Topological Sort and Tasks Scheduling (Kahn’s algorithm)

Topological Sort of a directed graph (a graph with unidirectional edges) is a linear ordering of its vertices such that for every directed edge (U, V) from vertex U to vertex VU comes before V in the ordering.

Given a directed graph, find the topological ordering of its vertices.

Example 1:

Input: Vertices=4, Edges=[3, 2], [3, 0], [2, 0], [2, 1]
Output: Following are the two valid topological sorts for the given graph:
1) 3, 2, 0, 1
2) 3, 2, 1, 0

Capture

Example 2:

Input: Vertices=5, Edges=[4, 2], [4, 3], [2, 0], [2, 1], [3, 1]
Output: Following are all valid topological sorts for the given graph:
1) 4, 2, 3, 0, 1
2) 4, 3, 2, 0, 1
3) 4, 3, 2, 1, 0
4) 4, 2, 3, 1, 0
5) 4, 2, 0, 3, 1

Capture

Here are a few fundamental concepts related to topological sort:

  1. Source: Any node that has no incoming edge and has only outgoing edges is called a source.
  2. Sink: Any node that has only incoming edges and no outgoing edge is called a sink.
  3. So, we can say that a topological ordering starts with one of the sources and ends at one of the sinks.
  4. A topological ordering is possible only when the graph has no directed cycles, i.e. if the graph is a Directed Acyclic Graph (DAG). If the graph has a cycle, some vertices will have cyclic dependencies which makes it impossible to find a linear ordering among vertices.

To find the topological sort of a graph we can traverse the graph in either a Breadth First Search (BFS) way using Kahn’s algorithm or Depth First Search (DFS). We will start with all the sources, and in a stepwise fashion, save all sources to a sorted list. We will then remove all sources and their edges from the graph. After the removal of the edges, we will have new sources, so we will repeat the above process until all vertices are visited.

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Time Complexity

In step ‘d’, each vertex will become a source only once and each edge will be accessed and removed once. Therefore, the time complexity of the above algorithm will be O(V+E), where ‘V’ is the total number of vertices and ‘E’ is the total number of edges in the graph.

Space Complexity

The space complexity will be O(V+E), since we are storing all of the edges for each vertex in an adjacency list.

Tasks Scheduling

There are ‘N’ tasks, labeled from ‘0’ to ‘N-1’. Each task can have some prerequisite tasks which need to be completed before it can be scheduled. Given the number of tasks and a list of prerequisite pairs, find out if it is possible to schedule all the tasks.

Example 1:

Input: Tasks=3, Prerequisites=[0, 1], [1, 2]
Output: true
Explanation: To execute task '1', task '0' needs to finish first. Similarly, task '1' needs to finish 
before '2' can be scheduled. A possible scheduling of tasks is: [0, 1, 2] 

Example 2:

Input: Tasks=3, Prerequisites=[0, 1], [1, 2], [2, 0]
Output: false
Explanation: The tasks have cyclic dependency, therefore they cannot be scheduled.

As you can see, we can represent the task and its dependency as an edge in a directed graph. There’s a little change in the above code that need to be changed to solve this problem so I let this as an exercise for you. Instead of returning a list, you just need to return a boolean.

Another application of Topological Sort is to check if a directed graph contains a cycle.