cp-algo/graph/shortest_path.hpp

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• Last update: 2025-12-12 22:29:00+01:00
• Include: #include "cp-algo/graph/shortest_path.hpp"

Depends on

• cp-algo/graph/base.hpp
• cp-algo/graph/concepts.hpp
• cp-algo/graph/edge_types.hpp
• cp-algo/structures/stack_union.hpp
• cp-algo/util/big_alloc.hpp

Verified with

• Minimum Diameter Spanning Tree (verify/graph/minimum_diameter_spanning_tree.test.cpp)
• Shortest Path (verify/graph/shortest_path.test.cpp)
• Tree Diameter (SPFA) (verify/graph/tree_diameter_spfa.test.cpp)

Code

#ifndef CP_ALGO_GRAPH_SHORTEST_PATH_HPP
#define CP_ALGO_GRAPH_SHORTEST_PATH_HPP
#include "base.hpp"
#include <algorithm>
#include <queue>
namespace cp_algo::graph {
    struct shortest_path_context {
        big_vector<int64_t> dist;
        big_vector<edge_index> pre;
        static constexpr int64_t inf = 1e18;
        shortest_path_context(int n)
            : dist(n, inf), pre(n) {}
    };

    struct dijkstra_context: shortest_path_context {
        struct que_t {
            int64_t dist;
            node_index v;
            bool operator<(que_t const& other) const {
                return dist > other.dist;
            }
        };
        big_priority_queue<que_t> pq;

        dijkstra_context(int n) : shortest_path_context(n) {}

        void push(node_index, edge_index, node_index v) {
            pq.push({dist[v], v});
        }

        std::optional<node_index> next_node() {
            while (!empty(pq)) {
                auto [dv, v] = pq.top();
                pq.pop();
                if (dv == dist[v]) {
                    return v;
                }
            }
            return std::nullopt;
        }
    };

    struct spfa_context: shortest_path_context {
        big_queue<node_index> que;
        big_vector<char> flags;
        static constexpr char in_queue = 1;
        static constexpr char invalidated = 2;

        spfa_context(int n) : shortest_path_context(n), flags(n) {}

        void push(node_index, edge_index, node_index v) {
            if (!(flags[v] & in_queue)) {
                que.push(v);
                flags[v] |= in_queue;
            }
        }

        std::optional<node_index> next_node() {
            while (!que.empty()) {
                node_index v = que.front();
                que.pop();
                flags[v] &= ~in_queue;
                if (!(flags[v] & invalidated)) {
                    return v;
                }
            }
            return std::nullopt;
        }
    };

    struct deep_spfa_context: spfa_context {
        struct traverse_edge {
            edge_index e;
            node_index v;
        };
        big_vector<big_basic_string<traverse_edge>> dependents;

        deep_spfa_context(int n) : spfa_context(n), dependents(n) {}

        void push(node_index u, edge_index e, node_index v) {
            invalidate_subtree(v);
            dependents[u].push_back({e, v});
            flags[v] &= ~invalidated;
            spfa_context::push(u, e, v);
        }

        void invalidate_subtree(node_index v) {
            big_vector<node_index> to_invalidate = {v};
            while (!empty(to_invalidate)) {
                node_index u = to_invalidate.back();
                to_invalidate.pop_back();
                flags[u] |= invalidated;
                flags[u] &= ~in_queue;
                for (auto [e, v]: dependents[u]) {
                    if (pre[v] == e) {
                        to_invalidate.push_back(v);
                    }
                }
                dependents[u].clear();
            }
        }
    };

    template<typename Context, weighted_graph_type graph>
    Context sssp_impl(graph const& g, node_index s) {
        Context context(g.n());
        context.dist[s] = 0;
        context.pre[s] = -1;
        context.push(s, -1, s);
        while(auto ov = context.next_node()) {
            node_index v = *ov;
            for(auto e: g.outgoing(v)) {
                node_index u = g.edge(e).traverse(v);
                auto w = g.edge(e).w;
                if(context.dist[v] + w < context.dist[u]) {
                    context.dist[u] = context.dist[v] + w;
                    context.pre[u] = e;
                    context.push(v, e, u);
                }
            }
        }
        return context;
    }

    template<weighted_graph_type graph>
    shortest_path_context dijkstra(graph const& g, node_index s) {
        return sssp_impl<dijkstra_context>(g, s);
    }
    template<weighted_graph_type graph>
    shortest_path_context spfa(graph const& g, node_index s) {
        return sssp_impl<spfa_context>(g, s);
    }
    template<weighted_graph_type graph>
    shortest_path_context deep_spfa(graph const& g, node_index s) {
        return sssp_impl<deep_spfa_context>(g, s);
    }
    
