算法阅读笔记

Bellman–Ford Algorithm

Bellman–Ford computes single-source shortest paths even with negative edge weights, and can detect a reachable negative-weight cycle.

类型算法分类Graphs深度高级典型复杂度O(VE)

简短面试回答

先从这个心智模型开始

Bellman–Ford computes single-source shortest paths even with negative edge weights, and can detect a reachable negative-weight cycle.

简单解释

它如何工作，不堆砌不必要的术语

Relax every edge V−1 times because a simple shortest path uses at most V−1 edges.

A further successful relaxation indicates a reachable negative cycle.

执行过程

把算法读成一连串决策

Define directed/undirected and weighted/unweighted assumptions.

Choose the graph representation.

Track algorithm state and handle disconnected/cyclic cases explicitly.

记住一个小例子

Read a tiny input by hand and write down the state after every important operation. For Bellman–Ford Algorithm, the goal is to make the invariant visible rather than to memorize code.

易读的 JavaScript 实现JavaScript

function bellmanFord(vertices, edges, start) {
  const dist = Array(vertices).fill(Infinity);
  dist[start] = 0;
  for (let i = 1; i < vertices; i++) {
    for (const [u, v, w] of edges) {
      if (dist[u] !== Infinity && dist[u] + w < dist[v]) {
        dist[v] = dist[u] + w;
      }
    }
  }
  return dist;
}

如何阅读这段代码

不要死记语法。跟踪每一行中发生变化的状态，并把它与上面描述的不变量联系起来。

复杂度

O(VE)

Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.

何时使用

适用场景

Use the algorithm only when its graph assumptions match the problem.

常见错误

通常哪里会出错

Forgetting disconnected components.
Mixing weighted and unweighted shortest-path algorithms.
Not protecting against revisits/cycles.

面试中如何解释

专业回答结构

Compare it with Dijkstra: Bellman–Ford is slower O(VE) but supports negative weights and negative-cycle detection.

问题→核心思路→不变量 / 状态→Big-O→权衡

快速面试检查

What problem does Bellman–Ford Algorithm solve?

Bellman–Ford computes single-source shortest paths even with negative edge weights, and can detect a reachable negative-weight cycle.

关于复杂度应该提到什么？

Its typical cost here is O(VE), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.

最常见的错误是什么？

Forgetting disconnected components.

继续阅读Graph representationDFS/BFSPriority queues

function bellmanFord(vertices, edges, start) { const dist = Array(vertices).fill(Infinity); dist[start] = 0; for (let i = 1; i < vertices; i++) { for (const [u, v, w] of edges) { if (dist[u] !== Infinity && dist[u] + w < dist[v]) { dist[v] = dist[u] + w; } } } return dist; }