Kahn's Algorithm BFS Topological Sort
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 a directed acyclic graph with numVertices vertices and a list of directed edges, return a topological ordering using Kahn's algorithm (BFS-based topological sort).
Example 1
- Input
- numVertices = 4, edges = [[3,0],[3,1],[2,1],[1,0]]
- Output
- [3,2,1,0]
- Why
- Vertex 3 has no incoming edges, then 2, then 1, then 0 in BFS topological order.
Example 2
- Input
- numVertices = 6, edges = [[5,0],[5,2],[4,0],[4,1],[2,3],[3,1]]
- Output
- [5,4,2,3,1,0]
- Why
- Process nodes with no incoming edges in BFS order.
Constraints
- 1 <= numVertices <= 100
- 0 <= edges.length <= numVertices * (numVertices - 1) / 2
How to think about it
Updated 2026-09-09Every directed acyclic graph contains at least one node with in-degree 0. Enqueue these initial sources; popping a node outputs it next in the linear order. Decrementing the in-degrees of its targets simulates deleting the node and its outgoing arrows, continuously revealing the next tier of dependency-free vertices.
Approaches, worst first
DFS post-order traversal stack
time O(V + E) · space O(V + E)
Traverse unvisited vertices with recursive depth-first search, appending each vertex to an ordering list only after all directed descendants finish. Reversing the completed post-order list yields a valid topological sequence, but requires recursion stack overhead and explicit cycle-state flags.
In-degree array with queue BFSWrite this one
time O(V + E) · space O(V + E)
Compute in-degrees for all vertices from 0 to numVertices - 1. Enqueue those with in-degree 0. Dequeue u, append to ordering list, and for each outgoing edge `u -> v`, decrement in-degree of v. Whenever in-degree of v becomes 0, push v to the queue.
Where people lose marks · 3
- If a cycle exists, the queue will empty before processing all vertices, resulting in an output array shorter than numVertices.
- Multiple independent nodes can have in-degree 0 concurrently; any arbitrary tie-breaking order among them produces a valid topological sort.
- Edges array can be empty, where any permutation of `[0, ..., numVertices - 1]` is a valid answer.
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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