    template<weighted_graph_type graph>
    shortest_path_context single_source_shortest_path(graph const& g, node_index s) {
        bool negative_edges = false;
        for (auto e: g.edges()) {
            negative_edges |= e.w < 0;
        }
        return negative_edges ? deep_spfa(g, s) : dijkstra(g, s);
    }

    big_vector<edge_index> recover_path(auto const& g, auto const& pre, node_index s, node_index t) {
        big_vector<edge_index> path;
        node_index v = t;
        while(v != s) {
            path.push_back(pre[v]);
            v = g.edge(pre[v]).traverse(v);
        }
        std::ranges::reverse(path);
        return path;
    }

    template<weighted_graph_type graph>
    std::optional<std::pair<int64_t, big_vector<edge_index>>> shortest_path(graph const& g, node_index s, node_index t) {
        auto [dist, pre] = single_source_shortest_path(g, s);
        if (dist[t] == shortest_path_context::inf) {
            return std::nullopt;
        }
        return {{dist[t], recover_path(g, pre, s, t)}};
    }
}
#endif // CP_ALGO_GRAPH_SHORTEST_PATH_HPP

#line 1 "cp-algo/graph/shortest_path.hpp"


#line 1 "cp-algo/graph/base.hpp"


#line 1 "cp-algo/graph/edge_types.hpp"


#include <iostream>
#include <cstdint>
namespace cp_algo::graph {
    using node_index = int;
    struct edge_base {
        int xor_nodes;

        edge_base() {}
        edge_base(node_index from, node_index to): xor_nodes(from ^ to) {}

        // Given one endpoint, return the other
        node_index traverse(node_index from) const {
            return xor_nodes ^ from;
        }

        static auto read(node_index v0 = 0) {
            node_index u, v;
            std::cin >> u >> v;
            u -= v0;
            v -= v0;
            return std::pair{u, edge_base(u, v)};
        }
    };

    struct weighted_edge: edge_base {
        int64_t w;

        weighted_edge() {}
        weighted_edge(node_index from, node_index to, int64_t w): edge_base(from, to), w(w) {}

        static auto read(node_index v0 = 0) {
            auto [u, e] = edge_base::read(v0);
            int64_t w;
            std::cin >> w;
            return std::pair{u, weighted_edge(u, e.traverse(u), w)};
        }
    };

    template<typename edge>
    concept edge_type = std::is_base_of_v<edge_base, edge>;
    template<typename edge>
    concept weighted_edge_type = std::is_base_of_v<weighted_edge, edge>;
}

#line 1 "cp-algo/graph/concepts.hpp"


#line 4 "cp-algo/graph/concepts.hpp"
#include <type_traits>

namespace cp_algo::graph {
    // Shared graph mode enum for all graph headers
    enum graph_mode { directed, undirected };
    // Traits: true for types that expose `edge_t` and static `mode`
    template<typename T, typename = void>
    struct graph_traits : std::false_type {};

    template<typename T>
    struct graph_traits<T, std::void_t<typename T::edge_t, decltype(T::mode)>> : std::true_type {
        using edge_t = typename T::edge_t;
        static constexpr auto mode = T::mode;
        static constexpr bool is_directed = mode == directed;
        static constexpr bool is_undirected = mode == undirected;
        static constexpr bool is_weighted = weighted_edge_type<edge_t>;
    };

    // Concepts
    template<typename G>
    concept graph_type = graph_traits<G>::value;

    template<typename G>
    concept digraph_type = graph_type<G> && graph_traits<G>::is_directed;

    template<typename G>
    concept undirected_graph_type = graph_type<G> && graph_traits<G>::is_undirected;

    template<typename G>
    concept weighted_graph_type = graph_type<G> && graph_traits<G>::is_weighted;

    template<typename G>
    concept weighted_digraph_type = digraph_type<G> && graph_traits<G>::is_weighted;

    template<typename G>
    concept weighted_undirected_graph_type = undirected_graph_type<G> && graph_traits<G>::is_weighted;
}

#line 1 "cp-algo/structures/stack_union.hpp"


#line 1 "cp-algo/util/big_alloc.hpp"



