#define PROBLEM "https://judge.yosupo.jp/problem/minimum_steiner_tree"
#include "../../graph/minimum_steiner_tree.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"
#include <limits>
#include <numeric>
#include <optional>
#include <utility>
#include <vector>
namespace {
struct NaiveDsu {
std::vector<int> parent;
explicit NaiveDsu(int n) : parent(n) {
std::iota(parent.begin(), parent.end(), 0);
}
int leader(int v) {
if (parent[v] == v) return v;
return parent[v] = leader(parent[v]);
}
void merge(int u, int v) {
u = leader(u);
v = leader(v);
if (u != v) parent[u] = v;
}
};
template <class T>
std::pair<std::optional<T>, std::optional<int>> naive(
const m1une::graph::Graph<T>& graph,
std::vector<int> terminals
) {
std::sort(terminals.begin(), terminals.end());
terminals.erase(std::unique(terminals.begin(), terminals.end()), terminals.end());
if (terminals.empty()) return {T(0), 0};
const auto edges = graph.edges();
assert(edges.size() < std::numeric_limits<std::uint64_t>::digits);
std::optional<T> weighted;
std::optional<int> unweighted;
for (std::uint64_t mask = 0; mask < (std::uint64_t(1) << edges.size()); mask++) {
NaiveDsu dsu(graph.size());
T cost = T(0);
int count = 0;
for (int i = 0; i < int(edges.size()); i++) {
if ((mask >> i & 1) == 0) continue;
dsu.merge(edges[i].from, edges[i].to);
cost += edges[i].cost;
count++;
}
bool connected = true;
for (int v : terminals) {
if (dsu.leader(v) != dsu.leader(terminals[0])) connected = false;
}
if (!connected) continue;
if (!weighted || cost < *weighted) weighted = cost;
if (!unweighted || count < *unweighted) unweighted = count;
}
return {weighted, unweighted};
}
template <class Cost, class GraphCost, class EdgeCost>
std::optional<Cost> naive_with_vertex_cost(
const m1une::graph::Graph<GraphCost>& graph,
std::vector<int> terminals,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost
) {
std::sort(terminals.begin(), terminals.end());
terminals.erase(std::unique(terminals.begin(), terminals.end()), terminals.end());
if (terminals.empty()) return Cost(0);
const auto edges = graph.edges();
std::optional<Cost> answer;
for (std::uint64_t mask = 0; mask < (std::uint64_t(1) << edges.size()); mask++) {
NaiveDsu dsu(graph.size());
std::vector<char> used(graph.size(), false);
for (int terminal : terminals) used[terminal] = true;
Cost cost = Cost(0);
for (int i = 0; i < int(edges.size()); i++) {
if ((mask >> i & 1) == 0) continue;
dsu.merge(edges[i].from, edges[i].to);
used[edges[i].from] = true;
used[edges[i].to] = true;
cost += edge_cost(edges[i]);
}
for (int v = 0; v < graph.size(); v++) {
if (used[v]) cost += vertex_cost[v];
}
bool connected = true;
for (int v : terminals) {
if (dsu.leader(v) != dsu.leader(terminals[0])) connected = false;
}
if (connected && (!answer || cost < *answer)) answer = cost;
}
return answer;
}
template <class Cost, class GraphCost, class EdgeCost>
void validate_built_tree(
const m1une::graph::Graph<GraphCost>& graph,
const m1une::graph::MinimumSteinerTreeResult<Cost>& result,
const std::vector<int>& terminals,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost
) {
assert(std::is_sorted(result.edge_ids.begin(), result.edge_ids.end()));
assert(std::adjacent_find(result.edge_ids.begin(), result.edge_ids.end()) == result.edge_ids.end());
assert(std::is_sorted(result.vertices.begin(), result.vertices.end()));
assert(std::adjacent_find(result.vertices.begin(), result.vertices.end()) == result.vertices.end());
std::vector<m1une::graph::Edge<GraphCost>> edge_by_id(graph.edge_count());
for (const auto& edge : graph.edges()) edge_by_id[edge.id] = edge;
NaiveDsu dsu(graph.size());
Cost cost = Cost(0);
for (int id : result.edge_ids) {
assert(0 <= id && id < graph.edge_count());
assert(graph.is_edge_alive(id));
const auto& edge = edge_by_id[id];
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), edge.from));
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), edge.to));
assert(dsu.leader(edge.from) != dsu.leader(edge.to));
dsu.merge(edge.from, edge.to);
cost += edge_cost(edge);
}
for (int v : result.vertices) cost += vertex_cost[v];
for (int terminal : terminals) {
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), terminal));
}
if (result.vertices.empty()) {
assert(terminals.empty() && result.edge_ids.empty());
} else {
assert(result.edge_ids.size() + 1 == result.vertices.size());
for (int v : result.vertices) assert(dsu.leader(v) == dsu.leader(result.vertices[0]));
}
assert(cost == result.cost);
}
void test_examples_and_failures() {
m1une::graph::Graph<long long> graph(6);
graph.add_edge(0, 1, 4);
graph.add_edge(1, 2, 1);
graph.add_edge(1, 3, 2);
graph.add_edge(3, 4, 3);
graph.add_edge(0, 4, 20);
auto weighted = m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 2, 4});
auto unweighted = m1une::graph::minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4}
);
assert(weighted && *weighted == 10);
assert(unweighted && *unweighted == 3);
assert(!m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 5}));
assert(!m1une::graph::minimum_steiner_tree_unweighted(graph, std::vector<int>{0, 5}));
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>{2, 2}) == 0);
assert(*m1une::graph::minimum_steiner_tree_unweighted(graph, std::vector<int>()) == 0);
const std::vector<long long> vertex_cost = {2, 3, 7, 5, 11, 13};
auto vertex_weighted = m1une::graph::minimum_steiner_tree(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
auto vertex_weighted_unit_edges = m1une::graph::minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(vertex_weighted && *vertex_weighted == 38);
assert(vertex_weighted_unit_edges && *vertex_weighted_unit_edges == 26);
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>{2}, vertex_cost) == 7);
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>(), vertex_cost) == 0);
m1une::graph::Graph<int> unit_graph(2);
unit_graph.add_edge(0, 1, 100);
const std::vector<long long> large_vertex_cost = {3'000'000'000LL, 4'000'000'000LL};
auto independent_cost_type = m1une::graph::minimum_steiner_tree_unweighted(
unit_graph,
std::vector<int>{0, 1},
large_vertex_cost
);
assert(independent_cost_type && *independent_cost_type == 7'000'000'001LL);
auto built = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(built && built->cost == 38);
validate_built_tree(
graph,
*built,
std::vector<int>{0, 2, 4},
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
auto built_unit = m1une::graph::build_minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(built_unit && built_unit->cost == 26);
validate_built_tree(
graph,
*built_unit,
std::vector<int>{0, 2, 4},
vertex_cost,
[](const auto&) { return 1LL; }
);
m1une::graph::Graph<long long> zero_cycle(3);
zero_cycle.add_edge(0, 1, 0);
zero_cycle.add_edge(1, 2, 0);
zero_cycle.add_edge(2, 0, 0);
auto zero_cycle_tree = m1une::graph::build_minimum_steiner_tree(
zero_cycle,
std::vector<int>{0, 1, 2}
);
assert(zero_cycle_tree && zero_cycle_tree->cost == 0);
assert(zero_cycle_tree->edge_ids.size() == 2);
validate_built_tree(
zero_cycle,
*zero_cycle_tree,
std::vector<int>{0, 1, 2},
std::vector<long long>(3, 0),
[](const auto& edge) { return edge.cost; }
);
auto single_vertex_tree = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>{2},
vertex_cost
);
