#include #include #include #include #include #include #include #include using namespace std; const int LIM = 6000006; const long double EPS = 1e-9; using LD = long double; using pii = pair; // 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 &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 &input, vector &vertex, vector &edge, vector &area ) { Beachline beachline = Beachline(); priority_queue, greater> 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> adj[LIM]; point KP[2]; int KN[2]; LD ans; void process_voronoi_diagrams(vector P) { int i, j, k; vector vertex; vector 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 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 visit(M); priority_queue> 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; }