#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>

// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif

namespace cp_algo {
    template <typename T, size_t Align = 32>
    class big_alloc {
        static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
        static_assert(std::popcount(Align) == 1, "Align must be a power of two");
    public:
        using value_type = T;
        template <class U> struct rebind { using other = big_alloc<U, Align>; };
        constexpr bool operator==(const big_alloc&) const = default;
        constexpr bool operator!=(const big_alloc&) const = default;

        big_alloc() noexcept = default;
        template <typename U, std::size_t A>
        big_alloc(const big_alloc<U, A>&) noexcept {}

        [[nodiscard]] T* allocate(std::size_t n) {
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align = std::max<std::size_t>(alignof(T),  Align);
#if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) {
                void* raw = mmap(nullptr, padded,
                                PROT_READ | PROT_WRITE,
                                MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
                madvise(raw, padded, MADV_HUGEPAGE);
                madvise(raw, padded, MADV_POPULATE_WRITE);
                return static_cast<T*>(raw);
            }
#endif
            return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
        }

        void deallocate(T* p, std::size_t n) noexcept {
            if (!p) return;
            std::size_t padded = round_up(n * sizeof(T));
            std::size_t align  = std::max<std::size_t>(alignof(T),  Align);
    #if CP_ALGO_USE_MMAP
            if (padded >= MEGABYTE) { munmap(p, padded); return; }
    #endif
            ::operator delete(p, padded, std::align_val_t(align));
        }

    private:
        static constexpr std::size_t MEGABYTE = 1 << 20;
        static constexpr std::size_t round_up(std::size_t x) noexcept {
            return (x + Align - 1) / Align * Align;
        }
    };

    template<typename T> using big_vector = std::vector<T, big_alloc<T>>;
    template<typename T> using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
    template<typename T> using big_deque = std::deque<T, big_alloc<T>>;
    template<typename T> using big_stack = std::stack<T, big_deque<T>>;
    template<typename T> using big_queue = std::queue<T, big_deque<T>>;
    template<typename T> using big_priority_queue = std::priority_queue<T, big_vector<T>>;
    template<typename T> using big_forward_list = std::forward_list<T, big_alloc<T>>;
    using big_string = big_basic_string<char>;

    template<typename Key, typename Value, typename Compare = std::less<Key>>
    using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
    template<typename T, typename Compare = std::less<T>>
    using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
    template<typename T, typename Compare = std::less<T>>
    using big_set = std::set<T, Compare, big_alloc<T>>;
    template<typename Ref, typename V = void>

    using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}

// Deduction guide to make elements_of with big_generator default to big_alloc
namespace std::ranges {
    template<typename Ref, typename V>
    elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}


#line 5 "cp-algo/structures/stack_union.hpp"
#include <iterator>
#include <ranges>
namespace cp_algo::structures {
    template<class datatype>
    struct stack_union {
        stack_union(int n = 0): head(n), next(1), data(1) {}

        void push(int v, datatype const& vdata) {
            next.push_back(head[v]);
            head[v] = (int)std::size(next) - 1;
            data.push_back(vdata);
        }
        template<typename... Args>
        void emplace(int v, Args&&... vdata) {
            next.push_back(head[v]);
            head[v] = (int)std::size(next) - 1;
            data.emplace_back(std::forward<Args>(vdata)...);
        }

        void reserve(int m) {
            data.reserve(m);
            next.reserve(m);
        }

        size_t size() const {return std::size(head);}
        size_t nodes() const {return std::size(data);}

        template<typename Su>
        struct _iterator {
            using value_type = std::conditional_t<std::is_const_v<Su>, const datatype, datatype>;
            using difference_type = std::ptrdiff_t;

            Su* su = nullptr;
            int sv = 0;

            value_type& operator*() const { return su->data[sv]; }
            _iterator& operator++() { 
                sv = su->next[sv];
                return *this; 
            }
            _iterator operator++(int) { auto tmp = *this; ++*this; return tmp; }
            friend bool operator==(_iterator const& it, std::default_sentinel_t) { 
                return it.sv == 0;
            }
        };

        using iterator = _iterator<stack_union<datatype>>;
        using const_iterator = _iterator<const stack_union<datatype>>;

        auto operator[](this auto&& self, int v) {
            using Iter = _iterator<std::remove_reference_t<decltype(self)>>;
            return std::ranges::subrange(Iter{&self, self.head[v]}, std::default_sentinel);
        }

