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\u9898\u610f\uff1a\u7ed9\u51fa\u4e86\u5e73\u9762\u4e0a $n$ \u4e2a\u6ca1\u6709\u516c\u5171\u70b9\u7684\u51f8\u591a\u8fb9\u5f62\u3002\u5e73\u9762\u4e0a\u6709\u4e24\u4e2a\u70b9\uff0c\u4f60\u9700\u8981\u4ece\u5176\u4e2d\u4e00\u70b9\u8d70\u5230\u53e6\u4e00\u70b9\uff0c\u518d\u4ece\u8be5\u70b9\u8d70\u5230\u4e00\u4e2a\u4e0d\u88ab\u4efb\u4f55\u51f8\u591a\u8fb9\u5f62\u5305\u542b\u7684\u5730\u65b9\uff0c\u4ee3\u4ef7\u4e3a\u8def\u5f84\u4e0a\u7a7f\u8fc7\u7684\u51f8\u591a\u8fb9\u5f62\u8fb9\u754c\u6570\u91cf\u3002\u4f60\u53ef\u4ee5\u968f\u610f\u5b89\u6392\u4e24\u4e2a\u70b9\u7684\u4f4d\u7f6e\uff0c\u6c42\u4ee3\u4ef7\u7684\u6700\u5c0f\u503c\u6700\u5927\u662f\u591a\u5c11\u3002$n\\leq 10^5$\uff0c\u51f8\u591a\u8fb9\u5f62\u603b\u8fb9\u6570 $6\\times 10^5$\uff0c\u503c\u57df $10^9$\u3002<\/p>\n \u8fd9\u9898\u5c31\u9700\u8981\u7528\u5230\u51f8\u591a\u8fb9\u5f62\u7684\u626b\u63cf\u7ebf\uff0c\u5c06\u51f8\u591a\u8fb9\u5f62\u4e4b\u95f4\u7684\u5305\u542b\u5173\u7cfb\u8f6c\u5316\u4e3a\u4e00\u9897\u6811\u3002\u8f6c\u5316\u4e3a\u6811\u4e4b\u540e\uff0c\u5c31\u662f\u6c42\u4ece\u6811\u4e0a\u4e00\u70b9\u8d70\u5230\u53e6\u4e00\u70b9\uff0c\u518d\u4ece\u8be5\u70b9\u8d70\u5230\u8d85\u7ea7\u6e90\u70b9\u7684\u8def\u5f84\u957f\u5ea6\u3002\u663e\u7136\uff0c\u7b54\u6848\u5c31\u662f\u4ece\u76f4\u5f84\u7684\u4e00\u7aef\u8d70\u5230\u53e6\u4e00\u7aef\uff0c\u518d\u8d70\u5230\u6811\u7684\u6839\u7684\u8def\u5f84\u957f\u5ea6\uff0c\u6c42\u4e2a\u76f4\u5f84\u5c31\u505a\u5b8c\u4e86\u3002<\/p>\n \u672c\u6587\u4e3b\u8981\u4ecb\u7ecd\u4e86\u5982\u4f55\u4f7f\u7528\u626b\u63cf\u7ebf\u5c06\u5e73\u9762\u4e0a\u5706\u6216\u51f8\u591a\u8fb9\u5f62\u4e4b\u95f4\u7684\u5305\u542b\u5173\u7cfb\uff0c\u9ad8\u6548\u5730\u8f6c\u5316\u4e3a\u6811\u7ed3\u6784\uff08\u68ee\u6797\uff0c\u901a\u8fc7\u5f15\u5165\u8d85\u7ea7\u6e90\u70b9 $0$ \u8f6c\u5316\u4e3a\u4e00\u68f5\u6811\uff09\uff0c\u4ece\u800c\u5c06\u51e0\u4f55\u95ee\u9898\u8f6c\u5316\u4e3a\u6811\u4e0a\u95ee\u9898\uff0c\u6838\u5fc3\u601d\u60f3\u5c31\u662f\u5229\u7528\u56fe\u5f62\u4e4b\u95f4\u7684\u76f8\u5bf9\u5173\u7cfb\uff0c\u5c06\u56fe\u5f62\u62c6\u6210\u4e0a\u4e0b\u4e24\u90e8\u5206\uff0c\u5e76\u5728\u626b\u63cf\u7ebf\u8fc7\u7a0b\u4e2d\u4f7f\u7528 set \u7ef4\u62a4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a $\\mathcal{O}(n \\log n)$\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u5706\u7684\u626b\u63cf\u7ebf\u7b97\u6cd5 \u5f15\u5165 \u5e73\u9762\u4e0a\u6709 $n$ \u4e2a\u5706\uff0c\u5706\u4e4b\u95f4\u6ca1\u6709\u516c\u5171\u70b9\uff0c\u5373\u5706\u4e4b\u95f4\u8981\u4e48\u5305\u542b\uff0c\u8981\u4e48\u76f8\u79bb\u3002\u6b64\u65f6\u5706\u8981\u4e48\u90fd\u662f\u4e00\u4e2a […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,3,6],"tags":[],"class_list":["post-510","post","type-post","status-publish","format-standard","hentry","category-c","category-oi","category-6"],"_links":{"self":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/510","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/comments?post=510"}],"version-history":[{"count":8,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/510\/revisions"}],"predecessor-version":[{"id":560,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/posts\/510\/revisions\/560"}],"wp:attachment":[{"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/media?parent=510"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/categories?post=510"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.xgnd.net\/index.php\/wp-json\/wp\/v2\/tags?post=510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\uff08\u5706\u7684\u61d2\u5f97\u518d\u5199\u4e00\u904d\u4e86\uff09<\/del>\u3002<\/p>\n#include <bits\/stdc++.h>\nusing namespace std;\ntypedef long long ll;\nconst int maxn = 100050, mod = 998244353, INF = 1e9;\nconst double eps = 1e-9;\nint sweep_line_x = -1e9;\nvector<int> g[maxn];\n\nstruct Circle {\n int x, y, r;\n\n double get_y(bool up_side) const {\n int b = -2 * y;\n ll c = (ll)y * y + (sweep_line_x - x) * ll(sweep_line_x - x) - (ll)r * r;\n ll delta = (ll)b * b - 4 * c;\n return (-b + (up_side ? sqrt(delta) : -sqrt(delta))) \/ 2;\n }\n\n bool operator<(const Circle& rhs) const {\n return r < rhs.r;\n }\n};\n\nstruct Curve {\n bool up_side;\n Circle c;\n\n bool operator<(const Curve& rhs) const {\n return c.get_y(up_side) + eps < rhs.c.get_y(rhs.up_side);\n }\n};\n\nstruct Interval {\n int id;\n Curve up, down;\n\n bool operator<(const Interval& rhs) const {\n return down < rhs.down || up < rhs.up;\n }\n\n bool operator==(const Interval& rhs) const {\n return !(*this < rhs) && !(rhs < *this);\n }\n};\n\nint main() {\n ios::sync_with_stdio(0);\n int n;\n cin >> n;\n\n vector<Circle> c(n + 1);\n vector<pair<int, int>> events;\n for (int i = 1;i <= n;i++) {\n cin >> c[i].x >> c[i].y >> c[i].r;\n events.emplace_back(c[i].x - c[i].r, i);\n events.emplace_back(c[i].x + c[i].r, i);\n }\n\n c[0] = { 0,0,INF };\n sort(events.begin(), events.end());\n\n set<Interval> s;\n s.insert({ 0, {true, c[0]}, {false, c[0]} });\n\n for (auto &[x, id] : events) {\n sweep_line_x = x;\n Interval cur = { id, {true, c[id]}, {false, c[id]} };\n if (x == c[id].x - c[id].r) { \/\/\u5706\u7684\u5de6\u7aef\u70b9\n auto it = --s.lower_bound(cur);\n auto par = *it;\n s.erase(it);\n g[par.id].push_back(id);\n Interval up = par, down = par;\n up.down = { true, c[id] };\n down.up = { false, c[id] };\n s.insert(up);\n s.insert(down);\n s.insert(cur);\n }\n else { \/\/\u5706\u7684\u53f3\u7aef\u70b9\n auto it = s.lower_bound(cur);\n auto joined = *prev(it);\n joined.up = next(it)->up;\n s.erase(prev(it));\n s.erase(next(it));\n s.erase(it);\n s.insert(joined);\n }\n }\n\n return 0;\n}<\/code><\/pre>\n\u4f8b\u9898<\/h3>\n
\u5f80\u65e5\u91cd\u73b0<\/a><\/h4>\n
