Incremental Bridge Connectivity
(graph/connected-components/incremental-bridge-connectivity.hpp)

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• Last update: 2022-10-23 21:54:47+09:00
• Include: #include "graph/connected-components/incremental-bridge-connectivity.hpp"

概要

辺の追加クエリのみ存在するとき, 二重辺連結成分を効率的に管理するデータ構造.

使い方

• IncrementalBridgeConnectivity(sz): sz 頂点で初期化する.
• find(k): 頂点 k が属する二重辺連結成分(の代表元)を求める.
• bridge_size(): 現在の橋の個数を返す.
• add_edge(x, y): 頂点 x と y との間に無向辺を追加する.

計算量

ならし $O(n \log n)$

参考

Incremental Bridge-Connectivity - data-structures

Depends on

• Union Find (structure/union-find/union-find.hpp)

Required by

• graph/connected-components/three-edge-connected-components.hpp

Verified with

• test/verify/yosupo-three-edge-connected-components.test.cpp
• test/verify/yosupo-two-edge-connected-components-2.test.cpp

Code

#pragma once

#include "../../structure/union-find/union-find.hpp"

struct IncrementalBridgeConnectivity {
private:
  UnionFind cc, bcc;
  vector< int > bbf;
  size_t bridge;

  int size() {
    return bbf.size();
  }

  int par(int x) {
    return bbf[x] == size() ? size() : bcc.find(bbf[x]);
  }

  int lca(int x, int y) {
    unordered_set< int > used;
    for(;;) {
      if(x != size()) {
        if(!used.insert(x).second) return x;
        x = par(x);
      }
      swap(x, y);
    }
  }

  void compress(int x, int y) {
    while(bcc.find(x) != bcc.find(y)) {
      int nxt = par(x);
      bbf[x] = bbf[y];
      bcc.unite(x, y);
      x = nxt;
      --bridge;
    }
  }

  void link(int x, int y) {
    int v = x, pre = y;
    while(v != size()) {
      int nxt = par(v);
      bbf[v] = pre;
      pre = v;
      v = nxt;
    }
  }

public:
  IncrementalBridgeConnectivity() = default;

  explicit IncrementalBridgeConnectivity(int sz) : cc(sz), bcc(sz), bbf(sz, sz), bridge(0) {}

  int find(int k) {
    return bcc.find(k);
  }

  size_t bridge_size() const {
    return bridge;
  }

  void add_edge(int x, int y) {
    x = bcc.find(x);
    y = bcc.find(y);
    if(cc.find(x) == cc.find(y)) {
      int w = lca(x, y);
      compress(x, w);
      compress(y, w);
    } else {
      if(cc.size(x) > cc.size(y)) swap(x, y);
      link(x, y);
      cc.unite(x, y);
      ++bridge;
    }
  }
};

#line 2 "graph/connected-components/incremental-bridge-connectivity.hpp"

#line 2 "structure/union-find/union-find.hpp"

struct UnionFind {
  vector< int > data;

  UnionFind() = default;

  explicit UnionFind(size_t sz) : data(sz, -1) {}

  bool unite(int x, int y) {
    x = find(x), y = find(y);
    if(x == y) return false;
    if(data[x] > data[y]) swap(x, y);
    data[x] += data[y];
    data[y] = x;
    return true;
  }

  int find(int k) {
    if(data[k] < 0) return (k);
    return data[k] = find(data[k]);
  }

  int size(int k) {
    return -data[find(k)];
  }

  bool same(int x, int y) {
    return find(x) == find(y);
  }

  vector< vector< int > > groups() {
    int n = (int) data.size();
    vector< vector< int > > ret(n);
    for(int i = 0; i < n; i++) {
      ret[find(i)].emplace_back(i);
    }
    ret.erase(remove_if(begin(ret), end(ret), [&](const vector< int > &v) {
      return v.empty();
    }), end(ret));
    return ret;
  }
};
#line 4 "graph/connected-components/incremental-bridge-connectivity.hpp"

struct IncrementalBridgeConnectivity {
private:
  UnionFind cc, bcc;
  vector< int > bbf;
  size_t bridge;

  int size() {
    return bbf.size();
  }

  int par(int x) {
    return bbf[x] == size() ? size() : bcc.find(bbf[x]);
  }

  int lca(int x, int y) {
    unordered_set< int > used;
    for(;;) {
      if(x != size()) {
        if(!used.insert(x).second) return x;
        x = par(x);
      }
      swap(x, y);
    }
  }

  void compress(int x, int y) {
    while(bcc.find(x) != bcc.find(y)) {
      int nxt = par(x);
      bbf[x] = bbf[y];
      bcc.unite(x, y);
      x = nxt;
      --bridge;
    }
  }

  void link(int x, int y) {
    int v = x, pre = y;
    while(v != size()) {
      int nxt = par(v);
      bbf[v] = pre;
      pre = v;
      v = nxt;
    }
  }

public:
  IncrementalBridgeConnectivity() = default;

  explicit IncrementalBridgeConnectivity(int sz) : cc(sz), bcc(sz), bbf(sz, sz), bridge(0) {}

  int find(int k) {
    return bcc.find(k);
  }

  size_t bridge_size() const {
    return bridge;
  }

  void add_edge(int x, int y) {
    x = bcc.find(x);
    y = bcc.find(y);
    if(cc.find(x) == cc.find(y)) {
      int w = lca(x, y);
      compress(x, w);
      compress(y, w);
    } else {
      if(cc.size(x) > cc.size(y)) swap(x, y);
      link(x, y);
      cc.unite(x, y);
      ++bridge;
    }
  }
};

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