test/k_shortest_walk.test.cpp

test/k_shortest_walk.test.cpp · GitHub で見る

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#define PROBLEM "https://judge.yosupo.jp/problem/k_shortest_walk"
#include "../graph/k_shortest_walk.hpp"
#include "../components/directed_csr.hpp"
#include "../components/fast_io.hpp"
#include <cstdint>

int main() {
	FastInput in;
	FastOutput out;
	int n = 0, m = 0, s = 0, t = 0, k = 0;
	in.read(n), in.read(m), in.read(s), in.read(t), in.read(k);
	using Graph = DirectedCSR<std::uint32_t>;
	Graph graph(n, m, [&](auto emit) {
		for (int i = 0; i < m; ++i) {
			Graph::Edge edge{};
			in.read(edge.from), in.read(edge.to), in.read(edge.weight);
			emit(edge);
		}
	});
	const auto answer = k_shortest_walk(graph, s, t, k);
	for (int i = 0; i < k; ++i) out.println(i < int(answer.size()) ? answer[i] : -1);
}

#line 1 "test/k_shortest_walk.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/k_shortest_walk"
#line 1 "graph/k_shortest_walk.hpp"



#line 1 "components/dijkstra_queue.hpp"



#line 1 "components/RadixHeap.hpp"



#include <algorithm>
#include <bit>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>
#include <vector>

template <class Key = std::uint64_t, class T = int>
class RadixHeap {
	static_assert(std::is_integral_v<Key> && !std::is_same_v<Key, bool> && sizeof(Key) <= 8);
	using U = std::make_unsigned_t<Key>;
	using State = std::pair<Key, T>;
	std::vector<State> bucket[std::numeric_limits<U>::digits + 1];
	Key lower;
	std::size_t count = 0;
	int index(Key key) const { return std::bit_width(U(U(key) ^ U(lower))); }
	void refill() {
		assert(count);
		if (!bucket[0].empty()) return;
		int i = 1;
		while (bucket[i].empty()) ++i;
		lower = bucket[i][0].first;
		for (const auto &s : bucket[i]) lower = std::min(lower, s.first);
		for (auto &s : bucket[i]) bucket[index(s.first)].push_back(std::move(s));
		bucket[i].clear();
	}

  public:
	explicit RadixHeap(Key minimum = 0) : lower(minimum) {}
	RadixHeap(const RadixHeap &) = default;
	RadixHeap &operator=(const RadixHeap &) = default;
	RadixHeap(RadixHeap &&other) noexcept : lower(other.lower), count(std::exchange(other.count, 0)) {
		for (int i = 0; i <= std::numeric_limits<U>::digits; ++i) bucket[i] = std::move(other.bucket[i]);
	}
	RadixHeap &operator=(RadixHeap &&other) noexcept {
		if (this != &other) {
			lower = other.lower;
			count = std::exchange(other.count, 0);
			for (int i = 0; i <= std::numeric_limits<U>::digits; ++i) bucket[i] = std::move(other.bucket[i]);
		}
		return *this;
	}

	bool empty() const { return count == 0; }
	std::size_t size() const { return count; }
	Key last() const { return lower; }
	void clear(Key minimum = 0) {
		for (auto &b : bucket) b.clear();
		lower = minimum;
		count = 0;
	}
	void push(State s) {
		assert(s.first >= lower);
		bucket[index(s.first)].push_back(std::move(s));
		++count;
	}
	const State &top() {
		refill();
		return bucket[0].back();
	}
	void pop() {
		refill();
		bucket[0].pop_back();
		--count;
	}
	State pop_value() {
		refill();
		State s = std::move(bucket[0].back());
		bucket[0].pop_back();
		--count;
		return s;
	}
};