assert(single_vertex_tree && single_vertex_tree->vertices == std::vector<int>{2});
assert(single_vertex_tree->edge_ids.empty() && single_vertex_tree->cost == 7);
auto empty_tree = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>(),
vertex_cost
);
assert(empty_tree && empty_tree->vertices.empty() && empty_tree->edge_ids.empty());
int removed = graph.add_edge(4, 5, 1);
graph.erase_edge(removed);
assert(!m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 5}));
assert(!m1une::graph::build_minimum_steiner_tree(graph, std::vector<int>{0, 5}));
}
void test_randomized() {
std::uint64_t state = 0x243f6a8885a308d3ULL;
auto random = [&state]() {
state ^= state << 7;
state ^= state >> 9;
return state;
};
for (int trial = 0; trial < 100; trial++) {
const int n = 1 + int(random() % 7);
std::vector<std::pair<int, int>> pairs;
for (int u = 0; u < n; u++) {
for (int v = u + 1; v < n; v++) pairs.emplace_back(u, v);
}
for (int i = int(pairs.size()) - 1; i > 0; i--) {
std::swap(pairs[i], pairs[random() % (i + 1)]);
}
m1une::graph::Graph<long long> graph(n);
const int m = int(random() % (std::min<int>(10, pairs.size()) + 1));
for (int i = 0; i < m; i++) {
graph.add_edge(pairs[i].first, pairs[i].second, random() % 10);
}
if (m > 0 && trial % 7 == 0) graph.erase_edge(int(random() % m));
std::vector<int> terminals;
for (int v = 0; v < n; v++) {
if (random() % 3 == 0) terminals.push_back(v);
}
if (!terminals.empty() && trial % 5 == 0) terminals.push_back(terminals.back());
std::vector<long long> vertex_cost(n);
for (long long& cost : vertex_cost) cost = random() % 8;
auto expected = naive(graph, terminals);
auto weighted = m1une::graph::minimum_steiner_tree(graph, terminals);
auto unweighted = m1une::graph::minimum_steiner_tree_unweighted(graph, terminals);
assert(weighted == expected.first);
assert(unweighted == expected.second);
auto expected_vertex_weighted = naive_with_vertex_cost<long long>(
graph,
terminals,
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
auto expected_vertex_weighted_unit_edges = naive_with_vertex_cost<long long>(
graph,
terminals,
vertex_cost,
[](const auto&) { return 1LL; }
);
auto vertex_weighted = m1une::graph::minimum_steiner_tree(
graph,
terminals,
vertex_cost
);
auto vertex_weighted_unit_edges = m1une::graph::minimum_steiner_tree_unweighted(
graph,
terminals,
vertex_cost
);
assert(vertex_weighted == expected_vertex_weighted);
assert(vertex_weighted_unit_edges == expected_vertex_weighted_unit_edges);
const std::vector<long long> zero_vertex_cost(n, 0);
auto built_weighted = m1une::graph::build_minimum_steiner_tree(graph, terminals);
auto built_unweighted = m1une::graph::build_minimum_steiner_tree_unweighted(
graph,
terminals
);
auto built_vertex_weighted = m1une::graph::build_minimum_steiner_tree(
graph,
terminals,
vertex_cost
);
auto built_vertex_weighted_unit_edges =
m1une::graph::build_minimum_steiner_tree_unweighted(graph, terminals, vertex_cost);
assert(bool(built_weighted) == bool(expected.first));
assert(bool(built_unweighted) == bool(expected.second));
assert(bool(built_vertex_weighted) == bool(expected_vertex_weighted));
assert(
bool(built_vertex_weighted_unit_edges) == bool(expected_vertex_weighted_unit_edges)
);
if (built_weighted) {
assert(built_weighted->cost == *expected.first);
validate_built_tree(
graph,
*built_weighted,
terminals,
zero_vertex_cost,
[](const auto& edge) { return edge.cost; }
);
}
if (built_unweighted) {
assert(built_unweighted->cost == *expected.second);
validate_built_tree(
graph,
*built_unweighted,
terminals,
std::vector<int>(n, 0),
[](const auto&) { return 1; }
);
}
if (built_vertex_weighted) {
assert(built_vertex_weighted->cost == *expected_vertex_weighted);
validate_built_tree(
graph,
*built_vertex_weighted,
terminals,
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
}
if (built_vertex_weighted_unit_edges) {
assert(built_vertex_weighted_unit_edges->cost == *expected_vertex_weighted_unit_edges);
validate_built_tree(
graph,
*built_vertex_weighted_unit_edges,
terminals,
vertex_cost,
[](const auto&) { return 1LL; }
);
}
}
}
} // namespace
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_examples_and_failures();
test_randomized();
int vertex_count, edge_count;
fast_input >> vertex_count >> edge_count;
m1une::graph::Graph<long long> graph(vertex_count);
for (int edge = 0; edge < edge_count; edge++) {
int first, second;
long long weight;
fast_input >> first >> second >> weight;
graph.add_edge(first, second, weight);
}
int terminal_count;
fast_input >> terminal_count;
std::vector<int> terminals(terminal_count);
for (int& terminal : terminals) fast_input >> terminal;
auto result = m1une::graph::build_minimum_steiner_tree(graph, terminals);
assert(result.has_value());
fast_output << result->cost << ' ' << result->edge_ids.size() << '\n';
for (int index = 0; index < int(result->edge_ids.size()); index++) {
if (index != 0) fast_output << ' ';
fast_output << result->edge_ids[index];
}
fast_output << '\n';
}
#line 1 "verify/graph/minimum_steiner_tree.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/minimum_steiner_tree"
#line 1 "graph/minimum_steiner_tree.hpp"
#include <algorithm>
#include <bit>
#include <cassert>
#include <cstddef>
#include <functional>
#include <limits>
#include <optional>
#include <queue>
#include <type_traits>
#include <utility>
#include <vector>
#line 1 "graph/graph.hpp"
#include <array>
#line 8 "graph/graph.hpp"
namespace m1une {
namespace graph {
template <class T = int>
struct Edge {
using cost_type = T;
int from;
int to;
T cost;
int id;
bool alive;
Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
: from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}
int other(int v) const {
assert(v == from || v == to);
return from ^ to ^ v;
}
};
template <class T = int>
struct Graph {
using edge_type = Edge<T>;
using cost_type = T;
private:
struct EdgePositions {
std::array<std::pair<int, int>, 2> value{};
int size = 0;
void push_back(std::pair<int, int> position) {
assert(size < 2);
value[size++] = position;
}
};
int _n;
int _edge_count;
std::vector<std::vector<edge_type>> _g;
std::vector<EdgePositions> _edge_positions;
public:
Graph() : _n(0), _edge_count(0) {}
explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
assert(0 <= n);
}
int size() const {
return _n;
}
bool empty() const {
return _n == 0;
}
int edge_count() const {
return _edge_count;
}
int add_vertex() {
_g.emplace_back();
return _n++;
}
int add_directed_edge(int from, int to, T cost = T(1)) {
assert(0 <= from && from < _n);
assert(0 <= to && to < _n);
int id = _edge_count++;
int idx = int(_g[from].size());
_g[from].push_back(edge_type(from, to, cost, id));
_edge_positions.emplace_back();
_edge_positions.back().push_back({from, idx});
return id;
}
int add_edge(int u, int v, T cost = T(1)) {
assert(0 <= u && u < _n);
assert(0 <= v && v < _n);
int id = _edge_count++;
int u_idx = int(_g[u].size());
_g[u].push_back(edge_type(u, v, cost, id));
int v_idx = int(_g[v].size());
_g[v].push_back(edge_type(v, u, cost, id));
_edge_positions.emplace_back();
_edge_positions.back().push_back({u, u_idx});
_edge_positions.back().push_back({v, v_idx});
return id;
}
void set_edge_alive(int id, bool alive) {
assert(0 <= id && id < _edge_count);
