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

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• Last update: 2021-01-08 06:36:19+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。Kosaraju のアルゴリズムの方が高速なのでそちらを使いましょう。

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 $O(V + E)$

Required by

• 2-SAT (Graph/misc/TwoSAT.py)

Verified with

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

Code

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

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

    def build(self):
        k = 0
        idxs = [0] * self.n
        for v in range(self.n):
            if self.ord[v] == -1:
                k = self.dfs(v, k, idxs)
        self.labels = [self.lb_cnt - lb - 1 for lb in self.labels]

    def dfs(self, root, k, idxs):
        dfs_stack = [root]
        scc_stack = []
        while dfs_stack:
            v = dfs_stack[-1]
            if v < 0:
                v = ~v
                prv_v = dfs_stack[-2]
                self.low[prv_v] = min(self.low[prv_v], self.low[v])
                dfs_stack.pop()
                continue
            idx = idxs[v]
            if self.ord[v] == -1:
                self.ord[v] = self.low[v] = k
                k += 1
                scc_stack.append(v)
            if idx < len(self.graph[v]):
                nxt_v = self.graph[v][idx]
                idxs[v] += 1
                if self.ord[nxt_v] == -1:
                    dfs_stack.append(~nxt_v)
                    dfs_stack.append(nxt_v)
                elif self.labels[nxt_v] == -1:
                    self.low[v] = min(self.low[v], self.ord[nxt_v])
            else:
                if self.ord[v] == self.low[v]:
                    while True:
                        prv_v = scc_stack.pop()
                        self.labels[prv_v] = self.lb_cnt
                        if prv_v == v:
                            break
                    self.lb_cnt += 1
                dfs_stack.pop()
        return k

    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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