#line 10 "components/dijkstra_queue.hpp"
#include <functional>
#line 15 "components/dijkstra_queue.hpp"

namespace shortest_path_detail {
template<class Weight, class Distance>
bool reaches_limit(const Weight &weight, const Distance &limit) {
	if constexpr (std::numeric_limits<Weight>::is_integer && std::numeric_limits<Weight>::is_bounded &&
				  std::numeric_limits<Distance>::is_integer && std::numeric_limits<Distance>::is_bounded) {
		using Compare = std::common_type_t<Weight, Distance>;
		return Compare(weight) >= Compare(limit);
	} else return weight >= limit;
}

template<class T> class RadixQueue {
	static_assert(std::is_integral_v<T> && sizeof(T) <= 8);
	struct State { T key; int vertex; };
	State small_[32];
	std::vector<std::uint64_t> buckets_[33];
	std::uint32_t minimum_[33];
	std::uint32_t occupied_ = 0;
	std::uint64_t last_ = 0;
	std::size_t count_ = 0;
	int mode_ = 0;
	RadixHeap<std::uint64_t> wide_;
	void put(std::uint64_t state) {
		std::uint32_t key = state >> 32;
		int index = std::bit_width(key ^ std::uint32_t(last_));
		buckets_[index].push_back(state);
		if (index) {
			occupied_ |= std::uint32_t(1) << (index - 1);
			minimum_[index] = std::min(minimum_[index], key);
		}
	}
	void promote() {
		if (mode_ == 0) {
			for (std::size_t i = 0; i < count_; ++i) wide_.push({std::uint64_t(small_[i].key), small_[i].vertex});
		} else {
			for (auto &bucket : buckets_) {
				for (auto state : bucket) {
					std::uint64_t key = last_ + std::uint32_t(std::uint32_t(state >> 32) - std::uint32_t(last_));
					wide_.push({key, int(std::uint32_t(state))});
				}
				bucket.clear();
			}
		}
		mode_ = 2;
	}

  public:
	bool empty() const { return count_ == 0; }
	void push(T key, int vertex) {
		assert(key >= 0 && std::uint64_t(key) >= last_);
		if (mode_ == 0 && count_ < 32) {
			small_[count_++] = {key, vertex};
			return;
		}
		if (mode_ == 0) {
			bool narrow = std::uint64_t(key) - last_ <= std::uint64_t(std::numeric_limits<std::int32_t>::max());
			for (std::size_t i = 0; i < count_; ++i)
				narrow &= std::uint64_t(small_[i].key) - last_ <= std::uint64_t(std::numeric_limits<std::int32_t>::max());
			if (narrow) {
				std::fill_n(minimum_, 33, std::numeric_limits<std::uint32_t>::max());
				for (std::size_t i = 0; i < count_; ++i)
					put((std::uint64_t(std::uint32_t(small_[i].key)) << 32) | std::uint32_t(small_[i].vertex));
				mode_ = 1;
			} else promote();
		}
		if (mode_ == 1 && std::uint64_t(key) - last_ > std::uint64_t(std::numeric_limits<std::int32_t>::max())) promote();
		if (mode_ == 2) wide_.push({std::uint64_t(key), vertex});
		else put((std::uint64_t(std::uint32_t(key)) << 32) | std::uint32_t(vertex));
		++count_;
	}
	std::pair<T, int> pop() {
		assert(count_);
		if (mode_ == 0) {
			std::size_t best = 0;
			for (std::size_t i = 1; i < count_; ++i)
				if (small_[i].key < small_[best].key) best = i;
			State state = small_[best];
			small_[best] = small_[--count_];
			last_ = std::uint64_t(state.key);
			return {state.key, state.vertex};
		}
		if (mode_ == 2) {
			auto state = wide_.pop_value();
			last_ = state.first;
			--count_;
			return {T(state.first), state.second};
		}
		if (buckets_[0].empty()) {
			int index = std::countr_zero(occupied_) + 1;
			last_ += std::uint32_t(minimum_[index] - std::uint32_t(last_));
			minimum_[index] = std::numeric_limits<std::uint32_t>::max();
			occupied_ &= occupied_ - 1;
			if (buckets_[index].size() == 1) {
				auto state = buckets_[index].back();
				buckets_[index].clear();
				--count_;
				return {T(last_), int(std::uint32_t(state))};
			}
			for (auto state : buckets_[index]) put(state);
			buckets_[index].clear();
		}
		auto state = buckets_[0].back();
		buckets_[0].pop_back();
		--count_;
		return {T(last_), int(std::uint32_t(state))};
	}
};

template<class T> class BinaryQueue {
	using State = std::pair<T, int>;
	std::vector<State> heap_;

  public:
	bool empty() const { return heap_.empty(); }
	void push(T cost, int vertex) {
		heap_.emplace_back(std::move(cost), vertex);
		std::push_heap(heap_.begin(), heap_.end(), std::greater<State>());
	}
	State pop() {
		std::pop_heap(heap_.begin(), heap_.end(), std::greater<State>());
		State state = std::move(heap_.back());
		heap_.pop_back();
		return state;
	}
};

template<class T>
using DistanceQueue = std::conditional_t<std::is_integral_v<T> && sizeof(T) <= 8, RadixQueue<T>, BinaryQueue<T>>;
}