for (int i = 0; i < _edge_positions[id].size; ++i) {
auto [v, idx] = _edge_positions[id].value[i];
_g[v][idx].alive = alive;
}
}
void erase_edge(int id) {
set_edge_alive(id, false);
}
void revive_edge(int id) {
set_edge_alive(id, true);
}
bool is_edge_alive(int id) const {
assert(0 <= id && id < _edge_count);
assert(_edge_positions[id].size != 0);
auto [v, idx] = _edge_positions[id].value[0];
return _g[v][idx].alive;
}
const std::vector<edge_type>& operator[](int v) const {
assert(0 <= v && v < _n);
return _g[v];
}
std::vector<edge_type>& operator[](int v) {
assert(0 <= v && v < _n);
return _g[v];
}
const std::vector<std::vector<edge_type>>& adjacency() const {
return _g;
}
std::vector<std::vector<edge_type>>& adjacency() {
return _g;
}
std::vector<edge_type> edges(bool include_inactive = false) const {
std::vector<edge_type> result;
result.reserve(_edge_count);
std::vector<char> used(_edge_count, false);
for (int v = 0; v < _n; v++) {
for (const auto& e : _g[v]) {
if (!include_inactive && !e.alive) continue;
if (0 <= e.id && e.id < _edge_count) {
if (used[e.id]) continue;
used[e.id] = true;
}
result.push_back(e);
}
}
return result;
}
Graph reversed() const {
Graph result(_n);
result._edge_count = _edge_count;
result._edge_positions.assign(_edge_count, {});
for (int v = 0; v < _n; v++) {
for (const auto& e : _g[v]) {
int idx = int(result._g[e.to].size());
result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
}
}
return result;
}
};
} // namespace graph
} // namespace m1une
#line 17 "graph/minimum_steiner_tree.hpp"
namespace m1une {
namespace graph {
template <class Cost>
struct MinimumSteinerTreeResult {
Cost cost;
std::vector<int> edge_ids;
std::vector<int> vertices;
};
namespace internal {
inline std::vector<int> steiner_terminals(int n, std::vector<int> terminals) {
for (int v : terminals) assert(0 <= v && v < n);
std::sort(terminals.begin(), terminals.end());
terminals.erase(std::unique(terminals.begin(), terminals.end()), terminals.end());
assert(terminals.size() < std::numeric_limits<std::size_t>::digits);
return terminals;
}
template <class Cost>
struct MinimumSteinerTreeDp {
Cost cost;
Cost inf;
std::size_t states;
std::size_t width;
std::vector<Cost> dp;
std::vector<int> terminals;
};
template <class Cost, class GraphCost, class EdgeCost>
std::optional<MinimumSteinerTreeDp<Cost>> minimum_steiner_tree_dp(
const Graph<GraphCost>& g,
std::vector<int> terminals,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost,
Cost inf
) {
const int n = g.size();
assert(vertex_cost.size() == std::size_t(n));
for (Cost cost : vertex_cost) assert(Cost(0) <= cost);
terminals = steiner_terminals(n, std::move(terminals));
const int k = int(terminals.size());
if (k == 0) return MinimumSteinerTreeDp<Cost>{Cost(0), inf, 1, std::size_t(n), {}, {}};
assert(Cost(0) < inf);
for (int v = 0; v < n; v++) {
for (const auto& edge : g[v]) {
if (edge.alive) assert(Cost(0) <= edge_cost(edge));
}
}
const std::size_t states = std::size_t(1) << k;
const std::size_t width = std::size_t(n);
assert(width <= std::numeric_limits<std::size_t>::max() / states);
std::vector<Cost> dp(states * width, inf);
for (int i = 0; i < k; i++) {
const int terminal = terminals[i];
if (vertex_cost[terminal] < inf) {
dp[(std::size_t(1) << i) * width + std::size_t(terminal)] = vertex_cost[terminal];
}
}
using QueueEntry = std::pair<Cost, int>;
for (std::size_t mask = 1; mask < states; mask++) {
const std::size_t mask_offset = mask * width;
for (std::size_t sub = (mask - 1) & mask; sub != 0; sub = (sub - 1) & mask) {
const std::size_t other = mask ^ sub;
if (sub > other) continue;
const std::size_t sub_offset = sub * width;
const std::size_t other_offset = other * width;
for (int v = 0; v < n; v++) {
const std::size_t vertex = std::size_t(v);
const Cost left = dp[sub_offset + vertex];
const Cost right = dp[other_offset + vertex];
if (left == inf || right == inf) continue;
assert(vertex_cost[v] <= right);
const Cost extra = right - vertex_cost[v];
if (left > inf - extra) continue;
const Cost candidate = left + extra;
Cost& current = dp[mask_offset + vertex];
if (candidate < current) current = candidate;
}
}
std::priority_queue<QueueEntry, std::vector<QueueEntry>, std::greater<QueueEntry>> queue;
for (int v = 0; v < n; v++) {
const Cost distance = dp[mask_offset + std::size_t(v)];
if (distance != inf) queue.emplace(distance, v);
}
while (!queue.empty()) {
auto [distance, v] = queue.top();
queue.pop();
if (distance != dp[mask_offset + std::size_t(v)]) continue;
for (const auto& edge : g[v]) {
if (!edge.alive) continue;
const Cost cost = edge_cost(edge);
if (cost >= inf || vertex_cost[edge.to] > inf - cost) continue;
const Cost extra = cost + vertex_cost[edge.to];
if (distance > inf - extra) continue;
const Cost candidate = distance + extra;
Cost& current = dp[mask_offset + std::size_t(edge.to)];
if (current <= candidate) continue;
current = candidate;
queue.emplace(candidate, edge.to);
}
}
}
const auto answer_begin = dp.begin() + (states - 1) * width;
const Cost answer = *std::min_element(answer_begin, dp.end());
if (answer == inf) return std::nullopt;
return MinimumSteinerTreeDp<Cost>{
answer,
inf,
states,
width,
std::move(dp),
std::move(terminals)
};
}
template <class T>
std::optional<MinimumSteinerTreeDp<int>> minimum_steiner_tree_unweighted_dp(
const Graph<T>& g,
std::vector<int> terminals
) {
const int n = g.size();
terminals = steiner_terminals(n, std::move(terminals));
const int k = int(terminals.size());
if (k == 0) return MinimumSteinerTreeDp<int>{0, n, 1, std::size_t(n), {}, {}};
const std::size_t states = std::size_t(1) << k;
const std::size_t width = std::size_t(n);
assert(width <= std::numeric_limits<std::size_t>::max() / states);
const int inf = n;
std::vector<int> dp(states * width, inf);
for (int i = 0; i < k; i++) {
dp[(std::size_t(1) << i) * width + std::size_t(terminals[i])] = 0;
}
for (std::size_t mask = 1; mask < states; mask++) {
const std::size_t mask_offset = mask * width;
for (std::size_t sub = (mask - 1) & mask; sub != 0; sub = (sub - 1) & mask) {
const std::size_t other = mask ^ sub;
if (sub > other) continue;
const std::size_t sub_offset = sub * width;
const std::size_t other_offset = other * width;
for (int v = 0; v < n; v++) {
const std::size_t vertex = std::size_t(v);
const int candidate = dp[sub_offset + vertex] + dp[other_offset + vertex];
int& current = dp[mask_offset + vertex];
if (candidate < current) current = candidate;
}
}
std::vector<int> bucket_head(n, -1);
std::vector<int> entry_vertex;
std::vector<int> entry_next;
entry_vertex.reserve(2 * width);
entry_next.reserve(2 * width);
auto push = [&](int distance, int v) {
entry_vertex.push_back(v);
entry_next.push_back(bucket_head[distance]);
bucket_head[distance] = int(entry_vertex.size()) - 1;
};
for (int v = 0; v < n; v++) {
const int distance = dp[mask_offset + std::size_t(v)];
if (distance != inf) push(distance, v);
}
for (int distance = 0; distance < n; distance++) {
for (int entry = bucket_head[distance]; entry != -1; entry = entry_next[entry]) {
const int v = entry_vertex[entry];
if (dp[mask_offset + std::size_t(v)] != distance) continue;
for (const auto& edge : g[v]) {
if (!edge.alive) continue;
int& current = dp[mask_offset + std::size_t(edge.to)];
