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:heavy_check_mark: graph/shortest_path/nonzero_group_product_shortest_path.hpp

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#include "graph/base.hpp"
#include "graph/shortest_path/dijkstra.hpp"

// given: group-labeled undirected graph G, and s.
// return: shortest path length from s to v, for each v.
// remark: path is not a walk. directed case is NP hard.
// https://arxiv.org/abs/1906.04062
template <typename WT, typename Monoid, typename GT>
vc<WT> nonzero_group_product_shortest_path(
    GT& G, vc<typename Monoid::value_type> edge_label, int s) {
  static_assert(!GT::is_directed);
  const int N = G.N, M = G.M;

  using X = typename Monoid::value_type;

  auto get = [&](int eid, int frm) -> X {
    auto& e = G.edges[eid];
    X x = edge_label[e.id];
    return (e.frm == frm ? x : Monoid::inverse(x));
  };

  // shortest path tree
  vc<WT> dist(N, infty<WT>);
  vc<X> phi(N, Monoid::unit());
  vc<int> par(N, -1);
  vc<int> depth(N);
  dist[s] = 0;
  pqg<pair<WT, int>> que;
  que.emplace(0, s);
  while (len(que)) {
    auto [dv, v] = POP(que);
    if (dv != dist[v]) continue;
    for (auto& e: G[v]) {
      if (chmin(dist[e.to], dv + e.cost)) {
        phi[e.to] = Monoid::op(phi[v], get(e.id, v));
        que.emplace(dist[e.to], e.to);
        par[e.to] = v, depth[e.to] = depth[v] + 1;
      }
    }
  }

  vc<bool> cons(M);
  FOR(i, M) {
    int a = G.edges[i].frm, b = G.edges[i].to;
    X x = Monoid::op(phi[a], edge_label[i]);
    cons[i] = (x == phi[b]);
  }

  vc<WT> h(M, infty<WT>);
  vc<WT> q(N, infty<WT>);
  for (auto& e: G.edges) {
    if (dist[e.frm] == infty<WT>) continue;
    if (!cons[e.id] && chmin(h[e.id], dist[e.frm] + dist[e.to] + e.cost)) {
      que.emplace(h[e.id], e.id);
    }
  }

  vc<int> root(N);
  FOR(v, N) root[v] = v;

  auto get_root = [&](int v) -> int {
    while (root[v] != v) { v = root[v] = root[root[v]]; }
    return v;
  };

  while (len(que)) {
    auto [x, eid] = POP(que);
    if (x != h[eid]) continue;
    int a = G.edges[eid].frm, b = G.edges[eid].to;
    a = get_root(a), b = get_root(b);
    vc<int> B;
    while (a != b) {
      if (depth[a] < depth[b]) swap(a, b);
      B.eb(a), a = get_root(par[a]);
    }
    for (auto& w: B) {
      root[w] = a, q[w] = x - dist[w];
      for (auto& e: G[w]) {
        if (cons[e.id] && chmin(h[e.id], q[w] + dist[e.to] + e.cost)) {
          que.emplace(h[e.id], e.id);
        }
      }
    }
  }

  vc<WT> ANS(N, infty<WT>);
  FOR(v, N) ANS[v] = (phi[v] == Monoid::unit() ? q[v] : dist[v]);
  return ANS;
}
#line 1 "graph/shortest_path/nonzero_group_product_shortest_path.hpp"

#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 3 "graph/shortest_path/dijkstra.hpp"

template <typename T, typename GT>
pair<vc<T>, vc<int>> dijkstra_dense(GT& G, int s) {
  const int N = G.N;
  vc<T> dist(N, infty<T>);
  vc<int> par(N, -1);
  vc<bool> done(N);
  dist[s] = 0;
  while (1) {
    int v = -1;
    T mi = infty<T>;
    FOR(i, N) {
      if (!done[i] && chmin(mi, dist[i])) v = i;
    }
    if (v == -1) break;
    done[v] = 1;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[v] + e.cost)) par[e.to] = v;
    }
  }
  return {dist, par};
}

template <typename T, typename GT, bool DENSE = false>
pair<vc<T>, vc<int>> dijkstra(GT& G, int v) {
  if (DENSE) return dijkstra_dense<T>(G, v);
  auto N = G.N;
  vector<T> dist(N, infty<T>);
  vector<int> par(N, -1);
  using P = pair<T, int>;

  priority_queue<P, vector<P>, greater<P>> que;

  dist[v] = 0;
  que.emplace(0, v);
  while (!que.empty()) {
    auto [dv, v] = que.top();
    que.pop();
    if (dv > dist[v]) continue;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
        par[e.to] = e.frm;
        que.emplace(dist[e.to], e.to);
      }
    }
  }
  return {dist, par};
}

