算法阅读笔记

Dijkstra's Algorithm

Dijkstra finds single-source shortest paths in a graph with non-negative edge weights by repeatedly finalizing the unsettled node with the smallest known distance.

类型算法分类Graphs深度高级典型复杂度O((V+E) log V) with heap

简短面试回答

先从这个心智模型开始

Dijkstra finds single-source shortest paths in a graph with non-negative edge weights by repeatedly finalizing the unsettled node with the smallest known distance.

简单解释

它如何工作，不堆砌不必要的术语

Initialize source distance to 0 and others to infinity.

Use a min-priority queue; when an edge u→v gives a cheaper route, relax v and update its distance.

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 Dijkstra's Algorithm, the goal is to make the invariant visible rather than to memorize code.

易读的 JavaScript 实现JavaScript

// Pseudocode-like JavaScript. Use a real min-heap in production.
function dijkstra(graph, start) {
  const dist = new Map([[start, 0]]);
  const pq = [[0, start]];

  while (pq.length) {
    pq.sort((a, b) => a[0] - b[0]);
    const [cost, node] = pq.shift();
    if (cost !== dist.get(node)) continue;

    for (const [next, weight] of graph[node] ?? []) {
      const candidate = cost + weight;
      if (candidate < (dist.get(next) ?? Infinity)) {
        dist.set(next, candidate);
        pq.push([candidate, next]);
      }
    }
  }
  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((V+E) log V) with heap

Do not quote this complexity mechanically.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

Using it with negative edge weights.
Forgetting stale entries when using a heap without decrease-key.

面试中如何解释

专业回答结构

State the non-negative-weight requirement and explain relaxation. With a binary heap and adjacency list, a common complexity is O((V+E) log V).

问题→核心思路→不变量 / 状态→Big-O→权衡

快速面试检查

What problem does Dijkstra's Algorithm solve?

Dijkstra finds single-source shortest paths in a graph with non-negative edge weights by repeatedly finalizing the unsettled node with the smallest known distance.

关于复杂度应该提到什么？

Its typical cost here is O((V+E) log V) with heap, 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？

Using it with negative edge weights.

继续阅读Graph representationDFS/BFSPriority queues

// Pseudocode-like JavaScript. Use a real min-heap in production. function dijkstra(graph, start) { const dist = new Map([[start, 0]]); const pq = [[0, start]]; while (pq.length) { pq.sort((a, b) => a[0] - b[0]); const [cost, node] = pq.shift(); if (cost !== dist.get(node)) continue; for (const [next, weight] of graph[node] ?? []) { const candidate = cost + weight; if (candidate < (dist.get(next) ?? Infinity)) { dist.set(next, candidate); pq.push([candidate, next]); } } } return dist; }