結果
| 問題 |
No.114 遠い未来
|
| コンテスト | |
| ユーザー |
 ruthen71
|
| 提出日時 | 2024-07-28 03:05:02 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 3,298 ms / 5,000 ms |
| コード長 | 9,612 bytes |
| コンパイル時間 | 2,791 ms |
| コンパイル使用メモリ | 149,204 KB |
| 実行使用メモリ | 13,312 KB |
| 最終ジャッジ日時 | 2024-07-28 03:05:18 |
| 合計ジャッジ時間 | 13,702 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 25 |
ソースコード
// #include <iostream>
//
// #include "graph/read_graph.hpp"
// #include "graph/minimum_steriner_tree.hpp"
//
// int main() {
// int N, M, T;
// std::cin >> N >> M >> T;
// auto g = read_graph<long long>(N, M, true);
// std::vector<int> terminals(T);
// for (int i = 0; i < T; i++) {
// std::cin >> terminals[i];
// terminals[i]--;
// }
// if (T <= 15) {
// auto dp = minimum_steiner_tree(g, terminals, 1'000'000'000'000'000'000LL);
// std::cout << dp.back()[terminals[0]] << '\n';
// } else {
// std::cout << minimum_steiner_tree_mst(g, terminals, 1'000'000'000'000'000'000LL) << '\n';
// }
// return 0;
// }
#include <iostream>
#include <string>
#include <vector>
template <class T> struct Edge {
int from, to;
T cost;
int id;
Edge() = default;
Edge(int from, int to, T cost = 1, int id = -1) : from(from), to(to), cost(cost), id(id) {}
friend std::ostream& operator<<(std::ostream& os, const Edge<T>& e) {
// output format: "{ id : from -> to, cost }"
return os << "{ " << e.id << " : " << e.from << " -> " << e.to << ", " << e.cost << " }";
}
};
template <class T> using Edges = std::vector<Edge<T>>;
template <class T> using Graph = std::vector<std::vector<Edge<T>>>;
template <class T> Graph<T> read_graph(const int n, const int m, const bool weight = false, const bool directed = false, const int offset = 1) {
Graph<T> g(n);
for (int i = 0; i < m; i++) {
int a, b;
std::cin >> a >> b;
a -= offset, b -= offset;
if (weight) {
T c;
std::cin >> c;
if (!directed) g[b].push_back(Edge(b, a, c, i));
g[a].push_back(Edge(a, b, c, i));
} else {
// c = 1
if (!directed) g[b].push_back(Edge(b, a, T(1), i));
g[a].push_back(Edge(a, b, T(1), i));
}
}
return g;
}
template <class T> Graph<T> read_parent(const int n, const bool weight = false, const bool directed = false, const int offset = 1) {
Graph<T> g(n);
for (int i = 1; i < n; i++) {
int p;
std::cin >> p;
p -= offset;
if (weight) {
T c;
std::cin >> c;
if (!directed) g[i].push_back(Edge(i, p, c, i - 1));
g[p].push_back(Edge(p, i, c, i - 1));
} else {
// c = 1
if (!directed) g[i].push_back(Edge(i, p, T(1), i - 1));
g[p].push_back(Edge(p, i, T(1), i - 1));
}
}
return g;
}
std::tuple<Graph<int>, std::vector<std::vector<int>>, std::vector<std::pair<int, int>>> read_grid(const int h, const int w, std::string rel = ".#") {
std::vector<std::string> s(h);
std::vector id(h, std::vector<int>(w, -1));
std::vector<std::pair<int, int>> loc;
int n = 0;
for (int i = 0; i < h; i++) {
std::cin >> s[i];
for (int j = 0; j < w; j++) {
if (s[i][j] == rel[1]) {
id[i][j] = n++;
loc.emplace_back(i, j);
}
}
}
int m = 0;
Graph<int> g(n);
for (int i = 0; i < h; i++) {
for (int j = 0; j < w; j++) {
if (s[i][j] == rel[1]) {
if (i + 1 < h and s[i + 1][j] == rel[1]) {
g[id[i][j]].push_back(Edge(id[i][j], id[i + 1][j], 1, m));
g[id[i + 1][j]].push_back(Edge(id[i + 1][j], id[i][j], 1, m++));
}
if (j + 1 < w and s[i][j + 1] == rel[1]) {
g[id[i][j]].push_back(Edge(id[i][j], id[i][j + 1], 1, m));
g[id[i][j + 1]].push_back(Edge(id[i][j + 1], id[i][j], 1, m++));
}
}
}
}
return {g, id, loc};
}
#include <queue>
#include <set>
#include <algorithm>
#include <cassert>
struct UnionFind {
int n;
std::vector<int> parents;
UnionFind() {}
UnionFind(int n) : n(n), parents(n, -1) {}
int leader(int x) { return parents[x] < 0 ? x : parents[x] = leader(parents[x]); }
bool merge(int x, int y) {
x = leader(x), y = leader(y);
if (x == y) return false;
if (parents[x] > parents[y]) std::swap(x, y);
parents[x] += parents[y];
parents[y] = x;
return true;
}
bool same(int x, int y) { return leader(x) == leader(y); }