#include <bits\/stdc++.h>\nusing namespace std;\ntypedef long long ll;\n#define int ll\nconst int maxn = 100050, mod = 998244353, INF = 1e9;\nconst double eps = 1e-9;\nint sweep_line_x = -1e9;\nvector<int> g[maxn];\nint dep[maxn], fa[maxn][22];\n\nstruct Circle {\n int x, y, r;\n\n double get_y(bool up_side) const {\n int b = -2 * y;\n ll c = (ll)y * y + (sweep_line_x - x) * ll(sweep_line_x - x) - (ll)r * r;\n ll delta = b * (ll)b - 4 * c;\n return (-b + (up_side ? sqrt(delta) : -sqrt(delta))) \/ 2;\n }\n\n bool operator<(const Circle& rhs) const {\n return r < rhs.r;\n }\n};\n\nstruct Curve {\n bool up_side;\n Circle c;\n\n bool operator<(const Curve& rhs) const {\n return c.get_y(up_side) + eps < rhs.c.get_y(rhs.up_side);\n }\n};\n\nstruct Interval {\n int id;\n Curve up, down;\n\n bool operator<(const Interval& rhs) const {\n return down < rhs.down || up < rhs.up;\n }\n\n bool operator==(const Interval& rhs) const {\n return !(*this < rhs) && !(rhs < *this);\n }\n};\n\nvoid dfs(int x, int fath) {\n fa[x][0] = fath;\n dep[x] = dep[fath] + 1;\n for (int i = 1;i <= 20;i++) {\n fa[x][i] = fa[fa[x][i - 1]][i - 1];\n }\n for (int to : g[x]) {\n if (to == fath)continue;\n dfs(to, x);\n }\n}\n\nint lca(int x, int y) {\n if (dep[x] < dep[y]) swap(x, y);\n\n for (int i = 20;i >= 0;i--) {\n if (dep[fa[x][i]] >= dep[y]) {\n x = fa[x][i];\n }\n if (x == y) return x;\n }\n for (int i = 20;i >= 0;i--) {\n if (fa[x][i] != fa[y][i]) {\n x = fa[x][i];\n y = fa[y][i];\n }\n }\n return fa[x][0];\n}\n\nstruct Query {\n pair<int, int> a, b;\n};\n\nvoid solve() {\n int n, m;\n cin >> n >> m;\n\n vector<Circle> c(n + 1);\n vector<pair<int, int>> events, p;\n vector<Query> q(m + 1);\n for (int i = 1;i <= n;i++) {\n cin >> c[i].x >> c[i].y >> c[i].r ;\n events.emplace_back(c[i].x - c[i].r, i);\n events.emplace_back(c[i].x + c[i].r, i);\n }\n\n for (int i = 1;i <= m;i++) {\n pair<int, int> a, b;\n cin >> a.first >> a.second >> b.first >> b.second;\n q[i] = { a,b };\n p.push_back(a);\n p.push_back(b);\n }\n\n sort(p.begin(), p.end());\n p.erase(unique(p.begin(), p.end()), p.end());\n\n vector<int> bel(p.size() + 1, 0);\n\n for (int i = 0;i < p.size();i++) {\n events.emplace_back(p[i].first, i + n + 1);\n }\n\n c[0] = { 0,0,INF };\n sort(events.begin(), events.end());\n\n set<Interval> s;\n s.insert({ 0, {true, c[0]}, {false, c[0]} });\n\n for (auto &[x, id] : events) {\n sweep_line_x = x;\n if (id > n) { \/\/\u8981\u8be2\u95ee\u7684\u70b9\n id -= n + 1;\n Circle now = { p[id].first,p[id].second,0 };\n Interval cur = { id, {true, now}, {false, now} };\n auto it = --s.lower_bound(cur);\n auto par = *it;\n bel[id] = par.id;\n continue;\n }\n Interval cur = { id, {true, c[id]}, {false, c[id]} };\n if (x == c[id].x - c[id].r) { \/\/\u5706\u7684\u5de6\u7aef\u70b9\n auto it = --s.lower_bound(cur);\n auto par = *it;\n s.erase(it);\n g[par.id].push_back(id);\n Interval up = par, down = par;\n up.down = { true, c[id] };\n down.up = { false, c[id] };\n s.insert(up);\n s.insert(down);\n s.insert(cur);\n }\n else { \/\/\u5706\u7684\u53f3\u7aef\u70b9\n auto it = s.lower_bound(cur);\n