強連結成分分解 (Kosarajuのアルゴリズム)
(Graph/Decomposition/SCC_Kosaraju.py)

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• Last update: 2021-01-09 01:09:35+09:00

概要

有向グラフを強連結成分分解するアルゴリズム。

使い方

StronglyConnectedComponents(n: int)
頂点数 n の辺のないグラフを構築する。計算量 $O(V)$

• add_edge(v: int, nxt_v: int) -> None
  頂点 v から頂点 nxt_v への有向辺を追加する。計算量 $O(1)$

• build() -> None
  強連結成分分解を行う。build の実行後には、インスタンス変数である labels に各頂点の強連結成分の番号 (縮約後の頂点番号) が格納される。縮約後の頂点番号はトポロジカルソート順である。計算量 $O(V + E)$

• construct_dag() -> Tuple[List[List[int]], List[int]]
  縮約後の頂点による有向グラフと、縮約後の頂点が含む元の頂点を返す。計算量 $O(V + E)$

Verified with

• TestCase/AOJ/GRL_3_C.Kosaraju.test.py
• TestCase/LibraryChecker/scc.Kosaraju.test.py

Code

class StronglyConnectedComponents:
    def __init__(self, n):
        self.n = n
        self.graph = [[] for _ in range(n)]
        self.rev_graph = [[] for _ in range(n)]
        self.labels = [-1] * n
        self.lb_cnt = 0

    def add_edge(self, v, nxt_v):
        self.graph[v].append(nxt_v)
        self.rev_graph[nxt_v].append(v)

    def build(self):
        self.post_order = []
        self.used = [False] * self.n
        for v in range(self.n):
            if not self.used[v]:
                self._dfs(v)

        for v in reversed(self.post_order):
            if self.labels[v] == -1:
                self._rev_dfs(v)
                self.lb_cnt += 1

    def _dfs(self, v):
        stack = [v, 0]
        while stack:
            v, idx = stack[-2:]
            if not idx and self.used[v]:
                stack.pop()
                stack.pop()
            else:
                self.used[v] = True
                if idx < len(self.graph[v]):
                    stack[-1] += 1
                    stack.append(self.graph[v][idx])
                    stack.append(0)
                else:
                    stack.pop()
                    self.post_order.append(stack.pop())

    def _rev_dfs(self, v):
        stack = [v]
        self.labels[v] = self.lb_cnt
        while stack:
            v = stack.pop()
            for nxt_v in self.rev_graph[v]:
                if self.labels[nxt_v] != -1:
                    continue
                stack.append(nxt_v)
                self.labels[nxt_v] = self.lb_cnt

    def construct_dag(self):
        self.dag = [[] for i in range(self.lb_cnt)]
        self.groups = [[] for i in range(self.lb_cnt)]
        for v, lb in enumerate(self.labels):
            for nxt_v in self.graph[v]:
                nxt_lb = self.labels[nxt_v]
                if lb == nxt_lb:
                    continue
                self.dag[lb].append(nxt_lb)
            self.groups[lb].append(v)
        return self.dag, self.groups

Traceback (most recent call last):
  File "/opt/hostedtoolcache/Python/3.12.4/x64/lib/python3.12/site-packages/onlinejudge_verify/documentation/build.py", line 71, in _render_source_code_stat
    bundled_code = language.bundle(stat.path, basedir=basedir, options={'include_paths': [basedir]}).decode()
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/opt/hostedtoolcache/Python/3.12.4/x64/lib/python3.12/site-packages/onlinejudge_verify/languages/python.py", line 96, in bundle
    raise NotImplementedError
NotImplementedError

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