【Bell-Ford 算法】CLRS Exercise 24.1-4，24.1-6

本文是一篇笔记，大部分内容取自 CLRS 第三版，第 24.1 节。

Exercise 24.1-4

Modify the Bellman-Ford algorithm so that it sets \(v.d\) to \(-\infty\) for all vertices \(v\) for which there is a negative-weight cycle on some path from the source to \(v\).

Theorem 24.4 (Correctness of the Bellman-Ford algorithm)

Let Bellman-Ford be run on a weighted, directed graph \(G = (V, E)\) with source \(s\) and weight function \(w : E \to \mathbb{R}.\) If \(G\) contains no negative-weight cycles that are reachable from \(s\), then the algorithm returns true, we have \(v.d = \delta(s, v)\) for all vertices \(v \in V\), and the predecessor subgraph \(G_{\pi}\) is a shortest-paths tree rooted at \(s\). If \(G\) does contain a negative-weight cycle reachable from \(s\), then the algorithm returns false.

Proof 只证明后半部分，即「If \(G\) does contain a negative-weight cycle reachable from \(s\), then the algorithm returns false.」

Suppose that graph \(G\) contains a negative-weight cycle that is reachable from the source \(s\); let this cycle be \(c = < v_0, v_1, \dots, v_k >\), where \(v_0 = v_k\). Then,
\begin{equation}
\sum_{i=1}^{k} w(v_{i - 1}, v_i) < 0. \label{E:1}
\end{equation}
Assume for the purpose of contradiction that the Bellman-Ford algorithm returns true. Thus, \(v_i.d \le v_{i - 1}.d + w(v_{i - 1}, v_i)\) for \(i = 1, 2, \dots, k\). Summing the inequalities around cycle \(c\) gives us
\begin{aligned}
\sum_{i = 1}^{k} v_i.d &\le \sum_{i = 1}^{k} (v_{i - 1}.d + w(v_{i - 1}, v_i)) \\
&= \sum_{i = 1}^{k} v_{i - 1}.d + \sum_{i = 1}^{k} w(v_{i - 1}, v_i).
\end{aligned}
Since \(v_0 = v_k\), each vertex in \(c\) appears exactly once in each of the summations \(\sum_{i = 1}^{k} v_i.d\) and \(\sum_{i =1}^k v_{i-1}.d\), and so
\begin{equation*}
\sum_{i = 1}^{k} v_i.d = \sum_{i = 1}^{k} v_{i - 1}.d
\end{equation*}
Moreover, the fact that cycle \(c\) is reachable from \(s\) implies that \(v_i.d\) is finite for \(i = 1, 2, \dots, k\). Thus,

\[0 \le \sum_{i = 1}^{k} w(v_{i - 1}, v_i), \]

which contradicts inequality \eqref{E:1}. We conclude that the Bellman-Ford algorithm returns true if graph \(G\) contains no negative-weight cycles reachable from the source, and false otherwise.

从上述证明过程可以看出，任意「negative-weighted cycle reachable from the source」中至少有一条边 \((u, v)\) 在伪代码的第 6 行使得 \(v.d > u.d + w(u,v)\) 成立。换言之，每个「negative-weighted cycle reachable from the source」的所有边之中，至少有一条边会在伪代码的第 6 行被探测到。

exercise 24.1-4 的答案：
对于伪代码第 6 行发现的每条边 \((u, v)\)，从 \(v\) 或 \(u\) 开始 DFS 或 BFS，把经过的点的最短路长度都置为 \(-\infty\)。

Exercise 24.1-6

Suppose that a weighted, directed graph \(G = (V, E)\) has a negative-weight cycle. Give an efficient algorithm to list the vertices of one such cycle. Prove that your algorithm is correct.

若边 \((u, v)\) 在第 \(|V|\) 轮迭代仍能被松弛，则从 \(s\) 到 \(v\) 的“最短路”至少有 \(|V|\) 条边，也就是说从 \(s\) 到 \(v\) 的“最短路”一定经过 negative-weight cycle。
从 \(v\) 开始沿着 predecessor 向上走，第一个重复经过的节点 \(t\) 一定在 negative-weight cycle 中。再从 \(t\) 开始沿着 predecessor 走一遍直到再次经过 \(t\)，即可找出这个环。

References

http://courses.csail.mit.edu/6.046/fall01/handouts/ps9sol.pdf