        big_vector<int> head, next;
        big_vector<datatype> data;
    };
}

#line 7 "cp-algo/graph/base.hpp"
namespace cp_algo::graph {
    using edge_index = int;
    template<edge_type _edge_t = edge_base, graph_mode _mode = undirected>
    struct graph {
        using edge_t = _edge_t;
        static constexpr auto mode = _mode;
        using incidence_list = structures::stack_union<edge_index>;
        graph(int n, int v0 = 0): v0(v0), adj(n) {}

        graph transpose() const {
            static_assert(mode == directed, "transpose is only defined for directed graphs");
            graph<edge_t, mode> gt(n(), v0);
            for(auto v: nodes()) {
                for(auto e: outgoing(v)) {
                    gt.add_edge(edge(e).traverse(v), edge(e));
                }
            }
            return gt;
        }
        edge_index add_edge(node_index u, edge_t e) {
            edge_index idx = (edge_index)size(E);
            E.push_back(e);
            adj.push(u, idx);
            if constexpr (mode == undirected) {
                adj.push(e.traverse(u), idx);
            }
            return idx;
        }
        edge_index add_edge(node_index u, auto... Args) {
            return add_edge(u, edge_t(u, Args...));
        }
        void read_edges(node_index m) {
            adj.reserve(mode == undirected ? 2 * m : m);
            for(edge_index i = 0; i < m; i++) {
                auto [u, e] = edge_t::read(v0);
                add_edge(u, e);
            }
        }
        auto outgoing(node_index v) const {return adj[v];}
        auto edges() const {return E | std::views::all;}
        auto nodes() const {return std::views::iota(node_index(0), n());}
        auto edge_indices() const {return std::views::iota(edge_index(0), m());}
        auto&& incidence_lists(this auto&& self) {return self.adj;}
        auto&& edge(this auto&& self, edge_index e) {return self.E[e];}
        node_index n() const {return (node_index)incidence_lists().size();}
        edge_index m() const {return (edge_index)edges().size();}
    private:
        node_index v0;
        big_vector<edge_t> E;
        incidence_list adj;
    };
    // aliases for most standard cases
    template<edge_type edge_t = edge_base>
    using digraph = graph<edge_t, directed>;
    template<weighted_edge_type edge_t = weighted_edge, graph_mode mode = undirected>
    using weighted_graph = graph<edge_t, mode>;
    template<weighted_edge_type edge_t = weighted_edge>
    using weighted_digraph = digraph<edge_t>;
}

#line 4 "cp-algo/graph/shortest_path.hpp"
#include <algorithm>
#line 6 "cp-algo/graph/shortest_path.hpp"
namespace cp_algo::graph {
    struct shortest_path_context {
        big_vector<int64_t> dist;
        big_vector<edge_index> pre;
        static constexpr int64_t inf = 1e18;
        shortest_path_context(int n)
            : dist(n, inf), pre(n) {}
    };

    struct dijkstra_context: shortest_path_context {
        struct que_t {
            int64_t dist;
            node_index v;
            bool operator<(que_t const& other) const {
                return dist > other.dist;
            }
        };
        big_priority_queue<que_t> pq;

        dijkstra_context(int n) : shortest_path_context(n) {}

        void push(node_index, edge_index, node_index v) {
            pq.push({dist[v], v});
        }

        std::optional<node_index> next_node() {
            while (!empty(pq)) {
                auto [dv, v] = pq.top();
                pq.pop();
                if (dv == dist[v]) {
                    return v;
                }
            }
            return std::nullopt;
        }
    };

    struct spfa_context: shortest_path_context {
        big_queue<node_index> que;
        big_vector<char> flags;
        static constexpr char in_queue = 1;
        static constexpr char invalidated = 2;

        spfa_context(int n) : shortest_path_context(n), flags(n) {}

        void push(node_index, edge_index, node_index v) {
            if (!(flags[v] & in_queue)) {
                que.push(v);
                flags[v] |= in_queue;
            }
        }

        std::optional<node_index> next_node() {
            while (!que.empty()) {
                node_index v = que.front();
                que.pop();
                flags[v] &= ~in_queue;
                if (!(flags[v] & invalidated)) {
                    return v;
                }
            }
            return std::nullopt;
        }
    };

    struct deep_spfa_context: spfa_context {
        struct traverse_edge {
            edge_index e;
            node_index v;
        };
        big_vector<big_basic_string<traverse_edge>> dependents;

        deep_spfa_context(int n) : spfa_context(n), dependents(n) {}

        void push(node_index u, edge_index e, node_index v) {
            invalidate_subtree(v);
            dependents[u].push_back({e, v});
            flags[v] &= ~invalidated;
            spfa_context::push(u, e, v);
        }

