开一个关于图的连通性的专题,内容包括:
- 无向图的割点、桥、双连通分量
- 有向图的强连通分量
- 最大团
- 极大团
- 全局最小割
- 最短路相关
Introduction to Algorithms 3rd edition, page 621,
Let $G = (V, E)$ be a connected, undirected graph. An articulation point of $G$ is a vertex whose removal disconnects $G$. A bridge of $G$ is an edge whose removal disconnects $G$. A biconnected component of $G$ is a maximal set of edges such that any two edges in the set lie on a common simple cycle. Figure 22.10 illustrates these definitions.
We can determine articulation points, bridges, and biconnected components using depth-first search. Let $G_{\pi} = (V, E_{\pi})$ be a depth-first tree of $G$.
a. Prove that the root of $G_{\pi}$ is an articulation point of $G$ if and only if it has at least two children in $G_{\pi}$.
b. Let $v$ be a nonroot vertex of $G_{\pi}$. Prove that $v$ is an articulation point of $G$ if and only if $v$ has a child $s$ such that there is no back edge from $s$ or any descendant of $s$ to a proper ancestor of $v$.
c. Let
\begin{aligned}
v.low = \min
\begin{cases}
v.d, \\
w.d : \text{$(u, w)$ is a back edge for some descendant $u$ of $v$}.
\end{cases}
\end{aligned}
Show how to compute $v.low$ for all vertices $v\in V$ in $O(E)$ time.
d. Show how to compute all articulation points in $O(E)$ time.
e. Prove that an edge of $G$ is a bridge if and only if it does not lie on any simple cycle of $G$.
f. Show how to compute all the bridges of $G$ in $O(E)$ time.
g. Prove that the biconnected components of $G$ partition the nonbridge edges of $G$.
h. Give an $O(E)$-time algorithm to label each edge $e$ of $G$ with a positive integer $e.bcc$ such that $e.bcc = e'.bcc$ if and only if $e$ and $e'$ are in the same biconnected component.
Problem 22-3 Euler tour
An Euler tour of a strongly connected, directed graph $G = (V, E)$ is a cycle that traverses each edge of $G$ exactly once, although it may visit a vertex more than once.
a. Show that $G$ has an Euler tour if and only if in-degree($v$) = out-degree($v$) for each vertex $v\in V$.
b. Describe an $O(E)$-time algorithm to find an Euler tour of $G$ if one exists. (Hint: Merge edge-disjoint-cycles.)
Problem 22.5-7
A directed graph $G = (V, E)$ is semiconnected if, for all pairs of vertices $u, v\in V$, we have $u \leadsto v$ or $v \leadsto u$. Give an efficient algorithm to determine whether or not $G$ is semiconnected. Prove that your algorithm is correct, and analyze its running time.
求 $G$ 的强连通分量，缩点后得到一个 DAG。问题可归约为求这个 DAG 是否为 semiconnected。一个 DAG 是 semiconnected 当且仅当其中有一个生成子图是一条链，换言之，设 DAG 中有 $n$ 个节点，这些节点按拓扑序依次是 $v_1, v_2, \dots, v_n$，对于每个 $i =1, 2, 3, \dots, n - 1$，要有一条从 $v_i$ 到 $v_{i+1}$ 的有向边。
至于代码实现，以 Tarjan 的强连通分量算法为例，我们可以在 DFS 的过程中，对于每个点 $u$ 计算出从 $u$ 出发的边所能到达的其他强连通分量的编号的最大值。
SCC(int n, const Graph &g,
function<int(const Arc &e)> get_v = [](const Arc &e) { return e; })
: n(n),
g(g),
get_v(move(get_v)),
scc_id(n + 1, -1),
scc_cnt(0) {} // SCCs are indexed from 0
bool dfs(int u) {
static int time;
static stack<int> s;
static vector<int> low(n + 1), order(n + 1);
static vector<bool> in_stack(n + 1);
static vector<int> max_scc_id(n + 1, -1);
order[u] = low[u] = ++time;
s.push(u);
in_stack[u] = true;
for (auto &e : g[u]) {
int v = get_v(e);
if (!order[v]) {
if (!dfs(v)) {
return false;
}
low[u] = min(low[u], low[v]);
} else if (in_stack[v]) {
low[u] = min(low[u], order[v]);
}
if (!in_stack[v]) {
max_scc_id[u] = max(max_scc_id[u], scc_id[v]);
}
}
int max_scc_ID = -1;
if (order[u] == low[u]) {
while (true) {
int v = s.top();
s.pop();
in_stack[v] = false;
scc_id[v] = scc_cnt;
max_scc_ID = max(max_scc_ID, max_scc_id[v]);
if (u == v) break;
}
if (max_scc_ID != scc_cnt - 1) {
return false;
}
++scc_cnt;
}
return true;
}
};