Shortest Path in Unweighted Graph
A medium Graph problem included in Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 1
- Core for
- 11 roles
- Platform
- GeeksforGeeks
The problem
Given an undirected graph with n nodes and a list of edges, find the shortest path distance from a source node to a destination node. All edges have unit weight (distance 1).
Example 1
- Input
- n = 6, edges = [[0,1],[0,2],[1,3],[2,3],[3,4],[4,5]], source = 0, destination = 5
- Output
- 4
- Why
- Shortest path: 0->2->3->4->5 has length 4.
Example 2
- Input
- n = 3, edges = [[0,1],[1,2]], source = 0, destination = 2
- Output
- 2
- Why
- Path: 0->1->2 has length 2.
Constraints
- 1 <= n <= 500
- 0 <= edges.length <= 5000
- source != destination
How to think about it
Updated 2026-09-09When all edge weights are uniform (1), distance equals level depth. A standard breadth-first search visits vertices in strict ascending order of distance from the source. The very first time destination is dequeued, its recorded distance is guaranteed to be minimal.
Approaches, worst first
DFS exhaustive path exploration
time O(V!) · space O(V)
Traverse all simple paths from source to destination using recursive backtracking with a visited set. Track the minimum path length across all complete routes. Without level-order expansion, DFS explores exponentially many paths through dense graphs before confirming the minimum distance.
Breadth-first search with distance arrayWrite this one
time O(V + E) · space O(V + E)
Build an adjacency list. Initialize a `dist` array to -1 and set `dist[source] = 0`. Push source to a queue. Pop nodes and iterate neighbors: if `dist[v] === -1`, set `dist[v] = dist[u] + 1` and enqueue. Break as soon as destination is reached.
Where people lose marks · 3
- If destination is in a disconnected component from source, the queue exhausts without visiting destination; return -1.
- Undirected edges must be inserted in both directions in the adjacency list; adding only directed edges drops reverse paths.
- Setting distance upon popping instead of when pushing to the queue causes identical vertices to be enqueued multiple times.
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.
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