        void invalidate_subtree(node_index v) {
            big_vector<node_index> to_invalidate = {v};
            while (!empty(to_invalidate)) {
                node_index u = to_invalidate.back();
                to_invalidate.pop_back();
                flags[u] |= invalidated;
                flags[u] &= ~in_queue;
                for (auto [e, v]: dependents[u]) {
                    if (pre[v] == e) {
                        to_invalidate.push_back(v);
                    }
                }
                dependents[u].clear();
            }
        }
    };

    template<typename Context, weighted_graph_type graph>
    Context sssp_impl(graph const& g, node_index s) {
        Context context(g.n());
        context.dist[s] = 0;
        context.pre[s] = -1;
        context.push(s, -1, s);
        while(auto ov = context.next_node()) {
            node_index v = *ov;
            for(auto e: g.outgoing(v)) {
                node_index u = g.edge(e).traverse(v);
                auto w = g.edge(e).w;
                if(context.dist[v] + w < context.dist[u]) {
                    context.dist[u] = context.dist[v] + w;
                    context.pre[u] = e;
                    context.push(v, e, u);
                }
            }
        }
        return context;
    }

    template<weighted_graph_type graph>
    shortest_path_context dijkstra(graph const& g, node_index s) {
        return sssp_impl<dijkstra_context>(g, s);
    }
    template<weighted_graph_type graph>
    shortest_path_context spfa(graph const& g, node_index s) {
        return sssp_impl<spfa_context>(g, s);
    }
    template<weighted_graph_type graph>
    shortest_path_context deep_spfa(graph const& g, node_index s) {
        return sssp_impl<deep_spfa_context>(g, s);
    }
    
    template<weighted_graph_type graph>
    shortest_path_context single_source_shortest_path(graph const& g, node_index s) {
        bool negative_edges = false;
        for (auto e: g.edges()) {
            negative_edges |= e.w < 0;
        }
        return negative_edges ? deep_spfa(g, s) : dijkstra(g, s);
    }

    big_vector<edge_index> recover_path(auto const& g, auto const& pre, node_index s, node_index t) {
        big_vector<edge_index> path;
        node_index v = t;
        while(v != s) {
            path.push_back(pre[v]);
            v = g.edge(pre[v]).traverse(v);
        }
        std::ranges::reverse(path);
        return path;
    }

    template<weighted_graph_type graph>
    std::optional<std::pair<int64_t, big_vector<edge_index>>> shortest_path(graph const& g, node_index s, node_index t) {
        auto [dist, pre] = single_source_shortest_path(g, s);
        if (dist[t] == shortest_path_context::inf) {
            return std::nullopt;
        }
        return {{dist[t], recover_path(g, pre, s, t)}};
    }
}