// 多点スタート。[dist, par, root]
template <typename T, typename GT>
tuple<vc<T>, vc<int>, vc<int>> dijkstra(GT& G, vc<int> vs) {
  assert(G.is_prepared());
  int N = G.N;
  vc<T> dist(N, infty<T>);
  vc<int> par(N, -1);
  vc<int> root(N, -1);

  using P = pair<T, int>;

  priority_queue<P, vector<P>, greater<P>> que;

  for (auto&& v: vs) {
    dist[v] = 0;
    root[v] = v;
    que.emplace(T(0), v);
  }

  while (!que.empty()) {
    auto [dv, v] = que.top();
    que.pop();
    if (dv > dist[v]) continue;
    for (auto&& e: G[v]) {
      if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
        root[e.to] = root[e.frm];
        par[e.to] = e.frm;
        que.push(mp(dist[e.to], e.to));
      }
    }
  }
  return {dist, par, root};
}
#line 4 "graph/shortest_path/nonzero_group_product_shortest_path.hpp"

// given: group-labeled undirected graph G, and s.
// return: shortest path length from s to v, for each v.
// remark: path is not a walk. directed case is NP hard.
// https://arxiv.org/abs/1906.04062
template <typename WT, typename Monoid, typename GT>
vc<WT> nonzero_group_product_shortest_path(
    GT& G, vc<typename Monoid::value_type> edge_label, int s) {
  static_assert(!GT::is_directed);
  const int N = G.N, M = G.M;

  using X = typename Monoid::value_type;

  auto get = [&](int eid, int frm) -> X {
    auto& e = G.edges[eid];
    X x = edge_label[e.id];
    return (e.frm == frm ? x : Monoid::inverse(x));
  };

  // shortest path tree
  vc<WT> dist(N, infty<WT>);
  vc<X> phi(N, Monoid::unit());
  vc<int> par(N, -1);
  vc<int> depth(N);
  dist[s] = 0;
  pqg<pair<WT, int>> que;
  que.emplace(0, s);
  while (len(que)) {
    auto [dv, v] = POP(que);
    if (dv != dist[v]) continue;
    for (auto& e: G[v]) {
      if (chmin(dist[e.to], dv + e.cost)) {
        phi[e.to] = Monoid::op(phi[v], get(e.id, v));
        que.emplace(dist[e.to], e.to);
        par[e.to] = v, depth[e.to] = depth[v] + 1;
      }
    }
  }

  vc<bool> cons(M);
  FOR(i, M) {
    int a = G.edges[i].frm, b = G.edges[i].to;
    X x = Monoid::op(phi[a], edge_label[i]);
    cons[i] = (x == phi[b]);
  }

  vc<WT> h(M, infty<WT>);
  vc<WT> q(N, infty<WT>);
  for (auto& e: G.edges) {
    if (dist[e.frm] == infty<WT>) continue;
    if (!cons[e.id] && chmin(h[e.id], dist[e.frm] + dist[e.to] + e.cost)) {
      que.emplace(h[e.id], e.id);
    }
  }

  vc<int> root(N);
  FOR(v, N) root[v] = v;

  auto get_root = [&](int v) -> int {
    while (root[v] != v) { v = root[v] = root[root[v]]; }
    return v;
  };

  while (len(que)) {
    auto [x, eid] = POP(que);
    if (x != h[eid]) continue;
    int a = G.edges[eid].frm, b = G.edges[eid].to;
    a = get_root(a), b = get_root(b);
    vc<int> B;
    while (a != b) {
      if (depth[a] < depth[b]) swap(a, b);
      B.eb(a), a = get_root(par[a]);
    }
    for (auto& w: B) {
      root[w] = a, q[w] = x - dist[w];
      for (auto& e: G[w]) {
        if (cons[e.id] && chmin(h[e.id], q[w] + dist[e.to] + e.cost)) {
          que.emplace(h[e.id], e.id);
        }
      }
    }
  }

  vc<WT> ANS(N, infty<WT>);
  FOR(v, N) ANS[v] = (phi[v] == Monoid::unit() ? q[v] : dist[v]);
  return ANS;
}
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