fastio::wt(' ');
  fastio::wt(p.y);
}
#endif

// A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1
template <typename T>
int ccw(Point<T> A, Point<T> B, Point<T> C) {
  T x = (B - A).det(C - A);
  if (x > 0) return 1;
  if (x < 0) return -1;
  return 0;
}

template <typename REAL, typename T, typename U>
REAL dist(Point<T> A, Point<U> B) {
  REAL dx = REAL(A.x) - REAL(B.x);
  REAL dy = REAL(A.y) - REAL(B.y);
  return sqrt(dx * dx + dy * dy);
}

// ax+by+c
template <typename T>
struct Line {
  T a, b, c;

  Line(T a, T b, T c) : a(a), b(b), c(c) {}
  Line(Point<T> A, Point<T> B) { a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x; }
  Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  template <typename U>
  U eval(Point<U> P) {
    return a * P.x + b * P.y + c;
  }

  template <typename U>
  T eval(U x, U y) {
    return a * x + b * y + c;
  }

  // 同じ直線が同じ a,b,c で表現されるようにする
  void normalize() {
    static_assert(is_same_v<T, int> || is_same_v<T, long long>);
    T g = gcd(gcd(abs(a), abs(b)), abs(c));
    a /= g, b /= g, c /= g;
    if (b < 0) { a = -a, b = -b, c = -c; }
    if (b == 0 && a < 0) { a = -a, b = -b, c = -c; }
  }

  bool is_parallel(Line other) { return a * other.b - b * other.a == 0; }
  bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; }
};

template <typename T>
struct Segment {
  Point<T> A, B;

  Segment(Point<T> A, Point<T> B) : A(A), B(B) {}
  Segment(T x1, T y1, T x2, T y2) : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  bool contain(Point<T> C) {
    T det = (C - A).det(B - A);
    if (det != 0) return 0;
    return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0;
  }

  Line<T> to_Line() { return Line(A, B); }
};

template <typename REAL>
struct Circle {
  Point<REAL> O;
  REAL r;
  Circle(Point<REAL> O, REAL r) : O(O), r(r) {}
  Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {}
  template <typename T>
  bool contain(Point<T> p) {
    REAL dx = p.x - O.x, dy = p.y - O.y;
    return dx * dx + dy * dy <= r * r;
  }
};
#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "random/shuffle.hpp"

template <typename T>
void shuffle(vc<T>& A) {
  FOR(i, len(A)) {
    int j = RNG(0, i + 1);
    if (i != j) swap(A[i], A[j]);
  }
}
#line 2 "ds/hashmap.hpp"

// u64 -> Val

template <typename Val>
struct HashMap {
  // n は入れたいものの個数で ok

  HashMap(u32 n = 0) { build(n); }
  void build(u32 n) {
    u32 k = 8;
    while (k < n * 2) k *= 2;
    cap = k / 2, mask = k - 1;
    key.resize(k), val.resize(k), used.assign(k, 0);
  }

  // size を保ったまま. size=0 にするときは build すること.

  void clear() {
    used.assign(len(used), 0);
    cap = (mask + 1) / 2;
  }
  int size() { return len(used) / 2 - cap; }

  int index(const u64& k) {
    int i = 0;
    for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
    return i;
  }

  Val& operator[](const u64& k) {
    if (cap == 0) extend();
    int i = index(k);
    if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
    return val[i];
  }

  Val get(const u64& k, Val default_value) {
    int i = index(k);
    return (used[i] ? val[i] : default_value);
  }

  bool count(const u64& k) {
    int i = index(k);
    return used[i] && key[i] == k;
  }