if (distance + 1 >= current) continue;
current = distance + 1;
push(current, edge.to);
}
}
}
}
const auto answer_begin = dp.begin() + (states - 1) * width;
const int answer = *std::min_element(answer_begin, dp.end());
if (answer == inf) return std::nullopt;
return MinimumSteinerTreeDp<int>{
answer,
inf,
states,
width,
std::move(dp),
std::move(terminals)
};
}
template <class Cost, class GraphCost, class EdgeCost>
MinimumSteinerTreeResult<Cost> restore_minimum_steiner_tree(
const Graph<GraphCost>& g,
const MinimumSteinerTreeDp<Cost>& data,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost
) {
MinimumSteinerTreeResult<Cost> result;
result.cost = data.cost;
if (data.terminals.empty()) return result;
const int n = g.size();
const std::size_t cells = data.states * data.width;
std::vector<char> state(cells, 0);
std::vector<char> selected_edge(g.edge_count(), false);
std::function<bool(std::size_t, int)> restore = [&](std::size_t mask, int start) {
const std::size_t position = mask * data.width + std::size_t(start);
if (state[position] == 2) return true;
if (state[position] == 1) return false;
state[position] = 1;
std::vector<int> search_parent(n, -2), search_edge(n, -1), stack;
search_parent[start] = -1;
stack.push_back(start);
int seed = -1;
std::size_t seed_split = 0;
while (!stack.empty() && seed == -1) {
const int v = stack.back();
stack.pop_back();
const std::size_t vertex_position = mask * data.width + std::size_t(v);
const Cost current = data.dp[vertex_position];
if (v != start && state[vertex_position] == 2) {
seed = v;
break;
}
if ((mask & (mask - 1)) == 0) {
const int terminal_index = int(std::countr_zero(mask));
if (v == data.terminals[terminal_index] && current == vertex_cost[v]) {
seed = v;
break;
}
}
for (std::size_t sub = (mask - 1) & mask; sub != 0; sub = (sub - 1) & mask) {
const std::size_t other = mask ^ sub;
if (sub > other) continue;
const Cost left = data.dp[sub * data.width + std::size_t(v)];
const Cost right = data.dp[other * data.width + std::size_t(v)];
if (left == data.inf || right == data.inf || right < vertex_cost[v]) continue;
const Cost extra = right - vertex_cost[v];
if (left > data.inf - extra || left + extra != current) continue;
seed = v;
seed_split = sub;
break;
}
if (seed != -1) break;
for (const auto& edge : g[v]) {
if (!edge.alive || search_parent[edge.to] != -2) continue;
const Cost cost = edge_cost(edge);
if (cost >= data.inf || vertex_cost[v] > data.inf - cost) continue;
const Cost extra = cost + vertex_cost[v];
const Cost previous = data.dp[mask * data.width + std::size_t(edge.to)];
if (previous == data.inf || previous > data.inf - extra) continue;
if (previous + extra != current) continue;
search_parent[edge.to] = v;
search_edge[edge.to] = edge.id;
stack.push_back(edge.to);
}
}
if (seed == -1) {
state[position] = 0;
return false;
}
if (seed_split != 0) {
const bool restored_left = restore(seed_split, seed);
const bool restored_right = restore(mask ^ seed_split, seed);
assert(restored_left && restored_right);
if (!restored_left || !restored_right) {
state[position] = 0;
return false;
}
}
for (int v = seed; v != -1; v = search_parent[v]) {
state[mask * data.width + std::size_t(v)] = 2;
if (search_parent[v] == -1) continue;
const int id = search_edge[v];
assert(0 <= id && id < g.edge_count());
selected_edge[id] = true;
}
return true;
};
int root = -1;
const std::size_t full_mask = data.states - 1;
for (int v = 0; v < n; v++) {
if (data.dp[full_mask * data.width + std::size_t(v)] == data.cost) {
root = v;
break;
}
}
assert(root != -1);
const bool restored = restore(full_mask, root);
assert(restored);
(void)restored;
std::vector<Edge<GraphCost>> edge_by_id(g.edge_count());
std::vector<char> has_edge(g.edge_count(), false);
for (const auto& edge : g.edges()) {
edge_by_id[edge.id] = edge;
has_edge[edge.id] = true;
}
std::vector<int> parent(n), component_size(n, 1);
for (int v = 0; v < n; v++) parent[v] = v;
auto leader = [&](auto&& self, int v) -> int {
if (parent[v] == v) return v;
return parent[v] = self(self, parent[v]);
};
std::vector<char> tree_edge(g.edge_count(), false);
for (int id = 0; id < g.edge_count(); id++) {
if (!selected_edge[id]) continue;
assert(has_edge[id]);
const auto& edge = edge_by_id[id];
int u = leader(leader, edge.from);
int v = leader(leader, edge.to);
if (u == v) continue;
if (component_size[u] < component_size[v]) std::swap(u, v);
parent[v] = u;
component_size[u] += component_size[v];
tree_edge[id] = true;
}
std::vector<std::vector<std::pair<int, int>>> tree(n);
std::vector<int> degree(n, 0);
std::vector<char> in_tree(n, false), is_terminal(n, false);
for (int terminal : data.terminals) {
in_tree[terminal] = true;
is_terminal[terminal] = true;
}
for (int id = 0; id < g.edge_count(); id++) {
if (!tree_edge[id]) continue;
const auto& edge = edge_by_id[id];
tree[edge.from].emplace_back(edge.to, id);
tree[edge.to].emplace_back(edge.from, id);
degree[edge.from]++;
degree[edge.to]++;
in_tree[edge.from] = true;
in_tree[edge.to] = true;
}
std::queue<int> leaves;
for (int v = 0; v < n; v++) {
if (in_tree[v] && !is_terminal[v] && degree[v] <= 1) leaves.push(v);
}
std::vector<char> removed_vertex(n, false), removed_edge(g.edge_count(), false);
while (!leaves.empty()) {
const int v = leaves.front();
leaves.pop();
if (removed_vertex[v] || is_terminal[v] || degree[v] > 1) continue;
removed_vertex[v] = true;
for (auto [to, id] : tree[v]) {
if (removed_edge[id]) continue;
removed_edge[id] = true;
degree[v]--;
degree[to]--;
if (!is_terminal[to] && degree[to] <= 1) leaves.push(to);
break;
}
}
Cost restored_cost = Cost(0);
for (int id = 0; id < g.edge_count(); id++) {
if (!tree_edge[id] || removed_edge[id]) continue;
result.edge_ids.push_back(id);
restored_cost += edge_cost(edge_by_id[id]);
}
for (int v = 0; v < n; v++) {
if (!in_tree[v] || removed_vertex[v]) continue;
result.vertices.push_back(v);
restored_cost += vertex_cost[v];
}
if constexpr (std::is_integral_v<Cost>) assert(restored_cost == result.cost);
result.cost = restored_cost;
return result;
}
} // namespace internal
template <class T>
std::optional<T> minimum_steiner_tree(
const Graph<T>& g,
std::vector<int> terminals,
const std::vector<T>& vertex_cost,
T inf = std::numeric_limits<T>::max() / T(4)
) {
auto result = internal::minimum_steiner_tree_dp(
g,
std::move(terminals),
vertex_cost,
[](const Edge<T>& edge) { return edge.cost; },
inf
);
if (!result) return std::nullopt;
return result->cost;
}
template <class T>
std::optional<T> minimum_steiner_tree(
const Graph<T>& g,
std::vector<int> terminals,
T inf = std::numeric_limits<T>::max() / T(4)
) {
return minimum_steiner_tree(g, std::move(terminals), std::vector<T>(g.size(), T(0)), inf);
}
template <class GraphCost, class Cost>
std::optional<Cost> minimum_steiner_tree_unweighted(
const Graph<GraphCost>& g,
std::vector<int> terminals,
const std::vector<Cost>& vertex_cost,
Cost inf = std::numeric_limits<Cost>::max() / Cost(4)
) {
auto result = internal::minimum_steiner_tree_dp(
g,
std::move(terminals),
vertex_cost,
[](const Edge<GraphCost>&) { return Cost(1); },
inf
);
if (!result) return std::nullopt;
return result->cost;
}
template <class T>
std::optional<MinimumSteinerTreeResult<T>> build_minimum_steiner_tree(
