In this article, you will learn to implement a Topological sort algorithm by using Depth-First Search and In-degree algorithms

Topological sort is an algorithm which takes a directed acyclic graph and returns a list of vertices in the linear ordering where each vertex has to precede all vertices it directs to

There are no topological orderings exist in a directed cyclic graph and more than one of them can exist in one directed acyclic graph

Let's get started

Problem

  • Given a directed graph G = (V, E)

  • Write an algorithm to return a sorted vertex list in topological ordering

  • Return an empty list if no topological orderings exist

Example 1

  • Input: Given a directed graph G = (V, E) with V = 4 and E = [[0, 1], [2, 0], [2, 1], [3, 2]]

  • Output: [3, 2, 0, 1]

  • Explanation: [3, 2, 1, 0] is also a valid topological ordering

Example 2

  • Input: Given a directed graph G = (V, E) with V = 5 and E = [[0, 1], [1, 2], [2, 0], [1, 3], [2, 4]]

  • Output: []

  • Explanation: No topological orderings exist as [0, 1], [1, 2], and [2, 0] form a cycle. Vertex 0 can not both precede and follow vertex 2

Approach 1: DFS Iterative with Color

  • Run DFS Iterative with Color to find a vertex with no outgoing edges (Black vertex) and add it to a stack S. Stop finding when found a cycle

    Repeat for all other vertices in the graph

  • Convert S to a list to get the final result


GraphDirectedByAdjacencyList is defined in Graph Data Structure Tutorial

  • Output
There are no topological orderings as the input graph is cyclic  
3 2 0 1  
  • Time complexity: O(V+E)

  • Space complexity: O(V)

Approach 2: DFS Recursive with Color

  • Run DFS Recursive with Color to find a vertex with no outgoing edges (Black vertex) and add it to a stack S. Stop finding when found a cycle

    Repeat for all other vertices in the graph

  • Convert S to a list to get the final result


GraphDirectedByAdjacencyList is defined in Graph Data Structure Tutorial

  • Output
There are no topological orderings as the input graph is cyclic  
3 2 0 1  
  • Time complexity: O(V+E)

  • Space complexity: O(V)

Approach 3: In-degree

  • Find a vertex with no incoming edges (in-degree = 0) and add it to a queue Q

    Repeat for all other vertices in the graph

  • Convert Q to a list to get the final result


GraphDirectedByAdjacencyList is defined in Graph Data Structure Tutorial

  • Output
There are no topological orderings as the input graph is cyclic  
3 2 0 1  
  • Time complexity: O(V+E)

  • Space complexity: O(V)