#line 1 "components/directed_csr.hpp"



#line 8 "components/directed_csr.hpp"
#include <memory>
#include <span>
#include <stdexcept>
#line 14 "components/directed_csr.hpp"

namespace directed_csr_detail { struct Access; }

template<class T>
class DirectedCSR {
  public:
	struct Edge { int from, to; T weight; };
	using Arc = std::pair<int, T>;
	using weight_type = T;
	using distance_type = std::conditional_t<std::is_integral_v<T> && sizeof(T) < 8, std::int64_t, T>;

  private:
	friend struct directed_csr_detail::Access;
	std::vector<std::size_t> offset_;
	std::vector<Arc> arcs_;
	int single_gap_ = -2;
	bool unique_incoming_ = true;
	void reset(int n, std::size_t m) {
		assert(n >= 0);
		offset_.assign(std::size_t(n) + 1, 0);
		single_gap_ = -2;
		unique_incoming_ = m <= std::size_t(n);
	}
	void count_edge(const Edge &edge, std::vector<std::uint64_t> &seen) {
		assert(0 <= edge.from && edge.from < size() && 0 <= edge.to && edge.to < size());
		++offset_[edge.from];
		if (unique_incoming_) {
			auto mask = std::uint64_t(1) << (edge.to % 64);
			auto &word = seen[edge.to / 64];
			unique_incoming_ = !(word & mask);
			word |= mask;
		}
	}
	void finish(std::span<const Edge> edges) {
		const auto n = offset_.size() - 1;
		single_gap_ = -2;
		if (edges.size() >= n - (n != 0) && edges.size() <= n &&
			std::all_of(offset_.begin(), offset_.end() - 1, [](std::size_t x) { return x <= 1; })) {
			single_gap_ = -1;
			if (edges.size() != n) single_gap_ = int(std::find(offset_.begin(), offset_.end() - 1, 0) - offset_.begin());
			for (std::size_t v = 0; v <= n; ++v) offset_[v] = v - (single_gap_ >= 0 && v > std::size_t(single_gap_));
			arcs_.resize(edges.size());
			for (const auto &edge : edges) arcs_[edge.from - (single_gap_ >= 0 && edge.from > single_gap_)] = {edge.to, edge.weight};
			return;
		}
		std::size_t active = offset_[0] != 0;
		for (std::size_t v = 1; v < offset_.size(); ++v) {
			active += offset_[v] != 0;
			offset_[v] += offset_[v - 1];
		}
		arcs_.resize(edges.size());
		std::size_t i = edges.size();
#if defined(__GNUC__) || defined(__clang__)
		if (active >= 8192) {
			for (; i > 32; --i) {
				const auto from = edges[i - 33].from;
				__builtin_prefetch(&offset_[from], 1, 3);
				__builtin_prefetch(&arcs_[offset_[from] - 1], 1, 3);
				const auto &edge = edges[i - 1];
				arcs_[--offset_[edge.from]] = {edge.to, edge.weight};
			}
		}
#endif
		for (; i; --i) {
			const auto &edge = edges[i - 1];
			arcs_[--offset_[edge.from]] = {edge.to, edge.weight};
		}
	}
	void clear_edges() {
		single_gap_ = -2;
		unique_incoming_ = true;
		std::fill(offset_.begin(), offset_.end(), 0);
		arcs_.clear();
	}

