Bellman Ford Algorithm
A medium Graph problem included in Love Babbar 450, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 2
- Core for
- 11 roles
- Platform
- GeeksforGeeks
The problem
Given a weighted directed graph and a source vertex, find the shortest path distances from the source to all other vertices using the Bellman-Ford algorithm. The graph may contain edges with negative weights but no negative cycles reachable from the source.
Example 1
- Input
- n = 5, edges = [[0,1,-1],[0,2,4],[1,2,3],[1,3,2],[1,4,2],[3,2,5],[3,1,1],[4,3,-3]], source = 0
- Output
- [0,-1,2,-2,1]
- Why
- Bellman-Ford handles negative weights and finds shortest paths in this weighted graph.
Example 2
- Input
- n = 3, edges = [[0,1,5],[1,2,3]], source = 0
- Output
- [0,5,8]
- Why
- Simple positive-weight paths: 0->1 costs 5, 0->1->2 costs 8.
Constraints
- 1 <= n <= 100
- 0 <= edges.length <= n * (n - 1)
- -1000 <= edge weight <= 1000
How to think about it
Updated 2026-09-09Any simple shortest path in a graph of n vertices contains at most n - 1 edges. Relaxing every edge in the graph guarantees that paths of length 1 are optimal after round 1, paths of length 2 after round 2, and so forth up to n - 1. A further relaxation on round n that still reduces any distance exposes the presence of a reachable negative-weight cycle.
Approaches, worst first
Standard edge relaxation passes
time O(V * E) · space O(V)
Initialize `dist[source] = 0` and all others to infinity. Repeat n - 1 times: for each edge `(u, v, w)`, if `dist[u] != inf` and `dist[u] + w < dist[v]`, update `dist[v] = dist[u] + w`. Perform an nth pass to detect reachable negative cycles.
Shortest Path Faster Algorithm (SPFA)Write this one
time O(V * E) · space O(V)
Maintain a queue of vertices whose tentative distances improved and relax only their outgoing edges. While worst-case still O(V * E), average performance on random graphs drops close to O(E).
Where people lose marks · 3
- Relaxing an edge `(u, v, w)` when `dist[u]` is still infinity produces overflow or bogus distances if infinity representation arithmetic is not guarded.
- A negative cycle unreachable from the chosen source will not be detected by source-initiated Bellman-Ford passes.
- Not terminating early if an entire pass of relaxations produces zero distance updates across all edges.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
Companies that have asked it
Tags taken from the problem's own GeeksforGeeks page — not a copied list.
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