ALGORITHM READING NOTE

Bellman–Ford Algorithm

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

TYPEAlgorithmCATEGORYGraphsDEPTHAdvancedTYPICAL COMPLEXITYO(VE)

SHORT INTERVIEW ANSWER

Start with this mental model

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

SIMPLE EXPLANATION

How it works, without unnecessary jargon

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.

EXECUTION

Read the algorithm as a sequence of decisions

Define directed/undirected and weighted/unweighted assumptions.

Choose the graph representation.

Track algorithm state and handle disconnected/cyclic cases explicitly.

A small example to keep in mind

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.

READABLE JAVASCRIPT IMPLEMENTATIONJavaScript

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;
}

How to read this code

Do not memorize the syntax. Follow the state that changes on each line and connect it to the invariant described above.

COMPLEXITY

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Do not quote this complexity mechanically.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.

WHEN TO USE IT

Good fit

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

COMMON MISTAKES

What usually goes wrong

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

HOW TO EXPLAIN IT IN AN INTERVIEW

A professional answer shape

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

Problem→Core idea→Invariant / state→Big-O→Trade-off

QUICK INTERVIEW CHECK

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.

What should I mention about complexity?

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

What is the most common mistake?

Forgetting disconnected components.

Continue readingGraph 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; }