const Graph<T>& g,
std::vector<int> terminals,
const std::vector<T>& vertex_cost,
T inf = std::numeric_limits<T>::max() / T(4)
) {
auto data = internal::minimum_steiner_tree_dp(
g,
std::move(terminals),
vertex_cost,
[](const Edge<T>& edge) { return edge.cost; },
inf
);
if (!data) return std::nullopt;
return internal::restore_minimum_steiner_tree(
g,
*data,
vertex_cost,
[](const Edge<T>& edge) { return edge.cost; }
);
}
template <class T>
std::optional<MinimumSteinerTreeResult<T>> build_minimum_steiner_tree(
const Graph<T>& g,
std::vector<int> terminals,
T inf = std::numeric_limits<T>::max() / T(4)
) {
std::vector<T> vertex_cost(g.size(), T(0));
return build_minimum_steiner_tree(g, std::move(terminals), vertex_cost, inf);
}
template <class GraphCost, class Cost>
std::optional<MinimumSteinerTreeResult<Cost>> build_minimum_steiner_tree_unweighted(
const Graph<GraphCost>& g,
std::vector<int> terminals,
const std::vector<Cost>& vertex_cost,
Cost inf = std::numeric_limits<Cost>::max() / Cost(4)
) {
auto data = internal::minimum_steiner_tree_dp(
g,
std::move(terminals),
vertex_cost,
[](const Edge<GraphCost>&) { return Cost(1); },
inf
);
if (!data) return std::nullopt;
return internal::restore_minimum_steiner_tree(
g,
*data,
vertex_cost,
[](const Edge<GraphCost>&) { return Cost(1); }
);
}
template <class T>
std::optional<MinimumSteinerTreeResult<int>> build_minimum_steiner_tree_unweighted(
const Graph<T>& g,
std::vector<int> terminals
) {
auto data = internal::minimum_steiner_tree_unweighted_dp(g, std::move(terminals));
if (!data) return std::nullopt;
std::vector<int> vertex_cost(g.size(), 0);
return internal::restore_minimum_steiner_tree(
g,
*data,
vertex_cost,
[](const Edge<T>&) { return 1; }
);
}
template <class T>
std::optional<int> minimum_steiner_tree_unweighted(
const Graph<T>& g,
std::vector<int> terminals
) {
auto result = internal::minimum_steiner_tree_unweighted_dp(g, std::move(terminals));
if (!result) return std::nullopt;
return result->cost;
}
} // namespace graph
} // namespace m1une
#line 4 "verify/graph/minimum_steiner_tree.test.cpp"
#line 7 "verify/graph/minimum_steiner_tree.test.cpp"
#include <cstdint>
#line 1 "utilities/fast_io.hpp"
#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"
namespace m1une {
namespace utilities {
struct FastOutput;
namespace internal {
// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;
// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};
template <class T>
struct is_range<T, std::void_t<
decltype(std::begin(std::declval<T&>())),
decltype(std::end(std::declval<T&>()))
>> : std::true_type {};
template <class T>
inline constexpr bool is_range_v = is_range<T>::value;
template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));
template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;
template <class T, class = void>
struct range_stored_value {
using type = range_value_t<T>;
};
template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};
template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;
// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};
template <class T, std::size_t N>
struct is_char_array<T[N]>
: std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};
template <class T>
struct is_string_like
: std::bool_constant<
std::is_same_v<std::decay_t<T>, std::string>
|| std::is_same_v<std::decay_t<T>, const char*>
|| std::is_same_v<std::decay_t<T>, char*>
|| is_char_array<std::remove_reference_t<T>>::value
> {};
template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;
// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};
template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
: std::true_type {};
template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;
template <class T, class = void>
struct has_static_mod_raw : std::false_type {};
template <class T>
struct has_static_mod_raw<
T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
: std::true_type {};
template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;
// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
std::is_integral_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>
|| std::is_same_v<std::remove_cv_t<T>, __uint128_t>;
template <class T>
inline constexpr bool is_signed_v =
std::is_signed_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>;
template <class T>
struct make_unsigned {
using type = std::make_unsigned_t<T>;
};
template <>
struct make_unsigned<__int128_t> {
using type = __uint128_t;
};
template <>
struct make_unsigned<__uint128_t> {
using type = __uint128_t;
};
template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;
} // namespace internal
struct FastInput {
static constexpr int buffer_size = 1 << 20;
private:
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _length;
int _file_descriptor;
bool _streaming;
bool refill() {
_position = 0;
if (_streaming) {
ssize_t length;
do {
length = ::read(_file_descriptor, _buffer, buffer_size);
} while (length < 0 && errno == EINTR);
if (length <= 0) {
_length = 0;
return false;
}
_length = int(length);
} else {
_length = int(std::fread(_buffer, 1, buffer_size, _stream));
}
return _length != 0;
}
template <class T>
bool read_integer_from_stream(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-') {
negative = true;
c = read_char_raw();
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
result = negative ? result * 10 - (c - '0')
: result * 10 + (c - '0');
c = read_char_raw();
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + T(c - '0');
c = read_char_raw();
}
value = negative ? T(0) - result : result;
}
return true;
}
bool prepare_number() {
if (_length - _position >= 64) return true;
const int remaining = _length - _position;
if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
_position = 0;
_length = remaining + added;
if (_length < buffer_size) _buffer[_length] = '\0';
return _length != 0;
}
public:
explicit FastInput(std::FILE* stream = stdin)
: _stream(stream),
_position(0),
_length(0),
_file_descriptor(::fileno(stream)),
_streaming([&] {
struct stat status;
return _file_descriptor >= 0
&& ::fstat(_file_descriptor, &status) == 0
&& !S_ISREG(status.st_mode);
}()) {}
FastInput(const FastInput&) = delete;
FastInput& operator=(const FastInput&) = delete;
int read_char_raw() {
if (_position == _length && !refill()) return EOF;
return _buffer[_position++];
}
bool skip_spaces() {
int c = read_char_raw();
while (c != EOF && c <= ' ') c = read_char_raw();
if (c == EOF) return false;
--_position;
return true;
}
bool read(char& value) {
if (!skip_spaces()) return false;
value = char(read_char_raw());
return true;
}
bool read(std::string& value) {
if (!skip_spaces()) return false;
value.clear();
while (true) {
const int begin = _position;
while (_position < _length &&
static_cast<unsigned char>(_buffer[_position]) > ' ') {
++_position;
}
value.append(_buffer + begin, _position - begin);
if (_position < _length) {
++_position;
return true;
}
if (!refill()) return true;
}
}
bool read(bool& value) {
int x;
if (!read(x)) return false;
value = x != 0;
return true;
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>,
bool
>
read(T& value) {
if (_streaming) return read_integer_from_stream(value);
if (!prepare_number()) return false;
