2021/finals/antisocial.cpp

#include <algorithm>
#include <cassert>
#include <cmath>
#include <functional>
#include <iostream>
#include <queue>
#include <utility>
#include <vector>
using namespace std;

const int LIM = 6000006;
const long double EPS = 1e-9;

using LD = long double;
using pii = pair<int, int>;

// From https://github.com/zigui-ps/VoronoiDiagram
struct point {
  LD x, y;

  point(): x(0), y(0) {}
  point(LD _x, LD _y): x(_x), y(_y) {}

  LD operator/(point p) const { return x * p.y - y * p.x; }
  point operator+(point p) const { return point(x + p.x, y + p.y); }
  point operator-(point p) const { return point(x - p.x, y - p.y); }
  friend point operator*(LD b, point a) { return point(b * a.x, b * a.y); }

  LD size() const { return hypot(x, y); }
  LD sz2() const { return x * x + y * y; }
  point r90() const { return point(-y, x); }
  LD dot(point p) { return x * p.x + y * p.y; }
};

int dcmp(LD x) {
  return x < -EPS ? -1 : x > EPS ? 1 : 0;
}

point line_intersect(point a, point b, point u, point v) {
  return u + (((a - u) / b) / (v / b)) * v;
}

point get_circumcenter(point p0, point p1, point p2) {
  return line_intersect(
    0.5 * (p0 + p1), (p0 - p1).r90(), 0.5 * (p1 + p2), (p1 - p2).r90()
  );
}

point project_point_segment(point a, point b, point c) {
  LD r = (b - a).dot(b - a);
  if (fabs(r) < EPS) {
    return a;
  }
  r = (c - a).dot(b - a) / r;
  if (r < 0) {
    return a;
  }
  if (r > 1) {
    return b;
  }
  return a + r * (b - a);
}

bool point_in_polygon_incl(const vector<point> &p, point q) {
  bool c = 0;
  for (int i = 0; i < p.size(); i++) {
    int j = (i + 1) % p.size();
    if ((project_point_segment(p[i], p[j], q) - q).sz2() < EPS) {
      return true;
    }
    if (
      (p[i].y <= q.y && q.y < p[j].y || p[j].y <= q.y && q.y < p[i].y) &&
      q.x < p[i].x + (p[j].x - p[i].x) * (q.y - p[i].y) / (p[j].y - p[i].y)
    )  {
      c = !c;
    }
  }
  return c;
}

// https://www.youtube.com/watch?v=h_vvP4ah6Ck
LD parabola_intersect(point left, point right, LD sweepline) {
  auto f2 = [](point left, point right, LD sweepline) {
    int sign = left.x < right.x ? 1 : -1;
    point m = 0.5 * (left + right);
    point v = line_intersect(
      m, (right - left).r90(), point(0, sweepline), point(1, 0)
    );
    point w = line_intersect(m, (left - v).r90(), v, left - v);
    LD l1 = (v - w).size();
    LD l2 = sqrt(pow(sweepline - m.y, 2) - (m - w).sz2());
    LD l3 = (left - v).size();
    return v.x + (m.x - v.x) * l3 / (l1 + sign * l2);
  };
  if (fabs(left.y - right.y) < fabs(left.x - right.x) * EPS) {
    return f2(left, right, sweepline);
  }
  int sign = left.y < right.y ? -1 : 1;
  point v = line_intersect(
    left, right - left, point(0, sweepline), point(1, 0)
  );
  LD d1 = (0.5 * (left + right) - v).sz2(), d2 = (0.5 * (left - right)).sz2();
  return v.x + sign * sqrt(max(0.0L, d1 - d2));
}

struct Beachline {
  struct node {
    point pt;
    int idx;
    int end;
    node *link[2], *par, *prv, *nxt;

    node() {}
    node(point _pt, int idx):
      pt(_pt), idx(idx), end(0), link{0, 0}, par(0), prv(0), nxt(0) {}
  } *root;

  LD sweepline;

  Beachline() : sweepline(-1e20), root(NULL) { }

  inline int dir(node* x) {
    return x->par->link[0] != x;
  }

  //     p        n          p            n
  //    / \      / \        / \          / \
  //   n   d => a   p   or a   n   =>   p   d
  //  / \          / \        / \      / \
  // a   b        b   d      c   d    a   c
  void rotate(node* n) {
    node* p = n->par;
    int d = dir(n);
    p->link[d] = n->link[!d];
    if (n->link[!d]) {
      n->link[!d]->par = p;
    }
    n->par = p->par;
    if (p->par) {
      p->par->link[dir(p)] = n;
    }
    n->link[!d] = p;
    p->par = n;
  }

