Luzhiled's Library

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:x: Namori Graph
(graph/others/namori-graph.hpp)

概要

$n$ 頂点 $n$ 辺からなる連結無向グラフは, サイクルが $1$ 個だけあるグラフとなる。

このグラフを, とある漫画家のアイコンにちなんで なもりグラフ と呼ばれることがるが, 学術的には Unicyclic Graph, Pseudoforest が正しい。

ここでは, このグラフを 1 つのサイクル と, サイクル内の頂点に付属する木に分解する。またサイクルに含まれる頂点番号を, サイクルの頂点数を $k$ として $[0, k)$ にふりなおし, これを tree_id と呼ぶことにする。

また付属する木も同様に, 木の頂点数を $l$ として $[0, l)$ にふりなおす。

使い方

計算量

Depends on

Verified with

Code

#pragma once

#include "../graph-template.hpp"

/**
 * @brief Namori Graph
 * @docs docs/namori-graph.md
 */
template< typename T = int >
struct NamoriGraph : Graph< T > {
public:
  using Graph< T >::Graph;
  using Graph< T >::g;

  vector< Graph< T > > forest;
  Edges< T > loop_edges;

  struct Info {
    int tree_id, id;
  };

  Info operator[](const int &k) const {
    return (Info) {mark_id[k], id[k]};
  }

  int inv(int tree_id, int k) {
    return iv[tree_id][k];
  }

  void build() {
    int n = (int) g.size();
    vector< int > deg(n), used(n);
    queue< int > que;
    for(int i = 0; i < n; i++) {
      deg[i] = (int) g[i].size();
      if(deg[i] == 1) {
        que.emplace(i);
        used[i] = true;
      }
    }
    while(not que.empty()) {
      int idx = que.front();
      que.pop();
      for(auto &e : g[idx]) {
        if(used[e.to]) {
          continue;
        }
        --deg[e.to];
        if(deg[e.to] == 1) {
          que.emplace(e.to);
          used[e.to] = true;
        }
      }
    }
    int mx = 0;
    for(auto &edges : g) {
      for(auto &e : edges) mx = max(mx, e.idx);
    }
    vector< int > edge_used(mx + 1);
    vector< int > loop;
    for(int v = 0; v < n; v++) {
      if(!used[v]) {
        for(bool update = true; update;) {
          update = false;
          loop.emplace_back(v);
          for(auto &e : g[v]) {
            if(used[e.to] or edge_used[e.idx]) {
              continue;
            }
            edge_used[e.idx] = true;
            loop_edges.emplace_back(v, e.to, e.cost, e.idx);
            v = e.to;
            update = true;
            break;
          }
        }
        break;
      }
    }
    loop.pop_back();
    mark_id.resize(n);
    id.resize(n);
    for(int i = 0; i < (int) loop.size(); i++) {
      int pre = loop[(i + loop.size() - 1) % loop.size()];
      int nxt = loop[(i + 1) % loop.size()];
      int sz = 0;
      mark_id[loop[i]] = i;
      iv.emplace_back();
      id[loop[i]] = sz++;
      iv.back().emplace_back(loop[i]);
      for(auto &e : g[loop[i]]) {
        if(e.to != pre and e.to != nxt) {
          mark_dfs(e.to, loop[i], i, sz);
        }
      }
      Graph< T > tree(sz);
      for(auto &e : g[loop[i]]) {
        if(e.to != pre and e.to != nxt) {
          tree.g[id[loop[i]]].emplace_back(id[loop[i]], id[e.to], e.cost, e.idx);
          tree.g[id[e.to]].emplace_back(id[e.to], id[loop[i]], e.cost, e.idx);
          build_dfs(e.to, loop[i], tree);
        }
      }
      forest.emplace_back(tree);
    }
  }

private:
  vector< vector< int > > iv;
  vector< int > mark_id, id;

  void mark_dfs(int idx, int par, int k, int &l) {
    mark_id[idx] = k;
    id[idx] = l++;
    iv.back().emplace_back(idx);
    for(auto &e : g[idx]) {
      if(e.to != par) {
        mark_dfs(e.to, idx, k, l);
      }
    }
  }

  void build_dfs(int idx, int par, Graph< T > &tree) {
    for(auto &e : g[idx]) {
      if(e.to != par) {
        tree.g[id[idx]].emplace_back(id[idx], id[e.to], e.cost, e.idx);
        tree.g[id[e.to]].emplace_back(id[e.to], id[idx], e.cost, e.idx);
        build_dfs(e.to, idx, tree);
      }
    }
  }
};
#line 2 "graph/others/namori-graph.hpp"

#line 2 "graph/graph-template.hpp"

/**
 * @brief Graph Template(グラフテンプレート)
 */
template< typename T = int >
struct Edge {
  int from, to;
  T cost;
  int idx;

  Edge() = default;