auto joined = *prev(it);\n joined.up = next(it)->up;\n s.erase(prev(it));\n s.erase(next(it));\n s.erase(it);\n s.insert(joined);\n }\n }\n\n dfs(0, 0);\n\n for (int i = 1;i <= m;i++) {\n pair<int,int> a=q[i].a,b=q[i].b;\n int x=lower_bound(p.begin(),p.end(),a)-p.begin();\n int y=lower_bound(p.begin(),p.end(),b)-p.begin();\n x=bel[x],y=bel[y];\n cout<<dep[x]+dep[y]-2*dep[lca(x,y)]<<'\\n';\n }\n}\n\nsigned main() {\n ios::sync_with_stdio(0);\n int t = 1;\n \/\/ cin>>t;\n while (t--) {\n solve();\n }\n\n return 0;\n}<\/code><\/pre>\n\u51f8\u591a\u8fb9\u5f62\u7684\u626b\u63cf\u7ebf<\/h2>\n
\u53ea\u662f\u7ec6\u8282\u7a0d\u5fae\u591a\u4e00\u70b9<\/del>\u3002<\/p>\n\u5b9e\u73b0<\/h3>\n
#include <bits\/stdc++.h>\nusing namespace std;\ntypedef long long ll;\n#define int ll\nconst int maxn = 600050, mod = 998244353, INF = 1e9;\nconst double eps = 1e-9;\nint sweep_line_x = -1e9;\nvector<int> g[maxn];\nint fa[maxn];\n\nstruct Segment {\n int id;\n pair<int, int> p, q;\n bool up_side;\n\n Segment() {}\n Segment(int id, pair<int, int> p, pair<int, int> q) : id(id), p(p), q(q)\n {\n if (p.first > q.first)up_side = true;\n else up_side = false;\n }\n double get_y(int x) const {\n if (p.first == q.first) return p.second;\n return p.second + 1.0 * (x - p.first) * (q.second - p.second) \/ (q.first - p.first);\n }\n\n bool operator<(const Segment& rhs) const {\n if (fabs(get_y(sweep_line_x) - rhs.get_y(sweep_line_x)) < eps) {\n if (up_side == rhs.up_side) return q.first < rhs.q.first;\n return up_side < rhs.up_side;\n }\n return get_y(sweep_line_x) + eps < rhs.get_y(sweep_line_x);\n }\n};\n\nstruct Event {\n int x;\n Segment l;\n bool is_left;\n bool operator<(const Event& rhs) const {\n if (x == rhs.x) {\n if (is_left == rhs.is_left) return l.p.second > rhs.l.p.second;\n return is_left > rhs.is_left;\n }\n return x < rhs.x;\n }\n};\n\nsigned main() {\n ios::sync_with_stdio(0);\n int n;\n cin >> n;\n\n vector<Segment> p;\n vector<Event> events;\n for (int i = 1;i <= n;i++) {\n int m;\n cin >> m;\n vector<pair<int, int>> tmp(m);\n for (int j = 0;j < m;j++) {\n cin >> tmp[j].first >> tmp[j].second;\n }\n for (int j = 0;j < m;j++) {\n int k = (j + 1) % m;\n if (tmp[j].first == tmp[k].first) continue;\n p.push_back({ i, tmp[j], tmp[k] });\n }\n }\n for (int i = 0; i < p.size(); i++) {\n if (p[i].p.first > p[i].q.first) swap(p[i].p, p[i].q);\n events.push_back({ p[i].p.first, p[i], true });\n events.push_back({ p[i].q.first, p[i], false });\n }\n\n sort(events.begin(), events.end());\n\n memset(fa,-1,sizeof fa);\n set<Segment> s;\n\n for (auto& [x, l, f] : events) {\n sweep_line_x = x;\n int id = l.id;\n if (l.up_side and fa[id] == -1) {\n auto it = s.upper_bound(l);\n if (it != s.end()) {\n Segment par = (*it);\n if (par.up_side) fa[id] = par.id;\n else fa[id] = fa[par.id];\n }\n else {\n fa[id] = 0;\n }\n g[fa[id]].push_back(id);\n g[id].push_back(fa[id]);\n }\n if (f == 1)s.insert(l);\n else s.erase(l);\n }\n\n return 0;\n}<\/code><\/pre>\n\u4f8b\u9898<\/h3>\n