  void splay(node* x, node* f = NULL) {
    while (x->par != f) {
      if (x->par->par == f) {
        // no-op
      } else if (dir(x) == dir(x->par)) {
        rotate(x->par);
      } else {
        rotate(x);
      }
      rotate(x);
    }
    if (f == NULL) {
      root = x;
    }
  }

  void insert(node* n, node* p, int d) {
    splay(p);
    node *c = p->link[d];
    n->link[d] = c;
    if (c) {
      c->par = n;
    }
    p->link[d] = n;
    n->par = p;
    node *prv = !d ? p->prv : p, *nxt = !d ? p : p->nxt;
    n->prv = prv;
    if (prv) {
      prv->nxt = n;
    }
    n->nxt = nxt;
    if (nxt) {
      nxt->prv = n;
    }
  }

  void erase(node* n) {
    node *prv = n->prv, *nxt = n->nxt;
    if (!prv && !nxt) {
      if (n == root) {
        root = NULL;
      }
      return;
    }
    n->prv = NULL;
    if (prv) {
      prv->nxt = nxt;
    }
    n->nxt = NULL;
    if (nxt) {
      nxt->prv = prv;
    }
    splay(n);
    if (!nxt) {
      root->par = NULL;
      n->link[0] = NULL;
      root = prv;
    } else {
      splay(nxt, n);
      node* c = n->link[0];
      nxt->link[0] = c;
      c->par = nxt;
      n->link[0] = NULL;
      n->link[1] = NULL;
      nxt->par = NULL;
      root = nxt;
    }
  }

  bool get_event(node *cur, LD &next_sweep) {
    if (!cur->prv || !cur->nxt) {
      return false;
    }
    point u = (cur->pt - cur->prv->pt).r90();
    point v = (cur->nxt->pt - cur->pt).r90();
    if (dcmp(u / v) != 1) {
      return false;
    }
    point p = get_circumcenter(cur->pt, cur->prv->pt, cur->nxt->pt);
    next_sweep = p.y + (p - cur->pt).size();
    return true;
  }

  node* find_beachline(LD x) {
    node* cur = root;
    while (cur) {
      LD left = cur->prv
                  ? parabola_intersect(cur->prv->pt, cur->pt, sweepline)
                  : -1e30;
      LD right = cur->nxt
                  ? parabola_intersect(cur->pt, cur->nxt->pt, sweepline)
                  : 1e30;
      if (left <= x && x <= right) {
        splay(cur);
        return cur;
      }
      cur = cur->link[x > right];
    }
    return NULL;
  }
};

using BeachNode = Beachline::node;

static BeachNode arr[LIM];
static int sz;
static BeachNode* new_node(point point, int idx) {
  arr[sz] = BeachNode(point, idx);
  return arr + (sz++);
}

struct event {
  int type, idx, prv, nxt;
  BeachNode *cur;
  LD sweep;

  event(LD sweep, int idx) : type(0), sweep(sweep), idx(idx) {}
  event(LD sweep, BeachNode* cur) :
    type(1), sweep(sweep), prv(cur->prv->idx), cur(cur), nxt(cur->nxt->idx) {}

  bool operator>(const event& l) const { return sweep > l.sweep; }
};

void voronoi_diagram(
  vector<point> &input,
  vector<point> &vertex,
  vector<pii> &edge,
  vector<pii> &area
) {
  Beachline beachline = Beachline();
  priority_queue<event, vector<event>, greater<event>> events;

  auto add_edge = [&](int u, int v, int a, int b, BeachNode* c1, BeachNode* c2) {
    if (c1) {
      c1->end = edge.size() * 2;
    }
    if (c2) {
      c2->end = edge.size() * 2 + 1;
    }
    edge.emplace_back(u, v);
    area.emplace_back(a, b);
  };
  auto write_edge = [&](int idx, int v) {
    if (idx % 2 == 0) {
      edge[idx / 2].first = v;
    } else {
      edge[idx / 2].second = v;
    }
  };
  auto add_event = [&](BeachNode* cur) {
    LD nxt;
    if (beachline.get_event(cur, nxt)) {
      events.emplace(nxt, cur);
    }
  };

  int n = input.size(), cnt = 0;
  /*arr = new BeachNode[n * 4]; */sz = 0;
  sort(input.begin(), input.end(), [](const point &l, const point &r) {
    return l.y != r.y ? l.y < r.y : l.x < r.x;
  });

