# Topological sort using DFS and BFS

Topological sort is a graph traversal in which node v is visited only after all its dependencies are visited. The graph has to be directed and has no cycles, i.e. directed acyclic graph (DAG). Once the graph is built, you can use the algorithms to find the order. The result is not necessary unique. Two algorithms topological sort BFS and DFS are used to solve the problem. They both have linear time O(V+E), V is number of vertices, E is number of edges. Topological sort are usually applied to ranking and scheduling problems. Here is an example to find tournament ranking.

Table of Content

Question statement:
(Modified) In a tennis tournament of N players. The following condition always hold-If player P1 has won the match with P2 and player P2 has won from P3, then Player P1 has also defeated P3. Find winner and ranking of players.

How to implement topological sort DFS?

Topological sort dfs uses the data structure stack. For each node and its followers, call the recursion helper. For any node returns from call stack, put it in the stack. The reverse of the stack is the final sorted result.

## Doodle

How to implement topological sort BFS?

Topological sort BFS uses Kahn’s algorithm. First find a list of start nodes which has no incoming edges, ie indegree is 0. Insert them into a queue. Then pull out items from the queue, update and check its followers’ indegrees. If the indegree is 0, add to the queue. Repeat until all nodes are checked. The sequence pulled from the queue is the final sorted result. The time complexity O(V+E) and space complexity O(V). V is number of nodes (vertices), E is number of edges.

## Doodle

Output:
{P1=[P5], P2=[P5], P3=[P2], P4=[P3, P1], P5=[]}
Topological sort dfs: [P4, P3, P2, P1, P5]
Topological sort bfs: [P4, P3, P1, P2, P5]

O Notation:
Time complexity: O(V+E)
Space complexity: O(V)
V is number of nodes, E is number of edges