#ifndef CP_ALGO_GRAPH_SHORTEST_PATH_HPP
#define CP_ALGO_GRAPH_SHORTEST_PATH_HPP
#include "base.hpp"
#include <algorithm>
#include <queue>
namespace cp_algo::graph{struct shortest_path_context{big_vector<int64_t>dist;big_vector<edge_index>pre;static constexpr int64_t inf=1e18;shortest_path_context(int n):dist(n,inf),pre(n){}};struct dijkstra_context:shortest_path_context{struct que_t{int64_t dist;node_index v;bool operator<(que_t const&other)const{return dist>other.dist;}};big_priority_queue<que_t>pq;dijkstra_context(int n):shortest_path_context(n){}void push(node_index,edge_index,node_index v){pq.push({dist[v],v});}std::optional<node_index>next_node(){while(!empty(pq)){auto[dv,v]=pq.top();pq.pop();if(dv==dist[v]){return v;}}return std::nullopt;}};struct spfa_context:shortest_path_context{big_queue<node_index>que;big_vector<char>flags;static constexpr char in_queue=1;static constexpr char invalidated=2;spfa_context(int n):shortest_path_context(n),flags(n){}void push(node_index,edge_index,node_index v){if(!(flags[v]&in_queue)){que.push(v);flags[v]|=in_queue;}}std::optional<node_index>next_node(){while(!que.empty()){node_index v=que.front();que.pop();flags[v]&=~in_queue;if(!(flags[v]&invalidated)){return v;}}return std::nullopt;}};struct deep_spfa_context:spfa_context{struct traverse_edge{edge_index e;node_index v;};big_vector<big_basic_string<traverse_edge>>dependents;deep_spfa_context(int n):spfa_context(n),dependents(n){}void push(node_index u,edge_index e,node_index v){invalidate_subtree(v);dependents[u].push_back({e,v});flags[v]&=~invalidated;spfa_context::push(u,e,v);}void invalidate_subtree(node_index v){big_vector<node_index>to_invalidate={v};while(!empty(to_invalidate)){node_index u=to_invalidate.back();to_invalidate.pop_back();flags[u]|=invalidated;flags[u]&=~in_queue;for(auto[e,v]:dependents[u]){if(pre[v]==e){to_invalidate.push_back(v);}}dependents[u].clear();}}};template<typename Context,weighted_graph_type graph>Context sssp_impl(graph const&g,node_index s){Context context(g.n());context.dist[s]=0;context.pre[s]=-1;context.push(s,-1,s);while(auto ov=context.next_node()){node_index v=*ov;for(auto e:g.outgoing(v)){node_index u=g.edge(e).traverse(v);auto w=g.edge(e).w;if(context.dist[v]+w<context.dist[u]){context.dist[u]=context.dist[v]+w;context.pre[u]=e;context.push(v,e,u);}}}return context;}template<weighted_graph_type graph>shortest_path_context dijkstra(graph const&g,node_index s){return sssp_impl<dijkstra_context>(g,s);}template<weighted_graph_type graph>shortest_path_context spfa(graph const&g,node_index s){return sssp_impl<spfa_context>(g,s);}template<weighted_graph_type graph>shortest_path_context deep_spfa(graph const&g,node_index s){return sssp_impl<deep_spfa_context>(g,s);}template<weighted_graph_type graph>shortest_path_context single_source_shortest_path(graph const&g,node_index s){bool negative_edges=false;for(auto e:g.edges()){negative_edges|=e.w<0;}return negative_edges?deep_spfa(g,s):dijkstra(g,s);}big_vector<edge_index>recover_path(auto const&g,auto const&pre,node_index s,node_index t){big_vector<edge_index>path;node_index v=t;while(v!=s){path.push_back(pre[v]);v=g.edge(pre[v]).traverse(v);}std::ranges::reverse(path);return path;}template<weighted_graph_type graph>std::optional<std::pair<int64_t,big_vector<edge_index>>>shortest_path(graph const&g,node_index s,node_index t){auto[dist,pre]=single_source_shortest_path(g,s);if(dist[t]==shortest_path_context::inf){return std::nullopt;}return{{dist[t],recover_path(g,pre,s,t)}};}}
#endif