  public:
	explicit DirectedCSR(int n = 0) {
		assert(n >= 0);
		offset_.assign(std::size_t(n) + 1, 0);
	}
	DirectedCSR(int n, std::span<const Edge> edges) { build(n, edges); }
	template<class Producer>
	DirectedCSR(int n, std::size_t m, Producer &&produce) { build(n, m, std::forward<Producer>(produce)); }
	void build(int n, std::span<const Edge> edges) {
		reset(n, edges.size());
		try {
			std::vector<std::uint64_t> seen(unique_incoming_ ? (std::size_t(n) + 63) / 64 : 0);
			for (const auto &edge : edges) count_edge(edge, seen);
			finish(edges);
		}
		catch (...) { clear_edges(); throw; }
	}
	template<class Producer>
	void build(int n, std::size_t m, Producer &&produce) {
		reset(n, m);
		try {
			std::vector<std::uint64_t> seen(unique_incoming_ ? (std::size_t(n) + 63) / 64 : 0);
			auto edges = std::make_unique_for_overwrite<Edge[]>(m);
			std::size_t count = 0;
			auto emit = [&](Edge edge) {
				if (count == m) throw std::length_error("DirectedCSR edge count");
				count_edge(edge, seen);
				edges[count++] = std::move(edge);
			};
			std::forward<Producer>(produce)(emit);
			if (count != m) throw std::length_error("DirectedCSR edge count");
			finish(std::span<const Edge>(edges.get(), m));
		} catch (...) { clear_edges(); throw; }
	}
	int size() const { return offset_.empty() ? 0 : int(offset_.size() - 1); }
	bool empty() const { return size() == 0; }
	std::size_t edge_count() const { return arcs_.size(); }
	std::span<const Arc> operator[](int v) const {
		assert(0 <= v && v < size());
		return std::span<const Arc>(arcs_).subspan(offset_[v], offset_[v + 1] - offset_[v]);
	}
};

namespace directed_csr_detail {
template<class T>
struct SingleEdgeView {
	std::span<const typename DirectedCSR<T>::Arc> arcs;
	int n, gap;
	int size() const { return n; }
	std::span<const typename DirectedCSR<T>::Arc> operator[](int v) const {
		if (v == gap) return {};
		return arcs.subspan(v - (gap >= 0 && v > gap), 1);
	}
};

struct Access {
	template<class T>
	static bool unique_incoming(const DirectedCSR<T> &g) { return g.offset_.empty() || g.unique_incoming_; }
	template<class T>
	static bool has_single_edge_layout(const DirectedCSR<T> &g) { return !g.offset_.empty() && g.single_gap_ >= -1; }
	template<class T>
	static SingleEdgeView<T> single_edge_view(const DirectedCSR<T> &g) { return {g.arcs_, g.size(), g.single_gap_}; }
};
}


#line 7 "graph/k_shortest_walk.hpp"
#include <array>
#line 16 "graph/k_shortest_walk.hpp"

namespace k_shortest_walk_detail {
template<class T> class SidetrackHeap {
  public:
	struct Node { T cost; int to, left = 0, right = 0, rank = 1; };
	std::vector<Node> nodes{{T(0), 0, 0, 0, 0}};
	int make(Node node) {
		if (nodes.size() == std::size_t(std::numeric_limits<int>::max()))
			throw std::length_error("K shortest walk heap");
		int id = int(nodes.size());
		nodes.push_back(std::move(node));
		return id;
	}
	template<bool Persistent> int meld(int a, int b) {
		std::array<int, 64> path;
		int depth = 0;
		while (a && b) {
			if (nodes[b].cost < nodes[a].cost) std::swap(a, b);
			path[depth++] = a;
			a = nodes[a].right;
		}
		int root = a ? a : b;
		while (depth) {
			int id = path[--depth];
			if constexpr (Persistent) id = make(nodes[id]);
			auto &node = nodes[id];
			node.right = root;
			if ((nodes[node.left].rank) < nodes[root].rank) std::swap(node.left, node.right);
			node.rank = nodes[node.right].rank + 1;
			root = id;
		}
		return root;
	}
};
}