  BeachNode *tmp = beachline.root = new_node(input[0], 0), *t2;
  for (int i = 1; i < n; i++) {
    if (dcmp(input[i].y - input[0].y) == 0) {
      add_edge(-1, -1, i - 1, i, 0, tmp);
      beachline.insert(t2 = new_node(input[i], i), tmp, 1);
      tmp = t2;
    } else {
      events.emplace(input[i].y, i);
    }
  }

  while (events.size()) {
    event q = events.top();
    events.pop();
    BeachNode *prv, *cur, *nxt, *site;
    int v = vertex.size(), idx = q.idx;
    beachline.sweepline = q.sweep;
    if (q.type == 0) {
      point pt = input[idx];
      cur = beachline.find_beachline(pt.x);
      beachline.insert(site = new_node(pt, idx), cur, 0);
      beachline.insert(prv = new_node(cur->pt, cur->idx), site, 0);
      add_edge(-1, -1, cur->idx, idx, site, prv);
      add_event(prv);
      add_event(cur);
    } else {
      cur = q.cur;
      prv = cur->prv;
      nxt = cur->nxt;
      if (!prv || !nxt || prv->idx != q.prv || nxt->idx != q.nxt) {
        continue;
      }
      vertex.push_back(get_circumcenter(prv->pt, nxt->pt, cur->pt));
      write_edge(prv->end, v);
      write_edge(cur->end, v);
      add_edge(v, -1, prv->idx, nxt->idx, 0, prv);
      beachline.erase(cur);
      add_event(prv);
      add_event(nxt);
    }
  }
  // delete arr;
}

int XR, YR;
int N, M;
vector<pair<int, LD>> adj[LIM];
point KP[2];
int KN[2];
LD ans;

void process_voronoi_diagrams(vector<point> P) {
  int i, j, k;
  vector<point> vertex;
  vector<pii> edge, area;
  voronoi_diagram(P, vertex, edge, area);
  M = vertex.size();
  for (int i = 0; i < area.size(); i++) {
    int e1 = edge[i].first, e2 = edge[i].second;
    if (e1 < 0 || e2 < 0) {
      continue; // Infinite edge.
    }
    point v1 = vertex[e1], v2 = vertex[e2];
    if (
      min(v1.x, v2.x) < -EPS || max(v1.x, v2.x) > XR + EPS ||
      min(v1.y, v2.y) < -EPS || max(v1.y, v2.y) > YR + EPS
    ) {
      continue; // Edge outside rectangle.
    }
    // Add edge to graph with appropriate weight.
    int pi1 = area[i].first, pi2 = area[i].second;
    point p1 = P[pi1], p2 = P[pi2];
    LD d = (p1 - project_point_segment(v1, v2, p1)).size();
    adj[e1].emplace_back(e2, d);
    adj[e2].emplace_back(e1, d);
    // Consider any key points in triangles adjacent to this edge.
    point mid = 0.5 * (p1 + p2);
    for (int j : {0, 1}) {
      for (int k : {0, 1}) {
        point kp = KP[j], p = k ? p2 : p1;
        ans = min(ans, (kp - p).size());  // Must be this close to a point.
        if (point_in_polygon_incl({p, v1, mid}, kp)) {
          KN[j] = e1;
        }
        if (point_in_polygon_incl({p, mid, v2}, kp)) {
          KN[j] = e2;
        }
      }
    }
  }
}

LD solve() {
  vector<point> P;
  for (int i = 0; i < LIM; i++) {
    adj[i].clear();
  }
  // Input.
  cin >> XR >> YR;
  cin >> KP[0].x >> KP[0].y >> KP[1].x >> KP[1].y;
  cin >> N;
  for (int i = 0; i < N; i++) {
    int a, b;
    cin >> a >> b;
    P.emplace_back(a, b);
    // Additionally mirror each point across each side of the rectangle.
    P.emplace_back(-a, b);
    P.emplace_back(a, -b);
    P.emplace_back(2 * XR - a, b);
    P.emplace_back(a, 2 * YR - b);
  }
  // Compute Voronoi diagram and corresponding graph.
  ans = 1e18;
  KN[0] = KN[1] = -1;
  process_voronoi_diagrams(P);
  assert(KN[0] >= 0 && KN[1] >= 0);
  // MST.
  vector<bool> visit(M);
  priority_queue<pair<LD, int>> Q;
  Q.emplace(ans, KN[0]);
  while (!Q.empty()) {
    ans = min(ans, Q.top().first);
    int i = Q.top().second;
    Q.pop();
    if (i == KN[1]) {
      break;
    }
    if (visit[i]) {
      continue;
    }
    visit[i] = true;
    for (auto c : adj[i]) {
      if (!visit[c.first]) {
        Q.emplace(c.second, c.first);
      }
    }
  }
  return ans;
}

int main() {
  cout.precision(10);
  int T;
  cin >> T;
  for (int t = 1; t <= T; t++) {
    cout << "Case #" << t << ": " << fixed << solve() << endl;
  }
  return 0;
}