#line 1 "cp-algo/graph/shortest_path.hpp"
#line 1 "cp-algo/graph/base.hpp"
#line 1 "cp-algo/graph/edge_types.hpp"
#include <iostream>
#include <cstdint>
namespace cp_algo::graph{using node_index=int;struct edge_base{int xor_nodes;edge_base(){}edge_base(node_index from,node_index to):xor_nodes(from^to){}node_index traverse(node_index from)const{return xor_nodes^from;}static auto read(node_index v0=0){node_index u,v;std::cin>>u>>v;u-=v0;v-=v0;return std::pair{u,edge_base(u,v)};}};struct weighted_edge:edge_base{int64_t w;weighted_edge(){}weighted_edge(node_index from,node_index to,int64_t w):edge_base(from,to),w(w){}static auto read(node_index v0=0){auto[u,e]=edge_base::read(v0);int64_t w;std::cin>>w;return std::pair{u,weighted_edge(u,e.traverse(u),w)};}};template<typename edge>concept edge_type=std::is_base_of_v<edge_base,edge>;template<typename edge>concept weighted_edge_type=std::is_base_of_v<weighted_edge,edge>;}
#line 1 "cp-algo/graph/concepts.hpp"
#line 4 "cp-algo/graph/concepts.hpp"
#include <type_traits>
namespace cp_algo::graph{enum graph_mode{directed,undirected};template<typename T,typename=void>struct graph_traits:std::false_type{};template<typename T>struct graph_traits<T,std::void_t<typename T::edge_t,decltype(T::mode)>>:std::true_type{using edge_t=typename T::edge_t;static constexpr auto mode=T::mode;static constexpr bool is_directed=mode==directed;static constexpr bool is_undirected=mode==undirected;static constexpr bool is_weighted=weighted_edge_type<edge_t>;};template<typename G>concept graph_type=graph_traits<G>::value;template<typename G>concept digraph_type=graph_type<G>&&graph_traits<G>::is_directed;template<typename G>concept undirected_graph_type=graph_type<G>&&graph_traits<G>::is_undirected;template<typename G>concept weighted_graph_type=graph_type<G>&&graph_traits<G>::is_weighted;template<typename G>concept weighted_digraph_type=digraph_type<G>&&graph_traits<G>::is_weighted;template<typename G>concept weighted_undirected_graph_type=undirected_graph_type<G>&&graph_traits<G>::is_weighted;}
#line 1 "cp-algo/structures/stack_union.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#  define CP_ALGO_USE_MMAP 1
#  include <sys/mman.h>
#else
#  define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{template<typename T,size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename T>using big_stack=std::stack<T,big_deque<T>>;template<typename T>using big_queue=std::queue<T,big_deque<T>>;template<typename T>using big_priority_queue=std::priority_queue<T,big_vector<T>>;template<typename T>using big_forward_list=std::forward_list<T,big_alloc<T>>;using big_string=big_basic_string<char>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;template<typename T,typename Compare=std::less<T>>using big_multiset=std::multiset<T,Compare,big_alloc<T>>;template<typename T,typename Compare=std::less<T>>using big_set=std::set<T,Compare,big_alloc<T>>;template<typename Ref,typename V=void>using big_generator=std::generator<Ref,V,big_alloc<std::byte>>;}namespace std::ranges{template<typename Ref,typename V>elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>;}
#line 5 "cp-algo/structures/stack_union.hpp"
#include <iterator>
#include <ranges>
namespace cp_algo::structures{template<class datatype>struct stack_union{stack_union(int n=0):head(n),next(1),data(1){}void push(int v,datatype const&vdata){next.push_back(head[v]);head[v]=(int)std::size(next)-1;data.push_back(vdata);}template<typename... Args>void emplace(int v,Args&&... vdata){next.push_back(head[v]);head[v]=(int)std::size(next)-1;data.emplace_back(std::forward<Args>(vdata)...);}void reserve(int m){data.reserve(m);next.reserve(m);}size_t size()const{return std::size(head);}size_t nodes()const{return std::size(data);}template<typename Su>struct _iterator{using value_type=std::conditional_t<std::is_const_v<Su>,const datatype,datatype>;using difference_type=std::ptrdiff_t;Su*su=nullptr;int sv=0;value_type&operator*()const{return su->data[sv];}_iterator&operator++(){sv=su->next[sv];return*this;}_iterator operator++(int){auto tmp=*this;++*this;return tmp;}friend bool operator==(_iterator const&it,std::default_sentinel_t){return it.sv==0;}};using iterator=_iterator<stack_union<datatype>>;using const_iterator=_iterator<const stack_union<datatype>>;auto operator[](this auto&&self,int v){using Iter=_iterator<std::remove_reference_t<decltype(self)>>;return std::ranges::subrange(Iter{&self,self.head[v]},std::default_sentinel);}big_vector<int>head,next;big_vector<datatype>data;};}
#line 7 "cp-algo/graph/base.hpp"