template<class Distance = std::int64_t, class G>
std::vector<Distance> k_shortest_walk(const G &g, int start, int target, int k,
	Distance inf = std::numeric_limits<Distance>::max()) {
	using T = Distance;
	using W = std::remove_cvref_t<decltype(g[0][0].second)>;
	static_assert(std::numeric_limits<T>::is_integer && std::numeric_limits<T>::is_bounded &&
		sizeof(T) <= 16 && !std::is_same_v<T, bool>);
	static_assert(std::numeric_limits<W>::is_integer && std::numeric_limits<W>::is_bounded &&
		sizeof(W) <= 16 && !std::is_same_v<W, bool>);
	assert(k >= 0 && inf > 0);
	if (!k) return {};
	assert(std::size_t(g.size()) <= std::size_t(std::numeric_limits<int>::max()));
	int n = int(g.size());
	assert(0 <= start && start < n && 0 <= target && target < n);
	std::size_t m = 0;
	for (int v = 0; v < n; ++v) m += g[v].size();
	DirectedCSR<W> reverse(n, m, [&](auto emit) {
		for (int v = 0; v < n; ++v)
			for (const auto &[to, weight] : g[v]) {
				assert(0 <= to && to < n && weight >= 0);
				emit(typename DirectedCSR<W>::Edge{to, v, weight});
			}
	});
	std::vector<T> distance(n, inf);
	std::vector<int> parent(n, -1), order;
	order.reserve(n);
	distance[target] = 0;
	{
		shortest_path_detail::DistanceQueue<T> queue;
		queue.push(0, target);
		while (!queue.empty()) {
			auto [cost, v] = queue.pop();
			if (cost != distance[v]) continue;
			order.push_back(v);
			for (const auto &[to, weight] : reverse[v]) {
				if (shortest_path_detail::reaches_limit(weight, T(inf - cost))) continue;
				T next = cost + T(weight);
				if (next >= distance[to]) continue;
				distance[to] = next;
				parent[to] = v;
				queue.push(next, to);
			}
		}
	}
	if (distance[start] == inf) return {};
	std::vector<T> answer{distance[start]};
	if (k == 1) return answer;
	k_shortest_walk_detail::SidetrackHeap<T> heap;
	std::vector<int> roots(n), local;
	using Compare = std::common_type_t<T, W>;
	for (int v : order) {
		local.clear();
		bool skipped = false;
		for (const auto &[to, weight] : g[v]) {
			if (distance[to] == inf) continue;
			if (!skipped && to == parent[v] && Compare(weight) == Compare(distance[v] - distance[to])) {
				skipped = true;
				continue;
			}
			T extra;
			if (distance[to] >= distance[v]) {
				extra = distance[to] - distance[v];
				if (shortest_path_detail::reaches_limit(weight, T(inf - extra))) continue;
				extra += T(weight);
			} else {
				Compare difference = Compare(weight) - Compare(distance[v] - distance[to]);
				if (shortest_path_detail::reaches_limit(difference, inf)) continue;
				extra = T(difference);
			}
			local.push_back(heap.make({extra, to}));
		}
		for (std::size_t size = local.size(); size > 1;) {
			std::size_t next = 0, i = 0;
			for (; i + 1 < size; i += 2) local[next++] = heap.template meld<false>(local[i], local[i + 1]);
			if (i < size) local[next++] = local[i];
			size = next;
		}
		int inherited = parent[v] == -1 ? 0 : roots[parent[v]];
		roots[v] = heap.template meld<true>(local.empty() ? 0 : local[0], inherited);
	}
	shortest_path_detail::DistanceQueue<T> queue;
	auto push = [&](T cost, int id) {
		if (id && heap.nodes[id].cost < inf - cost) queue.push(cost + heap.nodes[id].cost, id);
	};
	push(distance[start], roots[start]);
	while (int(answer.size()) < k && !queue.empty()) {
		auto [cost, id] = queue.pop();
		answer.push_back(cost);
		if (int(answer.size()) == k) break;
		const auto &node = heap.nodes[id];
		push(cost, roots[node.to]);
		push(cost - node.cost, node.left);
		push(cost - node.cost, node.right);
	}
	return answer;
}


#line 1 "components/fast_io.hpp"



#line 8 "components/fast_io.hpp"
#include <charconv>
#line 11 "components/fast_io.hpp"
#include <cstdio>
#include <cstring>
#line 15 "components/fast_io.hpp"
#include <string>
#include <string_view>
#line 18 "components/fast_io.hpp"
#if defined(__unix__) || defined(__APPLE__)
#include <sys/mman.h>
#include <sys/stat.h>
#include <unistd.h>
#endif