int c = static_cast<unsigned char>(_buffer[_position++]);
while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);
bool negative = false;
if (c == '-') {
negative = true;
c = static_cast<unsigned char>(_buffer[_position++]);
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
const int first = c - '0';
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = negative ? result * 100 - (first * 10 + second)
: result * 100 + (first * 10 + second);
++_position;
} else {
result = negative ? result * 10 - first : result * 10 + first;
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
const unsigned first = unsigned(c - '0');
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = result * 100 + T(first * 10 + unsigned(second));
++_position;
} else {
result = result * 10 + T(first);
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = negative ? T(0) - result : result;
}
if (_position > _length) _position = _length;
return true;
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>, bool>
read(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-' || c == '+') {
negative = c == '-';
c = read_char_raw();
}
long double result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + (c - '0');
c = read_char_raw();
}
if (c == '.') {
long double place = 0.1L;
c = read_char_raw();
while ('0' <= c && c <= '9') {
result += (c - '0') * place;
place *= 0.1L;
c = read_char_raw();
}
}
if (c == 'e' || c == 'E') {
c = read_char_raw();
bool exponent_negative = false;
if (c == '-' || c == '+') {
exponent_negative = c == '-';
c = read_char_raw();
}
int exponent = 0;
while ('0' <= c && c <= '9') {
exponent = exponent * 10 + (c - '0');
c = read_char_raw();
}
long double scale = 1;
long double power = 10;
while (exponent > 0) {
if (exponent & 1) scale *= power;
power *= power;
exponent >>= 1;
}
result = exponent_negative ? result / scale : result * scale;
}
value = static_cast<T>(negative ? -result : result);
return true;
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>,
bool
>
read(T& value) {
long long x;
if (!read(x)) return false;
if constexpr (internal::has_static_mod_raw_v<T>) {
if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
value = T::raw(uint32_t(x));
} else {
value = T(x);
}
} else {
value = T(x);
}
return true;
}
template <class First, class Second>
bool read(std::pair<First, Second>& value) {
if (!read(value.first)) return false;
return read(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>,
bool
>
read(Range& range) {
using StoredValue = internal::range_stored_value_t<Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
for (auto&& value : range) {
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
bool x;
if (!read(x)) return false;
value = x;
} else {
if (!read(value)) return false;
}
}
return true;
}
template <class First, class Second, class... Rest>
bool read(First& first, Second& second, Rest&... rest) {
if (!read(first)) return false;
return read(second, rest...);
}
template <class T>
FastInput& operator>>(T& value) {
if (!read(value)) std::abort();
return *this;
}
};
struct FastOutput {
static constexpr int buffer_size = 1 << 20;
private:
inline static const auto digit_quads = [] {
std::array<char, 40000> result{};
for (int i = 0; i < 10000; i++) {
int value = i;
for (int j = 3; j >= 0; j--) {
result[4 * i + j] = char('0' + value % 10);
value /= 10;
}
}
return result;
}();
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _precision;
std::chars_format _float_format;
char _range_separator;
std::string* _capture = nullptr;
template <class T>
std::string format_cell(const T& value) {
std::string result;
struct CaptureGuard {
std::string*& target;
std::string* previous;
~CaptureGuard() { target = previous; }
} guard{_capture, _capture};
_capture = &result;
write(value);
return result;
}
template <class Matrix>
void write_aligned_matrix(const Matrix& matrix) {
std::vector<std::vector<std::string>> rows;
std::vector<std::size_t> widths;
for (const auto& row : matrix) {
auto& cells = rows.emplace_back();
std::size_t column = 0;
for (const auto& value : row) {
cells.push_back(format_cell(value));
if (column == widths.size()) widths.push_back(0);
widths[column] = std::max(widths[column], cells.back().size());
++column;
}
}
bool first = true;
for (const auto& row : rows) {
if (!first) write_char('\n');
first = false;
for (std::size_t column = 0; column < row.size(); ++column) {
if (column != 0) write_char(_range_separator);
for (std::size_t padding = row[column].size();
padding < widths[column]; ++padding) {
write_char(' ');
}
write(row[column]);
}
}
}
public:
explicit FastOutput(std::FILE* stream = stdout)
: _stream(stream),
_position(0),
_precision(6),
_float_format(std::chars_format::general),
_range_separator(' ') {
if (_stream == stdout
&& internal::standard_output_instance == nullptr) {
internal::standard_output_instance = this;
}
}
FastOutput(const FastOutput&) = delete;
FastOutput& operator=(const FastOutput&) = delete;
~FastOutput() {
flush();
if (internal::standard_output_instance == this) {
internal::standard_output_instance = nullptr;
}
}
void flush() {
if (_position != 0) {
std::fwrite(_buffer, 1, _position, _stream);
_position = 0;
}
std::fflush(_stream);
}
void write_char(char c) {
if (_capture != nullptr) {
_capture->push_back(c);
return;
}
if (_position == buffer_size) flush();
_buffer[_position++] = c;
}
void write(const char* s) {
while (*s != '\0') write_char(*s++);
}
void write(const std::string& s) {
if (_capture != nullptr) {
_capture->append(s);
return;
}
std::size_t position = 0;
while (position < s.size()) {
if (_position == buffer_size) flush();
const std::size_t copied =
std::min<std::size_t>(buffer_size - _position, s.size() - position);
std::memcpy(_buffer + _position, s.data() + position, copied);
_position += int(copied);
position += copied;
}
}
void write(char c) {
write_char(c);
}
void write(bool value) {
write_char(value ? '1' : '0');
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>>
write(T value) {
char digits[128];
auto [end, error] = std::to_chars(
digits,
digits + sizeof(digits),
value,
_float_format,
_precision
);
if (error != std::errc()) std::abort();
for (const char* pointer = digits; pointer != end; pointer++) {
write_char(*pointer);
}
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>
>
write(T value) {
using Raw = std::remove_cv_t<T>;
using Unsigned = internal::make_unsigned_t<Raw>;
Unsigned magnitude;
if constexpr (internal::is_signed_v<Raw>) {
if (value < 0) {
write_char('-');
magnitude = Unsigned(0) - Unsigned(value);
} else {
magnitude = Unsigned(value);
}
} else {
magnitude = value;
}
if (magnitude == 0) {
write_char('0');
return;
}
unsigned chunks[16];
int count = 0;
while (magnitude >= 10000) {
const Unsigned quotient = magnitude / 10000;
chunks[count++] = unsigned(magnitude - quotient * 10000);
magnitude = quotient;
}
if (_capture == nullptr && _position > buffer_size - 64) flush();
char captured[64];
char* const begin = _capture != nullptr ? captured : _buffer + _position;
char* destination = begin;
const unsigned leading = unsigned(magnitude);
const char* first = digit_quads.data() + 4 * leading;
int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
