Topological Sort (BFS - Kahn's)
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 1
- Core for
- 12 roles
The problem
Given a directed acyclic graph with numVertices vertices and a list of directed edges, return a topological ordering of the vertices using breadth-first search (Kahn's algorithm).
Example 1
- Input
- numVertices = 6, edges = [[5,0],[5,2],[4,0],[4,1],[2,3],[3,1]]
- Output
- [5,4,2,3,1,0]
- Why
- Vertices 5 and 4 have no incoming edges, then their dependents are processed in BFS order.
Example 2
- Input
- numVertices = 4, edges = [[1,0],[2,0],[3,1],[3,2]]
- Output
- [3,1,2,0]
- Why
- Vertex 3 comes first, then 1 and 2, then 0.
Constraints
- 1 <= numVertices <= 100
- 0 <= edges.length <= numVertices * (numVertices - 1) / 2
How to think about it
Updated 2026-09-29Every DAG must possess at least one vertex with zero incoming dependencies. Once that vertex is placed into the topological ordering, its outgoing edges can be conceptually stripped away, exposing new vertices whose prerequisite requirements have dropped to zero. A queue drives this elimination process forward naturally in BFS manner.
Approaches, worst first
Adjacency matrix indegree scanning
time O(V^2) · space O(V^2)
Build an adjacency matrix and count column sums to determine in-degrees. In each round, scan the entire matrix to find an unselected vertex with in-degree zero, append it to the topological order, and clear its outgoing row entries. Dense matrix lookups take quadratic time on sparse inputs.
Kahn algorithm with indegree queueWrite this one
time O(V + E) · space O(V + E)
Compute the in-degree of every vertex and seed a queue with all vertices having in-degree 0. Continuously dequeue a node, append it to the result array, and decrement the in-degree of each target neighbor. Any neighbor reaching an in-degree of 0 is enqueued.
Where people lose marks · 3
- If the graph contains a cycle, the output list will contain fewer than numVertices elements because cyclic nodes never reach in-degree 0.
- Decrementing in-degree multiple times for duplicate edges if parallel edges are not deduplicated during adjacency building.
- Assuming a unique order; BFS topological sorts can process sibling zero-indegree nodes in arbitrary order.
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.
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