namespace fast_io_detail { struct BulkInput; struct BulkOutput; }

class FastInput {
	friend struct fast_io_detail::BulkInput;
	static constexpr std::size_t capacity = 1 << 16;
	FILE *file = nullptr;
	std::unique_ptr<char[]> buffer;
	const char *current = "", *end = current;
	std::string scratch;
	void *mapping = nullptr;
	std::size_t mapped_size = 0;
	bool refill() {
		if (!file) return false;
		std::size_t n = std::fread(buffer.get(), 1, capacity, file);
		current = buffer.get();
		end = current + n;
		return n;
	}
	static const char *scan(const char *p, const char *end) {
		while (end - p >= 8) {
			std::uint64_t x;
			std::memcpy(&x, p, 8);
			if ((x - 0x2121212121212121ULL) & ~x & 0x8080808080808080ULL) break;
			p += 8;
		}
		while (p != end && static_cast<unsigned char>(*p) > ' ') ++p;
		return p;
	}

public:
	explicit FastInput(FILE *input = stdin) {
#if defined(__unix__) || defined(__APPLE__)
		struct stat status;
		int fd = fileno(input);
		auto offset = ftello(input);
		if (fd >= 0 && offset >= 0 && !fstat(fd, &status) && S_ISREG(status.st_mode) && status.st_size > offset) {
			mapped_size = status.st_size;
			mapping = mmap(nullptr, mapped_size, PROT_READ, MAP_PRIVATE, fd, 0);
			if (mapping != MAP_FAILED) {
				current = static_cast<const char *>(mapping) + offset;
				end = static_cast<const char *>(mapping) + mapped_size;
				return;
			}
			mapping = nullptr;
		}
#endif
		file = input;
		buffer = std::make_unique_for_overwrite<char[]>(capacity);
	}
	explicit FastInput(std::string_view input) : current(input.empty() ? "" : input.data()), end(current + input.size()) {}
	FastInput(const FastInput &) = delete;
	FastInput &operator=(const FastInput &) = delete;
	~FastInput() {
#if defined(__unix__) || defined(__APPLE__)
		if (mapping) munmap(mapping, mapped_size);
#endif
	}
	std::string_view token() {
		while (true) {
			while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
			if (current != end) break;
			if (!refill()) return {};
		}
		const char *begin = current;
		current = scan(current, end);
		if (current != end || !file) return {begin, std::size_t(current - begin)};
		scratch.assign(begin, end);
		while (refill()) {
			current = scan(current, end);
			scratch.append(buffer.get(), std::size_t(current - buffer.get()));
			if (current != end) break;
		}
		return scratch;
	}
	template <class T>
	bool read(T &value) {
		if constexpr (std::is_integral_v<T> && !std::is_same_v<T, bool>) {
			while (true) {
				while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
				if (current != end) break;
				if (!refill()) return false;
			}
			if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4 && std::endian::native == std::endian::little) {
				if (end - current >= 11) {
					std::uint64_t x;
					std::memcpy(&x, current, 8);
					if (!(((x + 0x4646464646464646ULL) | (x - 0x3030303030303030ULL)) & 0x8080808080808080ULL)) {
						x &= 0x0f0f0f0f0f0f0f0fULL;
						x = (x * 10 + (x >> 8)) & 0x00ff00ff00ff00ffULL;
						x = (x * 100 + (x >> 16)) & 0x0000ffff0000ffffULL;
						std::uint64_t v = std::uint32_t(x * 10000 + (x >> 32));
						int n = 8;
						for (; n < 10 && static_cast<unsigned char>(current[n]) - unsigned('0') < 10; ++n)
							v = v * 10 + (current[n] - '0');
						if (static_cast<unsigned char>(current[n]) <= ' ' && v <= 0xffffffffULL) {
							current += n;
							value = T(v);
							return true;
						}
					}
				}
			}
			T parsed{};
			auto result = std::from_chars(current, end, parsed);
			if (result.ec == std::errc{} && (result.ptr != end ? static_cast<unsigned char>(*result.ptr) <= ' ' : !file)) {
				current = result.ptr;
				value = parsed;
				return true;
			}
		}
		auto s = token();
		if (s.empty()) return false;
		if constexpr (std::is_same_v<T, bool>) {
			assert(s == "0" || s == "1");
			value = s[0] == '1';
		} else if constexpr (std::is_integral_v<T>) {
			auto result = std::from_chars(s.data(), s.data() + s.size(), value);
			assert(result.ec == std::errc{} && result.ptr == s.data() + s.size());
			(void)result;
		} else value = T(s);
		return true;
	}
};