for (; skip < 4; skip++) *destination++ = first[skip];
while (count--) {
const char* digits = digit_quads.data() + 4 * chunks[count];
std::memcpy(destination, digits, 4);
destination += 4;
}
if (_capture != nullptr) {
_capture->append(begin, destination - begin);
} else {
_position += int(destination - begin);
}
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>
>
write(const T& value) {
write(value.val());
}
template <class First, class Second>
void write(const std::pair<First, Second>& value) {
write(value.first);
write_char(' ');
write(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>
>
write(const Range& range) {
using StoredValue = internal::range_stored_value_t<const Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
bool first = true;
for (const auto& value : range) {
if (!first) write_char(nested ? '\n' : _range_separator);
first = false;
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
write(static_cast<bool>(value));
} else {
write(value);
}
}
}
template <class First, class... Rest>
void print(const First& first, const Rest&... rest) {
write(first);
((write_char(' '), write(rest)), ...);
}
void println() {
write_char('\n');
}
void set_precision(int precision) {
_precision = precision;
}
void set_fixed(int precision = 6) {
_float_format = std::chars_format::fixed;
_precision = precision;
}
void set_general(int precision = 6) {
_float_format = std::chars_format::general;
_precision = precision;
}
void set_range_separator(char separator) {
_range_separator = separator;
}
template <class Matrix>
void write_aligned(const Matrix& matrix) {
using Row = internal::range_stored_value_t<const Matrix>;
using Cell = internal::range_stored_value_t<const Row>;
static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
"write_aligned requires a two-dimensional range");
static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
"write_aligned requires scalar cells");
write_aligned_matrix(matrix);
}
template <class Matrix>
void println_aligned(const Matrix& matrix) {
write_aligned(matrix);
write_char('\n');
}
template <class... Args>
void println(const Args&... args) {
print(args...);
write_char('\n');
}
template <class T>
FastOutput& operator<<(const T& value) {
write(value);
return *this;
}
};
} // namespace utilities
} // namespace m1une
#line 10 "verify/graph/minimum_steiner_tree.test.cpp"
#include <numeric>
#line 14 "verify/graph/minimum_steiner_tree.test.cpp"
namespace {
struct NaiveDsu {
std::vector<int> parent;
explicit NaiveDsu(int n) : parent(n) {
std::iota(parent.begin(), parent.end(), 0);
}
int leader(int v) {
if (parent[v] == v) return v;
return parent[v] = leader(parent[v]);
}
void merge(int u, int v) {
u = leader(u);
v = leader(v);
if (u != v) parent[u] = v;
}
};
template <class T>
std::pair<std::optional<T>, std::optional<int>> naive(
const m1une::graph::Graph<T>& graph,
std::vector<int> terminals
) {
std::sort(terminals.begin(), terminals.end());
terminals.erase(std::unique(terminals.begin(), terminals.end()), terminals.end());
if (terminals.empty()) return {T(0), 0};
const auto edges = graph.edges();
assert(edges.size() < std::numeric_limits<std::uint64_t>::digits);
std::optional<T> weighted;
std::optional<int> unweighted;
for (std::uint64_t mask = 0; mask < (std::uint64_t(1) << edges.size()); mask++) {
NaiveDsu dsu(graph.size());
T cost = T(0);
int count = 0;
for (int i = 0; i < int(edges.size()); i++) {
if ((mask >> i & 1) == 0) continue;
dsu.merge(edges[i].from, edges[i].to);
cost += edges[i].cost;
count++;
}
bool connected = true;
for (int v : terminals) {
if (dsu.leader(v) != dsu.leader(terminals[0])) connected = false;
}
if (!connected) continue;
if (!weighted || cost < *weighted) weighted = cost;
if (!unweighted || count < *unweighted) unweighted = count;
}
return {weighted, unweighted};
}
template <class Cost, class GraphCost, class EdgeCost>
std::optional<Cost> naive_with_vertex_cost(
const m1une::graph::Graph<GraphCost>& graph,
std::vector<int> terminals,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost
) {
std::sort(terminals.begin(), terminals.end());
terminals.erase(std::unique(terminals.begin(), terminals.end()), terminals.end());
if (terminals.empty()) return Cost(0);
const auto edges = graph.edges();
std::optional<Cost> answer;
for (std::uint64_t mask = 0; mask < (std::uint64_t(1) << edges.size()); mask++) {
NaiveDsu dsu(graph.size());
std::vector<char> used(graph.size(), false);
for (int terminal : terminals) used[terminal] = true;
Cost cost = Cost(0);
for (int i = 0; i < int(edges.size()); i++) {
if ((mask >> i & 1) == 0) continue;
dsu.merge(edges[i].from, edges[i].to);
used[edges[i].from] = true;
used[edges[i].to] = true;
cost += edge_cost(edges[i]);
}
for (int v = 0; v < graph.size(); v++) {
if (used[v]) cost += vertex_cost[v];
}
bool connected = true;
for (int v : terminals) {
if (dsu.leader(v) != dsu.leader(terminals[0])) connected = false;
}
if (connected && (!answer || cost < *answer)) answer = cost;
}
return answer;
}
template <class Cost, class GraphCost, class EdgeCost>
void validate_built_tree(
const m1une::graph::Graph<GraphCost>& graph,
const m1une::graph::MinimumSteinerTreeResult<Cost>& result,
const std::vector<int>& terminals,
const std::vector<Cost>& vertex_cost,
EdgeCost edge_cost
) {
assert(std::is_sorted(result.edge_ids.begin(), result.edge_ids.end()));
assert(std::adjacent_find(result.edge_ids.begin(), result.edge_ids.end()) == result.edge_ids.end());
assert(std::is_sorted(result.vertices.begin(), result.vertices.end()));
assert(std::adjacent_find(result.vertices.begin(), result.vertices.end()) == result.vertices.end());
std::vector<m1une::graph::Edge<GraphCost>> edge_by_id(graph.edge_count());
for (const auto& edge : graph.edges()) edge_by_id[edge.id] = edge;
NaiveDsu dsu(graph.size());
Cost cost = Cost(0);
for (int id : result.edge_ids) {
assert(0 <= id && id < graph.edge_count());
assert(graph.is_edge_alive(id));
const auto& edge = edge_by_id[id];
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), edge.from));
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), edge.to));
assert(dsu.leader(edge.from) != dsu.leader(edge.to));
dsu.merge(edge.from, edge.to);
cost += edge_cost(edge);
}
for (int v : result.vertices) cost += vertex_cost[v];
for (int terminal : terminals) {
assert(std::binary_search(result.vertices.begin(), result.vertices.end(), terminal));
}
if (result.vertices.empty()) {
assert(terminals.empty() && result.edge_ids.empty());
} else {
assert(result.edge_ids.size() + 1 == result.vertices.size());
for (int v : result.vertices) assert(dsu.leader(v) == dsu.leader(result.vertices[0]));
}
assert(cost == result.cost);
}
void test_examples_and_failures() {
m1une::graph::Graph<long long> graph(6);
graph.add_edge(0, 1, 4);
graph.add_edge(1, 2, 1);
graph.add_edge(1, 3, 2);
graph.add_edge(3, 4, 3);
graph.add_edge(0, 4, 20);
auto weighted = m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 2, 4});
auto unweighted = m1une::graph::minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4}
);
assert(weighted && *weighted == 10);
assert(unweighted && *unweighted == 3);
assert(!m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 5}));
assert(!m1une::graph::minimum_steiner_tree_unweighted(graph, std::vector<int>{0, 5}));
