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#include "graph/manhattan_mst.hpp"
$2$ 次元座標上の $N$ 点を与えて、マンハッタン距離におけるMSTを構築する。返り値 $(x, es)$ について、 $x$ は木の重みの総和、 $es$ には使用した辺 $(u_i, v_i)$ が格納されている。
#pragma once #include <algorithm> #include <map> #include <numeric> #include <tuple> #include <utility> #include <vector> #include "../data_structure/dsu.hpp" #include "../graph/base.hpp" namespace ebi { template <class T> std::pair<T, std::vector<std::pair<int, int>>> manhattan_mst( std::vector<T> &xs, std::vector<T> &ys) { const int n = xs.size(); std::vector<int> ids(n); std::iota(ids.begin(), ids.end(), 0); std::vector<std::tuple<T, int, int>> edges; for (int s = 0; s < 2; s++) { for (int t = 0; t < 2; t++) { std::sort(ids.begin(), ids.end(), [&](int i, int j) -> bool { return xs[i] + ys[i] < xs[j] + ys[j]; }); std::map<T, int> sweep; for (int i : ids) { for (auto itr = sweep.lower_bound(-ys[i]); itr != sweep.end(); itr = sweep.erase(itr)) { int j = itr->second; if (xs[i] - xs[j] < ys[i] - ys[j]) break; edges.emplace_back( std::abs(ys[i] - ys[j]) + std::abs(xs[i] - xs[j]), i, j); } sweep[-ys[i]] = i; } std::swap(xs, ys); } for (auto &x : xs) x = -x; } std::sort(edges.begin(), edges.end(), [&](auto a, auto b) -> bool { return std::get<0>(a) < std::get<0>(b); }); dsu uf(n); std::vector<std::pair<int, int>> es; T sum = 0; for (auto [cost, i, j] : edges) { if (uf.same(i, j)) continue; uf.merge(i, j); sum += cost; es.emplace_back(i, j); } return {sum, es}; } } // namespace ebi
#line 2 "graph/manhattan_mst.hpp" #include <algorithm> #include <map> #include <numeric> #include <tuple> #include <utility> #include <vector> #line 2 "data_structure/dsu.hpp" #line 4 "data_structure/dsu.hpp" namespace ebi { struct dsu { private: std::vector<int> par; public: dsu(int n = 0) : par(n, -1) {} bool same(int x, int y) { return leader(x) == leader(y); } bool merge(int x, int y) { x = leader(x); y = leader(y); if (x == y) return false; if (par[x] > par[y]) std::swap(x, y); par[x] += par[y]; par[y] = x; return true; } int leader(int x) { if (par[x] < 0) return x; else return par[x] = leader(par[x]); } int size(int x) { return -par[leader(x)]; } int count_group() { int c = 0; for (int i = 0; i < int(par.size()); i++) { if (par[i] < 0) c++; } return c; } std::vector<std::vector<int>> groups() { int n = par.size(); std::vector result(n, std::vector<int>()); for (int i = 0; i < n; i++) { result[leader(i)].emplace_back(i); } result.erase(std::remove_if(result.begin(), result.end(), [](const std::vector<int> &v) -> bool { return v.empty(); }), result.end()); return result; } void clear() { for (int i = 0; i < int(par.size()); i++) { par[i] = -1; } } }; } // namespace ebi #line 2 "graph/base.hpp" #include <cassert> #include <iostream> #include <ranges> #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 12 "graph/manhattan_mst.hpp" namespace ebi { template <class T> std::pair<T, std::vector<std::pair<int, int>>> manhattan_mst( std::vector<T> &xs, std::vector<T> &ys) { const int n = xs.size(); std::vector<int> ids(n); std::iota(ids.begin(), ids.end(), 0); std::vector<std::tuple<T, int, int>> edges; for (int s = 0; s < 2; s++) { for (int t = 0; t < 2; t++) { std::sort(ids.begin(), ids.end(), [&](int i, int j) -> bool { return xs[i] + ys[i] < xs[j] + ys[j]; }); std::map<T, int> sweep; for (int i : ids) { for (auto itr = sweep.lower_bound(-ys[i]); itr != sweep.end(); itr = sweep.erase(itr)) { int j = itr->second; if (xs[i] - xs[j] < ys[i] - ys[j]) break; edges.emplace_back( std::abs(ys[i] - ys[j]) + std::abs(xs[i] - xs[j]), i, j); } sweep[-ys[i]] = i; } std::swap(xs, ys); } for (auto &x : xs) x = -x; } std::sort(edges.begin(), edges.end(), [&](auto a, auto b) -> bool { return std::get<0>(a) < std::get<0>(b); }); dsu uf(n); std::vector<std::pair<int, int>> es; T sum = 0; for (auto [cost, i, j] : edges) { if (uf.same(i, j)) continue; uf.merge(i, j); sum += cost; es.emplace_back(i, j); } return {sum, es}; } } // namespace ebi