class FastOutput {
	friend struct fast_io_detail::BulkOutput;
	inline static constexpr auto quads = [] {
		std::array<std::array<char, 4>, 10000> table{};
		for (int i = 0; i < 10000; ++i)
			table[i] = {char('0' + i / 1000), char('0' + i / 100 % 10), char('0' + i / 10 % 10), char('0' + i % 10)};
		return table;
	}();
	FILE *file;
	std::unique_ptr<char[]> buffer = std::make_unique_for_overwrite<char[]>(1 << 16);
	std::size_t capacity = 1 << 16, length = 0;
	char *reserve(std::size_t n) {
		if (n > capacity - length) {
			if (!flush()) throw std::runtime_error("output failed");
			if (n > capacity) {
				std::size_t next = std::max(n, 2 * capacity);
				buffer = std::make_unique_for_overwrite<char[]>(next);
				capacity = next;
			}
		}
		return buffer.get() + length;
	}

public:
	explicit FastOutput(FILE *output = stdout) : file(output) {}
	FastOutput(const FastOutput &) = delete;
	FastOutput &operator=(const FastOutput &) = delete;
	~FastOutput() { flush(); }
	bool flush() {
		std::size_t offset = 0;
		while (offset < length) {
			auto n = std::fwrite(buffer.get() + offset, 1, length - offset, file);
			if (!n) {
				std::memmove(buffer.get(), buffer.get() + offset, length - offset);
				length -= offset;
				return false;
			}
			offset += n;
		}
		length = 0;
		return std::fflush(file) == 0;
	}
	void write(char c) { *reserve(1) = c; ++length; }
	void write(__uint128_t value) {
		char digits[40];
		char *end = digits + sizeof(digits), *begin = end;
		do { *--begin = char('0' + value % 10); value /= 10; } while (value);
		write(std::string_view(begin, end - begin));
	}
	void write(__int128_t value) {
		__uint128_t magnitude = __uint128_t(value);
		if (value < 0) write('-'), magnitude = 0 - magnitude;
		write(magnitude);
	}
	void write(std::string_view s) {
		char *out = reserve(s.size());
		if (!s.empty()) std::memcpy(out, s.data(), s.size());
		length += s.size();
	}
	template <class T> requires std::is_integral_v<T>
	void write(T value) {
		char *out = reserve(32);
		if constexpr (std::is_same_v<T, bool>) *out = '0' + value, ++length;
		else if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4) {
			char *p = out;
			if (value >= 100000000) {
				auto high = value / 100000000, low = value % 100000000;
				p = std::to_chars(p, p + 2, high).ptr;
				std::memcpy(p, quads[low / 10000].data(), 4);
				std::memcpy(p + 4, quads[low % 10000].data(), 4);
				p += 8;
			} else if (value >= 10000) {
				p = std::to_chars(p, p + 4, value / 10000).ptr;
				std::memcpy(p, quads[value % 10000].data(), 4);
				p += 4;
			} else p = std::to_chars(p, p + 4, value).ptr;
			length += p - out;
		} else length += std::to_chars(out, out + 32, value).ptr - out;
	}
	template <class T> requires requires(const T &value, char *out) { value.decimal_capacity(); value.to_chars(out); }
	void write(const T &value) {
		char *out = reserve(value.decimal_capacity());
		length += value.to_chars(out) - out;
	}
	template <class T>
	void println(const T &value) { write(value); write('\n'); }
};


#line 6 "test/k_shortest_walk.test.cpp"

int main() {
	FastInput in;
	FastOutput out;
	int n = 0, m = 0, s = 0, t = 0, k = 0;
	in.read(n), in.read(m), in.read(s), in.read(t), in.read(k);
	using Graph = DirectedCSR<std::uint32_t>;
	Graph graph(n, m, [&](auto emit) {
		for (int i = 0; i < m; ++i) {
			Graph::Edge edge{};
			in.read(edge.from), in.read(edge.to), in.read(edge.weight);
			emit(edge);
		}
	});
	const auto answer = k_shortest_walk(graph, s, t, k);
	for (int i = 0; i < k; ++i) out.println(i < int(answer.size()) ? answer[i] : -1);
}