  // f(key, val)

  template <typename F>
  void enumerate_all(F f) {
    FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
  }

private:
  u32 cap, mask;
  vc<u64> key;
  vc<Val> val;
  vc<bool> used;

  u64 hash(u64 x) {
    static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
    x += FIXED_RANDOM;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & mask;
  }

  void extend() {
    vc<pair<u64, Val>> dat;
    dat.reserve(len(used) / 2 - cap);
    FOR(i, len(used)) {
      if (used[i]) dat.eb(key[i], val[i]);
    }
    build(2 * len(dat));
    for (auto& [a, b]: dat) (*this)[a] = b;
  }
};
#line 2 "random/hash_pair.hpp"

template <typename T>
u64 hash_pair(pair<T, T> X) {
  static ll hash_base = 0;
  if (hash_base == 0) hash_base = RNG_64();
  return hash_base * X.fi + X.se;
}
#line 6 "geo/closest_pair.hpp"

template <typename T>
pair<int, int> closest_pair(vc<Point<T>> points) {
  int N = len(points);
  assert(N >= 2);
  HashMap<int> MP(N);
  vc<int> I(N);
  iota(all(I), 0);
  shuffle(I);
  points = rearrange(points, I);

  auto calc = [&](int i, int j) -> T {
    return (points[j] - points[i]).dot(points[j] - points[i]);
  };

  T best = calc(0, 1);
  pair<int, int> res = {0, 1};
  T w = sqrtl(best);

  vc<int> nxt(N, -1);

  auto insert = [&](int i) -> void {
    u64 k = hash_pair<ll>({points[i].x / w, points[i].y / w});
    nxt[i] = MP.get(k, -1);
    MP[k] = i;
  };

  auto query = [&](int i) -> bool {
    ll a = points[i].x / w;
    ll b = points[i].y / w;
    bool upd = 0;
    FOR(dx, -1, 2) FOR(dy, -1, 2) {
      u64 k = hash_pair<ll>({a + dx, b + dy});
      int j = MP.get(k, -1);
      while (j != -1) {
        if (chmin(best, calc(i, j))) { upd = 1, res = {i, j}, w = sqrtl(best); }
        j = nxt[j];
      }
    }
    return upd;
  };

  if (best == T(0)) {
    res.fi = I[res.fi], res.se = I[res.se];
    return res;
  }

  insert(0), insert(1);
  FOR(i, 2, N) {
    if (query(i)) {
      if (best == T(0)) break;
      MP.build(N);
      FOR(j, i) insert(j);
    }
    insert(i);
  }
  res.fi = I[res.fi], res.se = I[res.se];
  return res;
}

pair<int, int> closest_pair_dc(vc<Point<ll>> point) {
  int N = len(point);
  assert(N >= 2);

  auto I = argsort(point);
  point = rearrange(point, I);

  ll best = -1;
  pair<int, int> best_pair = {-1, -1};

  auto upd = [&](int i, int j) -> void {
    Point<ll> p = point[i] - point[j];
    ll d = p.dot(p);
    if (best == -1 || best > d) { best = d, best_pair = {I[i], I[j]}; }
  };
  upd(0, 1);

  auto dfs = [&](auto &dfs, int L, int R) -> vc<int> {
    // return: [L,R) を y について sort したもの
    if (R == L + 1) return {L};
    int M = (L + R) / 2;
    vc<int> I0 = dfs(dfs, L, M);
    vc<int> I1 = dfs(dfs, M, R);
    vc<int> I;
    vc<int> near;
    int a = 0, b = 0;
    FOR(R - L) {
      int idx = [&]() -> int {
        if (a == len(I0)) return I1[b++];
        if (b == len(I1)) return I0[a++];
        int i = I0[a], j = I1[b];
        if (point[i].y < point[j].y) {
          ++a;
          return i;
        }
        ++b;
        return j;
      }();
      I.eb(idx);
      ll dx = point[M].x - point[idx].x;
      if (dx * dx > best) { continue; }
      FOR_R(k, len(near)) {
        int j = near[k];
        ll dy = point[idx].y - point[j].y;
        if (best == 0 || dy * dy > best) break;
        upd(idx, j);
      }
      near.eb(idx);
    }
    return I;
  };
  dfs(dfs, 0, N);
  return best_pair;
}

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