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:heavy_check_mark: graph/planar_graph.hpp

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#include "graph/base.hpp"
#include "geo/base.hpp"
#include "geo/angle_sort.hpp"

/*
・連結平面グラフになっていないときにどう動作するかは何も考えていない
・N=1 も扱わない
*/
template <typename XY>
struct Planar_Graph {
  using P = Point<XY>;
  int NV, NE, NF;
  // 頂点, 辺からなるグラフ. 有向辺を 2 つ入れておく
  Graph<int, 1> G;
  // 頂点属性
  vc<P> point; // 座標
  // 辺属性
  vc<int> left_face; // 有向辺の左にある面の番号
  vc<int> nxt_edge;  // 面を反時計回りにまわるときの次の辺
  // 面属性
  vc<int> first_edge;

  Planar_Graph(int N, vc<P> point) : NV(N), G(N), point(point) {
    assert(N > 1);
  }

  void add(int a, int b) { G.add(a, b), G.add(b, a); }
  void build() {
    G.build();
    NE = G.M / 2;
    nxt_edge.assign(G.M, -1);
    left_face.assign(G.M, -1);
    int v0 = 0;
    int e0 = 0;
    FOR(v, NV) {
      if (point[v] < point[v0]) v0 = v;
      vc<int> eid;
      vc<P> dir;
      for (auto& e: G[v]) {
        eid.eb(e.id);
        dir.eb(point[e.to] - point[e.frm]);
      }
      auto I = angle_sort(dir);
      assert(len(I) > 0);
      FOR(k, len(I)) {
        int i = (k == 0 ? I.back() : I[k - 1]);
        int j = I[k];
        i = eid[i], j = eid[j];
        nxt_edge[j ^ 1] = i;
      }
      if (v == v0) e0 = eid[I[0] ^ 1];
    }
    for (auto& x: nxt_edge) assert(x != -1);

    auto make_face = [&](int e) -> void {
      int p = len(first_edge);
      first_edge.eb(e);
      while (left_face[e] == -1) {
        left_face[e] = p;
        e = nxt_edge[e];
      }
    };

    make_face(e0);
    FOR(e, 2 * NE) {
      if (left_face[e] == -1) make_face(e);
    }
    NF = len(first_edge);
    assert(NV - NE + NF == 2);
  }

  // return {vs, es}
  // vs = [v0,v1,v2,v0], es = [e0,e1,e2]
  pair<vc<int>, vc<int>> get_face_data(int fid) {
    vc<int> eid = {first_edge[fid]};
    while (1) {
      int e = nxt_edge[eid.back()];
      if (e == first_edge[fid]) break;
      eid.eb(e);
    }
    vc<int> vid;
    for (auto& e: eid) vid.eb(G.edges[e].frm);
    vid.eb(vid[0]);
    return {vid, eid};
  }
};
#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#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+(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>
REAL dist(Point<T> A, Point<T> B) {
  A = A - B;
  T p = A.dot(A);
  return sqrt(REAL(p));
}

// 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) {
    static_assert(is_integral<T>::value);
    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;
  }
};

template <typename T>
struct Polygon {
  vc<Point<T>> points;
  T a;

  template <typename A, typename B>
  Polygon(vc<pair<A, B>> pairs) {
    for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b));
    build();
  }
  Polygon(vc<Point<T>> points) : points(points) { build(); }

  int size() { return len(points); }

  template <typename REAL>
  REAL area() {
    return a * 0.5;
  }

  template <enable_if_t<is_integral<T>::value, int> = 0>
  T area_2() {
    return a;
  }

  bool is_convex() {
    FOR(j, len(points)) {
      int i = (j == 0 ? len(points) - 1 : j - 1);
      int k = (j == len(points) - 1 ? 0 : j + 1);
      if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false;
    }
    return true;
  }

private:
  void build() {
    a = 0;
    FOR(i, len(points)) {
      int j = (i + 1 == len(points) ? 0 : i + 1);
      a += points[i].det(points[j]);
    }
    if (a < 0) {
      a = -a;
      reverse(all(points));
    }
  }
};
#line 2 "geo/angle_sort.hpp"

#line 4 "geo/angle_sort.hpp"

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<Point<T>>& P) {
  vector<int> lower, origin, upper;
  const Point<T> O = {0, 0};
  FOR(i, len(P)) {
    if (P[i] == O) origin.eb(i);
    elif ((P[i].y < 0) || (P[i].y == 0 && P[i].x > 0)) lower.eb(i);
    else upper.eb(i);
  }
  sort(all(lower), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  sort(all(upper), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  auto& I = lower;
  I.insert(I.end(), all(origin));
  I.insert(I.end(), all(upper));
  return I;
}

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<pair<T, T>>& P) {
  vc<Point<T>> tmp(len(P));
  FOR(i, len(P)) tmp[i] = Point<T>(P[i]);
  return angle_sort<T>(tmp);
}
#line 4 "graph/planar_graph.hpp"

/*
・連結平面グラフになっていないときにどう動作するかは何も考えていない
・N=1 も扱わない
*/
template <typename XY>
struct Planar_Graph {
  using P = Point<XY>;
  int NV, NE, NF;
  // 頂点, 辺からなるグラフ. 有向辺を 2 つ入れておく
  Graph<int, 1> G;
  // 頂点属性
  vc<P> point; // 座標
  // 辺属性
  vc<int> left_face; // 有向辺の左にある面の番号
  vc<int> nxt_edge;  // 面を反時計回りにまわるときの次の辺
  // 面属性
  vc<int> first_edge;

  Planar_Graph(int N, vc<P> point) : NV(N), G(N), point(point) {
    assert(N > 1);
  }

  void add(int a, int b) { G.add(a, b), G.add(b, a); }
  void build() {
    G.build();
    NE = G.M / 2;
    nxt_edge.assign(G.M, -1);
    left_face.assign(G.M, -1);
    int v0 = 0;
    int e0 = 0;
    FOR(v, NV) {
      if (point[v] < point[v0]) v0 = v;
      vc<int> eid;
      vc<P> dir;
      for (auto& e: G[v]) {
        eid.eb(e.id);
        dir.eb(point[e.to] - point[e.frm]);
      }
      auto I = angle_sort(dir);
      assert(len(I) > 0);
      FOR(k, len(I)) {
        int i = (k == 0 ? I.back() : I[k - 1]);
        int j = I[k];
        i = eid[i], j = eid[j];
        nxt_edge[j ^ 1] = i;
      }
      if (v == v0) e0 = eid[I[0] ^ 1];
    }
    for (auto& x: nxt_edge) assert(x != -1);

    auto make_face = [&](int e) -> void {
      int p = len(first_edge);
      first_edge.eb(e);
      while (left_face[e] == -1) {
        left_face[e] = p;
        e = nxt_edge[e];
      }
    };

    make_face(e0);
    FOR(e, 2 * NE) {
      if (left_face[e] == -1) make_face(e);
    }
    NF = len(first_edge);
    assert(NV - NE + NF == 2);
  }

  // return {vs, es}
  // vs = [v0,v1,v2,v0], es = [e0,e1,e2]
  pair<vc<int>, vc<int>> get_face_data(int fid) {
    vc<int> eid = {first_edge[fid]};
    while (1) {
      int e = nxt_edge[eid.back()];
      if (e == first_edge[fid]) break;
      eid.eb(e);
    }
    vc<int> vid;
    for (auto& e: eid) vid.eb(G.edges[e].frm);
    vid.eb(vid[0]);
    return {vid, eid};
  }
};
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