Freedom from Prison<\/a><\/h4>\n
#include <bits\/stdc++.h>\nusing namespace std;\ntypedef long long ll;\n#define int ll\nconst int maxn = 600050, mod = 998244353, INF = 1e9;\nconst double eps = 1e-9;\nint sweep_line_x = -1e9;\nvector<int> g[maxn];\nint d, t;\nint fa[maxn];\n\nstruct Segment {\n int id;\n pair<int, int> p, q;\n bool up_side;\n\n Segment() {}\n Segment(int id, pair<int, int> p, pair<int, int> q) : id(id), p(p), q(q)\n {\n if (p.first > q.first)up_side = true;\n else up_side = false;\n }\n double get_y(int x) const {\n if (p.first == q.first) return p.second;\n return p.second + 1.0 * (x - p.first) * (q.second - p.second) \/ (q.first - p.first);\n }\n\n bool operator<(const Segment& rhs) const {\n if (fabs(get_y(sweep_line_x) - rhs.get_y(sweep_line_x)) < eps) {\n if (up_side == rhs.up_side) return q.first < rhs.q.first;\n return up_side < rhs.up_side;\n }\n return get_y(sweep_line_x) + eps < rhs.get_y(sweep_line_x);\n }\n};\n\nstruct Event {\n int x;\n Segment l;\n bool is_left;\n bool operator<(const Event& rhs) const {\n if (x == rhs.x) {\n if (is_left == rhs.is_left) return l.p.second > rhs.l.p.second;\n return is_left > rhs.is_left;\n }\n return x < rhs.x;\n }\n};\n\nvoid dfs(int x, int dep, int fath) {\n if (dep > d) {\n d = dep;\n t = x;\n }\n for (int to : g[x]) {\n if (to == fath)continue;\n dfs(to, dep+1, x);\n }\n}\n\nsigned main() {\n ios::sync_with_stdio(0);\n int n;\n cin >> n;\n\n vector<Segment> p;\n vector<Event> events;\n for (int i = 1;i <= n;i++) {\n int m;\n cin >> m;\n vector<pair<int, int>> tmp(m);\n for (int j = 0;j < m;j++) {\n cin >> tmp[j].first >> tmp[j].second;\n }\n for (int j = 0;j < m;j++) {\n int k = (j + 1) % m;\n if (tmp[j].first == tmp[k].first) continue;\n p.push_back({ i, tmp[j], tmp[k] });\n }\n }\n for (int i = 0; i < p.size(); i++) {\n if (p[i].p.first > p[i].q.first) swap(p[i].p, p[i].q);\n events.push_back({ p[i].p.first, p[i], true });\n events.push_back({ p[i].q.first, p[i], false });\n }\n\n sort(events.begin(), events.end());\n\n memset(fa,-1,sizeof fa);\n set<Segment> s;\n\n for (auto& [x, l, f] : events) {\n sweep_line_x = x;\n int id = l.id;\n if (l.up_side and fa[id] == -1) {\n auto it = s.upper_bound(l);\n if (it != s.end()) {\n Segment par = (*it);\n if (par.up_side) fa[id] = par.id;\n else fa[id] = fa[par.id];\n }\n else {\n fa[id] = 0;\n }\n g[fa[id]].push_back(id);\n g[id].push_back(fa[id]);\n }\n if (f == 1)s.insert(l);\n else s.erase(l);\n }\n\n dfs(0, 0, -1);\n int st=t;\n dfs(t, 0, -1);\n int ans = d;\n\n vector<int> dis(maxn,INF);\n queue<int> q;\n dis[0] = 0;\n q.push(0);\n while (!q.empty()){\n int v = q.front();\n q.pop();\n for (auto const& to : g[v]){\n if (dis[to] > dis[v] + 1){\n dis[to] = dis[v] + 1;\n q.push(to);\n }\n }\n }\n ans += max(dis[t], dis[st]);\n cout << ans << '\\n';\n\n return 0;\n}<\/code><\/pre>\n\u603b\u7ed3<\/h2>\n