namespace cp_algo::graph{using edge_index=int;template<edge_type _edge_t=edge_base,graph_mode _mode=undirected>struct graph{using edge_t=_edge_t;static constexpr auto mode=_mode;using incidence_list=structures::stack_union<edge_index>;graph(int n,int v0=0):v0(v0),adj(n){}graph transpose()const{static_assert(mode==directed,"transpose is only defined for directed graphs");graph<edge_t,mode>gt(n(),v0);for(auto v:nodes()){for(auto e:outgoing(v)){gt.add_edge(edge(e).traverse(v),edge(e));}}return gt;}edge_index add_edge(node_index u,edge_t e){edge_index idx=(edge_index)size(E);E.push_back(e);adj.push(u,idx);if constexpr(mode==undirected){adj.push(e.traverse(u),idx);}return idx;}edge_index add_edge(node_index u,auto... Args){return add_edge(u,edge_t(u,Args...));}void read_edges(node_index m){adj.reserve(mode==undirected?2*m:m);for(edge_index i=0;i<m;i++){auto[u,e]=edge_t::read(v0);add_edge(u,e);}}auto outgoing(node_index v)const{return adj[v];}auto edges()const{return E|std::views::all;}auto nodes()const{return std::views::iota(node_index(0),n());}auto edge_indices()const{return std::views::iota(edge_index(0),m());}auto&&incidence_lists(this auto&&self){return self.adj;}auto&&edge(this auto&&self,edge_index e){return self.E[e];}node_index n()const{return(node_index)incidence_lists().size();}edge_index m()const{return(edge_index)edges().size();}private:node_index v0;big_vector<edge_t>E;incidence_list adj;};template<edge_type edge_t=edge_base>using digraph=graph<edge_t,directed>;template<weighted_edge_type edge_t=weighted_edge,graph_mode mode=undirected>using weighted_graph=graph<edge_t,mode>;template<weighted_edge_type edge_t=weighted_edge>using weighted_digraph=digraph<edge_t>;}
#line 4 "cp-algo/graph/shortest_path.hpp"
#include <algorithm>
#line 6 "cp-algo/graph/shortest_path.hpp"
namespace cp_algo::graph{struct shortest_path_context{big_vector<int64_t>dist;big_vector<edge_index>pre;static constexpr int64_t inf=1e18;shortest_path_context(int n):dist(n,inf),pre(n){}};struct dijkstra_context:shortest_path_context{struct que_t{int64_t dist;node_index v;bool operator<(que_t const&other)const{return dist>other.dist;}};big_priority_queue<que_t>pq;dijkstra_context(int n):shortest_path_context(n){}void push(node_index,edge_index,node_index v){pq.push({dist[v],v});}std::optional<node_index>next_node(){while(!empty(pq)){auto[dv,v]=pq.top();pq.pop();if(dv==dist[v]){return v;}}return std::nullopt;}};struct spfa_context:shortest_path_context{big_queue<node_index>que;big_vector<char>flags;static constexpr char in_queue=1;static constexpr char invalidated=2;spfa_context(int n):shortest_path_context(n),flags(n){}void push(node_index,edge_index,node_index v){if(!(flags[v]&in_queue)){que.push(v);flags[v]|=in_queue;}}std::optional<node_index>next_node(){while(!que.empty()){node_index v=que.front();que.pop();flags[v]&=~in_queue;if(!(flags[v]&invalidated)){return v;}}return std::nullopt;}};struct deep_spfa_context:spfa_context{struct traverse_edge{edge_index e;node_index v;};big_vector<big_basic_string<traverse_edge>>dependents;deep_spfa_context(int n):spfa_context(n),dependents(n){}void push(node_index u,edge_index e,node_index v){invalidate_subtree(v);dependents[u].push_back({e,v});flags[v]&=~invalidated;spfa_context::push(u,e,v);}void invalidate_subtree(node_index v){big_vector<node_index>to_invalidate={v};while(!empty(to_invalidate)){node_index u=to_invalidate.back();to_invalidate.pop_back();flags[u]|=invalidated;flags[u]&=~in_queue;for(auto[e,v]:dependents[u]){if(pre[v]==e){to_invalidate.push_back(v);}}dependents[u].clear();}}};template<typename Context,weighted_graph_type graph>Context sssp_impl(graph const&g,node_index s){Context context(g.n());context.dist[s]=0;context.pre[s]=-1;context.push(s,-1,s);while(auto ov=context.next_node()){node_index v=*ov;for(auto e:g.outgoing(v)){node_index u=g.edge(e).traverse(v);auto w=g.edge(e).w;if(context.dist[v]+w<context.dist[u]){context.dist[u]=context.dist[v]+w;context.pre[u]=e;context.push(v,e,u);}}}return context;}template<weighted_graph_type graph>shortest_path_context dijkstra(graph const&g,node_index s){return sssp_impl<dijkstra_context>(g,s);}template<weighted_graph_type graph>shortest_path_context spfa(graph const&g,node_index s){return sssp_impl<spfa_context>(g,s);}template<weighted_graph_type graph>shortest_path_context deep_spfa(graph const&g,node_index s){return sssp_impl<deep_spfa_context>(g,s);}template<weighted_graph_type graph>shortest_path_context single_source_shortest_path(graph const&g,node_index s){bool negative_edges=false;for(auto e:g.edges()){negative_edges|=e.w<0;}return negative_edges?deep_spfa(g,s):dijkstra(g,s);}big_vector<edge_index>recover_path(auto const&g,auto const&pre,node_index s,node_index t){big_vector<edge_index>path;node_index v=t;while(v!=s){path.push_back(pre[v]);v=g.edge(pre[v]).traverse(v);}std::ranges::reverse(path);return path;}template<weighted_graph_type graph>std::optional<std::pair<int64_t,big_vector<edge_index>>>shortest_path(graph const&g,node_index s,node_index t){auto[dist,pre]=single_source_shortest_path(g,s);if(dist[t]==shortest_path_context::inf){return std::nullopt;}return{{dist[t],recover_path(g,pre,s,t)}};}}

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