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>{2, 2}) == 0);
assert(*m1une::graph::minimum_steiner_tree_unweighted(graph, std::vector<int>()) == 0);
const std::vector<long long> vertex_cost = {2, 3, 7, 5, 11, 13};
auto vertex_weighted = m1une::graph::minimum_steiner_tree(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
auto vertex_weighted_unit_edges = m1une::graph::minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(vertex_weighted && *vertex_weighted == 38);
assert(vertex_weighted_unit_edges && *vertex_weighted_unit_edges == 26);
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>{2}, vertex_cost) == 7);
assert(*m1une::graph::minimum_steiner_tree(graph, std::vector<int>(), vertex_cost) == 0);
m1une::graph::Graph<int> unit_graph(2);
unit_graph.add_edge(0, 1, 100);
const std::vector<long long> large_vertex_cost = {3'000'000'000LL, 4'000'000'000LL};
auto independent_cost_type = m1une::graph::minimum_steiner_tree_unweighted(
unit_graph,
std::vector<int>{0, 1},
large_vertex_cost
);
assert(independent_cost_type && *independent_cost_type == 7'000'000'001LL);
auto built = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(built && built->cost == 38);
validate_built_tree(
graph,
*built,
std::vector<int>{0, 2, 4},
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
auto built_unit = m1une::graph::build_minimum_steiner_tree_unweighted(
graph,
std::vector<int>{0, 2, 4},
vertex_cost
);
assert(built_unit && built_unit->cost == 26);
validate_built_tree(
graph,
*built_unit,
std::vector<int>{0, 2, 4},
vertex_cost,
[](const auto&) { return 1LL; }
);
m1une::graph::Graph<long long> zero_cycle(3);
zero_cycle.add_edge(0, 1, 0);
zero_cycle.add_edge(1, 2, 0);
zero_cycle.add_edge(2, 0, 0);
auto zero_cycle_tree = m1une::graph::build_minimum_steiner_tree(
zero_cycle,
std::vector<int>{0, 1, 2}
);
assert(zero_cycle_tree && zero_cycle_tree->cost == 0);
assert(zero_cycle_tree->edge_ids.size() == 2);
validate_built_tree(
zero_cycle,
*zero_cycle_tree,
std::vector<int>{0, 1, 2},
std::vector<long long>(3, 0),
[](const auto& edge) { return edge.cost; }
);
auto single_vertex_tree = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>{2},
vertex_cost
);
assert(single_vertex_tree && single_vertex_tree->vertices == std::vector<int>{2});
assert(single_vertex_tree->edge_ids.empty() && single_vertex_tree->cost == 7);
auto empty_tree = m1une::graph::build_minimum_steiner_tree(
graph,
std::vector<int>(),
vertex_cost
);
assert(empty_tree && empty_tree->vertices.empty() && empty_tree->edge_ids.empty());
int removed = graph.add_edge(4, 5, 1);
graph.erase_edge(removed);
assert(!m1une::graph::minimum_steiner_tree(graph, std::vector<int>{0, 5}));
assert(!m1une::graph::build_minimum_steiner_tree(graph, std::vector<int>{0, 5}));
}
void test_randomized() {
std::uint64_t state = 0x243f6a8885a308d3ULL;
auto random = [&state]() {
state ^= state << 7;
state ^= state >> 9;
return state;
};
for (int trial = 0; trial < 100; trial++) {
const int n = 1 + int(random() % 7);
std::vector<std::pair<int, int>> pairs;
for (int u = 0; u < n; u++) {
for (int v = u + 1; v < n; v++) pairs.emplace_back(u, v);
}
for (int i = int(pairs.size()) - 1; i > 0; i--) {
std::swap(pairs[i], pairs[random() % (i + 1)]);
}
m1une::graph::Graph<long long> graph(n);
const int m = int(random() % (std::min<int>(10, pairs.size()) + 1));
for (int i = 0; i < m; i++) {
graph.add_edge(pairs[i].first, pairs[i].second, random() % 10);
}
if (m > 0 && trial % 7 == 0) graph.erase_edge(int(random() % m));
std::vector<int> terminals;
for (int v = 0; v < n; v++) {
if (random() % 3 == 0) terminals.push_back(v);
}
if (!terminals.empty() && trial % 5 == 0) terminals.push_back(terminals.back());
std::vector<long long> vertex_cost(n);
for (long long& cost : vertex_cost) cost = random() % 8;
auto expected = naive(graph, terminals);
auto weighted = m1une::graph::minimum_steiner_tree(graph, terminals);
auto unweighted = m1une::graph::minimum_steiner_tree_unweighted(graph, terminals);
assert(weighted == expected.first);
assert(unweighted == expected.second);
auto expected_vertex_weighted = naive_with_vertex_cost<long long>(
graph,
terminals,
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
auto expected_vertex_weighted_unit_edges = naive_with_vertex_cost<long long>(
graph,
terminals,
vertex_cost,
[](const auto&) { return 1LL; }
);
auto vertex_weighted = m1une::graph::minimum_steiner_tree(
graph,
terminals,
vertex_cost
);
auto vertex_weighted_unit_edges = m1une::graph::minimum_steiner_tree_unweighted(
graph,
terminals,
vertex_cost
);
assert(vertex_weighted == expected_vertex_weighted);
assert(vertex_weighted_unit_edges == expected_vertex_weighted_unit_edges);
const std::vector<long long> zero_vertex_cost(n, 0);
auto built_weighted = m1une::graph::build_minimum_steiner_tree(graph, terminals);
auto built_unweighted = m1une::graph::build_minimum_steiner_tree_unweighted(
graph,
terminals
);
auto built_vertex_weighted = m1une::graph::build_minimum_steiner_tree(
graph,
terminals,
vertex_cost
);
auto built_vertex_weighted_unit_edges =
m1une::graph::build_minimum_steiner_tree_unweighted(graph, terminals, vertex_cost);
assert(bool(built_weighted) == bool(expected.first));
assert(bool(built_unweighted) == bool(expected.second));
assert(bool(built_vertex_weighted) == bool(expected_vertex_weighted));
assert(
bool(built_vertex_weighted_unit_edges) == bool(expected_vertex_weighted_unit_edges)
);
if (built_weighted) {
assert(built_weighted->cost == *expected.first);
validate_built_tree(
graph,
*built_weighted,
terminals,
zero_vertex_cost,
[](const auto& edge) { return edge.cost; }
);
}
if (built_unweighted) {
assert(built_unweighted->cost == *expected.second);
validate_built_tree(
graph,
*built_unweighted,
terminals,
std::vector<int>(n, 0),
[](const auto&) { return 1; }
);
}
if (built_vertex_weighted) {
assert(built_vertex_weighted->cost == *expected_vertex_weighted);
validate_built_tree(
graph,
*built_vertex_weighted,
terminals,
vertex_cost,
[](const auto& edge) { return edge.cost; }
);
}
if (built_vertex_weighted_unit_edges) {
assert(built_vertex_weighted_unit_edges->cost == *expected_vertex_weighted_unit_edges);
validate_built_tree(
graph,
*built_vertex_weighted_unit_edges,
terminals,
vertex_cost,
[](const auto&) { return 1LL; }
);
}
}
}
} // namespace
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_examples_and_failures();
test_randomized();
int vertex_count, edge_count;
fast_input >> vertex_count >> edge_count;
m1une::graph::Graph<long long> graph(vertex_count);
for (int edge = 0; edge < edge_count; edge++) {
int first, second;
long long weight;
fast_input >> first >> second >> weight;
graph.add_edge(first, second, weight);
}
int terminal_count;
fast_input >> terminal_count;
std::vector<int> terminals(terminal_count);
for (int& terminal : terminals) fast_input >> terminal;
auto result = m1une::graph::build_minimum_steiner_tree(graph, terminals);
assert(result.has_value());
fast_output << result->cost << ' ' << result->edge_ids.size() << '\n';
for (int index = 0; index < int(result->edge_ids.size()); index++) {
if (index != 0) fast_output << ' ';
fast_output << result->edge_ids[index];
}
fast_output << '\n';
}
verify/graph/minimum_steiner_tree.test.cpp