  Edge(int from, int to, T cost = 1, int idx = -1) : from(from), to(to), cost(cost), idx(idx) {}

  operator int() const { return to; }
};

template< typename T = int >
struct Graph {
  vector< vector< Edge< T > > > g;
  int es;

  Graph() = default;

  explicit Graph(int n) : g(n), es(0) {}

  size_t size() const {
    return g.size();
  }

  void add_directed_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es++);
  }

  void add_edge(int from, int to, T cost = 1) {
    g[from].emplace_back(from, to, cost, es);
    g[to].emplace_back(to, from, cost, es++);
  }

  void read(int M, int padding = -1, bool weighted = false, bool directed = false) {
    for(int i = 0; i < M; i++) {
      int a, b;
      cin >> a >> b;
      a += padding;
      b += padding;
      T c = T(1);
      if(weighted) cin >> c;
      if(directed) add_directed_edge(a, b, c);
      else add_edge(a, b, c);
    }
  }

  inline vector< Edge< T > > &operator[](const int &k) {
    return g[k];
  }

  inline const vector< Edge< T > > &operator[](const int &k) const {
    return g[k];
  }
};

template< typename T = int >
using Edges = vector< Edge< T > >;
#line 4 "graph/others/namori-graph.hpp"

/**
 * @brief Namori Graph
 * @docs docs/namori-graph.md
 */
template< typename T = int >
struct NamoriGraph : Graph< T > {
public:
  using Graph< T >::Graph;
  using Graph< T >::g;

  vector< Graph< T > > forest;
  Edges< T > loop_edges;

  struct Info {
    int tree_id, id;
  };

  Info operator[](const int &k) const {
    return (Info) {mark_id[k], id[k]};
  }

  int inv(int tree_id, int k) {
    return iv[tree_id][k];
  }

  void build() {
    int n = (int) g.size();
    vector< int > deg(n), used(n);
    queue< int > que;
    for(int i = 0; i < n; i++) {
      deg[i] = (int) g[i].size();
      if(deg[i] == 1) {
        que.emplace(i);
        used[i] = true;
      }
    }
    while(not que.empty()) {
      int idx = que.front();
      que.pop();
      for(auto &e : g[idx]) {
        if(used[e.to]) {
          continue;
        }
        --deg[e.to];
        if(deg[e.to] == 1) {
          que.emplace(e.to);
          used[e.to] = true;
        }
      }
    }
    int mx = 0;
    for(auto &edges : g) {
      for(auto &e : edges) mx = max(mx, e.idx);
    }
    vector< int > edge_used(mx + 1);
    vector< int > loop;
    for(int v = 0; v < n; v++) {
      if(!used[v]) {
        for(bool update = true; update;) {
          update = false;
          loop.emplace_back(v);
          for(auto &e : g[v]) {
            if(used[e.to] or edge_used[e.idx]) {
              continue;
            }
            edge_used[e.idx] = true;
            loop_edges.emplace_back(v, e.to, e.cost, e.idx);
            v = e.to;
            update = true;
            break;
          }
        }
        break;
      }
    }
    loop.pop_back();
    mark_id.resize(n);
    id.resize(n);
    for(int i = 0; i < (int) loop.size(); i++) {
      int pre = loop[(i + loop.size() - 1) % loop.size()];
      int nxt = loop[(i + 1) % loop.size()];
      int sz = 0;
      mark_id[loop[i]] = i;
      iv.emplace_back();
      id[loop[i]] = sz++;
      iv.back().emplace_back(loop[i]);
      for(auto &e : g[loop[i]]) {
        if(e.to != pre and e.to != nxt) {
          mark_dfs(e.to, loop[i], i, sz);
        }
      }
      Graph< T > tree(sz);
      for(auto &e : g[loop[i]]) {
        if(e.to != pre and e.to != nxt) {
          tree.g[id[loop[i]]].emplace_back(id[loop[i]], id[e.to], e.cost, e.idx);
          tree.g[id[e.to]].emplace_back(id[e.to], id[loop[i]], e.cost, e.idx);
          build_dfs(e.to, loop[i], tree);
        }
      }
      forest.emplace_back(tree);
    }
  }

private:
  vector< vector< int > > iv;
  vector< int > mark_id, id;

  void mark_dfs(int idx, int par, int k, int &l) {
    mark_id[idx] = k;
    id[idx] = l++;
    iv.back().emplace_back(idx);
    for(auto &e : g[idx]) {
      if(e.to != par) {
        mark_dfs(e.to, idx, k, l);
      }
    }
  }

  void build_dfs(int idx, int par, Graph< T > &tree) {
    for(auto &e : g[idx]) {
      if(e.to != par) {
        tree.g[id[idx]].emplace_back(id[idx], id[e.to], e.cost, e.idx);
        tree.g[id[e.to]].emplace_back(id[e.to], id[idx], e.cost, e.idx);
        build_dfs(e.to, idx, tree);
      }
    }
  }
};
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