int size(int x) { return -parents[leader(x)]; }
std::vector<std::vector<int>> groups() {
std::vector<int> leader_buf(n), group_size(n);
for (int i = 0; i < n; i++) {
leader_buf[i] = leader(i);
group_size[leader_buf[i]]++;
}
std::vector<std::vector<int>> result(n);
for (int i = 0; i < n; i++) {
result[i].reserve(group_size[i]);
}
for (int i = 0; i < n; i++) {
result[leader_buf[i]].push_back(i);
}
result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }), result.end());
return result;
}
void init(int n) { parents.assign(n, -1); } // reset
};
// minimum steiner tree
// O(3 ^ k n + 2 ^ k m \log m) (n = |V|, m = |E|, k = |terminals|)
// https://www.slideshare.net/wata_orz/ss-12131479#50
// https://kopricky.github.io/code/Academic/steiner_tree.html
// https://atcoder.jp/contests/abc364/editorial/10547
template <class T> std::vector<std::vector<T>> minimum_steiner_tree(Graph<T>& g, std::vector<int>& terminals, const T inf) {
const int n = (int)(g.size());
const int k = (int)(terminals.size());
const int k2 = 1 << k;
// dp[bit][v] = ターミナルの部分集合が bit (0 ~ k - 1 に圧縮), 加えて頂点 v も含まれる最小シュタイナー木
std::vector dp(k2, std::vector<T>(n, inf));
for (int i = 0; i < k; i++) dp[1 << i][terminals[i]] = T(0);
for (int bit = 0; bit < (1 << k); bit++) {
// dp[bit][v] = min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v])
// 通常の実装
// for (int sub = bit; sub > 0; sub = (sub - 1) & bit) {
// 定数倍高速化
// bit の中で 1 要素だけ sub と bit ^ sub のどちらに属するか決める
int bit2 = bit ^ (bit & -bit);
for (int sub = bit2; sub > 0; sub = (sub - 1) & bit2) {
for (int v = 0; v < n; v++) {
dp[bit][v] = std::min(dp[bit][v], dp[sub][v] + dp[bit ^ sub][v]);
}
}
// dp[bit][v] = min(dp[bit][v], dp[bit][u] + cost(u, v))
using tp = std::pair<T, int>;
std::priority_queue<tp, std::vector<tp>, std::greater<tp>> que;
for (int u = 0; u < n; u++) que.emplace(dp[bit][u], u);
while (!que.empty()) {
auto [d, u] = que.top();
que.pop();
if (dp[bit][u] != d) continue;
for (auto&& e : g[u]) {
if (dp[bit][e.to] > d + e.cost) {
dp[bit][e.to] = d + e.cost;
que.emplace(dp[bit][e.to], e.to);
}
}
}
}
// dp[k2 - 1][i] = ターミナルと頂点 i を含む最小シュタイナー木
// dp[k2 - 1][terminals[0]] が基本的な答えになる
return dp;
}
// O(2 ^ {n - k} m \log n)
// https://yukicoder.me/problems/no/114/editorial
// n - k <= 20, n <= 64
template <class T> T minimum_steiner_tree_mst(Graph<T>& g, std::vector<int>& terminals, const T inf) {
const int n = (int)(g.size());
const int k = (int)(terminals.size());
assert(n <= 64);
// ターミナルに含まれない点集合 (others) を取得
unsigned long long st = 0;
for (int i = 0; i < k; i++) st |= 1LL << terminals[i];
std::vector<int> others;
for (int i = 0; i < n; i++)
if ((st >> i & 1) == 0) others.emplace_back(i);
// 辺のリスト
std::vector<Edge<T>> edges;
for (int v = 0; v < n; v++) {
for (auto&& e : g[v]) {
edges.push_back(e);
}
}
std::sort(edges.begin(), edges.end(), [&](Edge<T>& a, Edge<T>& b) -> bool { return a.cost < b.cost; });
// ターミナル + others の組合せを全列挙 -> Minimum Spanning Tree を求める
T ans = inf;
for (int bit = 0; bit < (1 << (n - k)); bit++) {
// 使う頂点集合
unsigned long long subv = st;
for (int i = 0; i < n - k; i++) {
if (bit >> i & 1) {
subv |= 1LL << others[i];
}
}
// Minimum Spanning Tree を計算
UnionFind uf(n);
T cur = 0;
int connected = 0;
for (auto&& e : edges) {
// subv に対する g の誘導部分グラフに含まれる辺のみ試す
if (!(subv >> e.from & 1) or !(subv >> e.to & 1)) continue;
if (!uf.same(e.from, e.to)) {
uf.merge(e.from, e.to);
cur += e.cost;
connected++;
}
}
// 全域木が作れたか判定
if (connected + 1 == k + __builtin_popcount(bit)) ans = std::min(ans, cur);
}
return ans;
}
int main() {
int N, M, T;
std::cin >> N >> M >> T;
auto g = read_graph<long long>(N, M, true);
std::vector<int> terminals(T);
for (int i = 0; i < T; i++) {
std::cin >> terminals[i];
terminals[i]--;
}
if (T <= 15) {
auto dp = minimum_steiner_tree(g, terminals, 1'000'000'000'000'000'000LL);
std::cout << dp.back()[terminals[0]] << '\n';
} else {
std::cout << minimum_steiner_tree_mst(g, terminals, 1'000'000'000'000'000'000LL) << '\n';
}
return 0;
}