6.10 Topological Sorting (with Examples) | How to find all Topological Orderings of a Graph

Summary

This video script introduces topological sorting, a linear ordering of vertices in a directed acyclic graph (DAG), where for every directed edge UV, vertex U comes before V. The script explains that a graph must be directed and acyclic to have a topological sort, and demonstrates the process of finding such an ordering by calculating the in-degree of vertices, removing those with in-degree zero, and updating the graph iteratively. It clarifies that a DAG can have multiple valid topological sorts and illustrates the method with examples, highlighting the impossibility of sorting graphs with cycles.

Takeaways

• 📚 Topological sorting is a linear ordering of vertices in a directed graph such that for every directed edge UV, vertex U comes before V in the ordering.
• ⛔ The graph must be directed and acyclic (DAG) for topological sorting to be possible; undirected graphs or graphs with cycles cannot have topological ordering.
• 🔍 To perform topological sorting, start by identifying the in-degree of each vertex, which is the number of incoming edges to the vertex.
• 🔄 Begin the sorting process by selecting vertices with an in-degree of 0 and deleting them along with their outgoing edges from the graph.
• 🔄 After deletion, update the in-degrees of the affected vertices and repeat the process until all vertices are ordered or no vertices with in-degree 0 are left.
• 🤔 If at any point no vertex with in-degree 0 is found and there are still vertices left, it indicates the presence of a cycle, making topological sorting impossible.
• 🔢 In-degree is recalculated after each deletion of a vertex and its edges to reflect changes in the graph structure.
• 🌐 A directed acyclic graph (DAG) can have more than one valid topological ordering, showcasing the flexibility in the sorting process.
• 📉 The process involves iteratively reducing the in-degree of vertices by removing those with an in-degree of 0 and their associated edges.
• 📈 The topological sort order is constructed incrementally, with vertices being added in the order they satisfy the in-degree of 0 condition.
• 🔗 The presence of a cycle in a graph is a critical factor that prevents the successful completion of a topological sort, as it leads to an impasse in the sorting process.

Q & A

What is topological sorting?

Topological sorting is a linear ordering of the vertices in a directed acyclic graph (DAG) such that for every directed edge UV, vertex U comes before vertex V in the ordering.

What is the prerequisite for a graph to have a topological ordering?

A graph must be directed and acyclic to have a topological ordering. If the graph is undirected or contains a cycle, topological sorting is not possible.

What is the significance of in-degree in topological sorting?

In-degree is the number of incoming edges to a vertex. It is used to determine the starting point for topological sorting, as the vertices with in-degree of 0 are the first to be placed in the ordering.

How does the presence of a cycle in a directed graph affect topological sorting?

If a directed graph contains a cycle, it is not possible to find a topological ordering for that graph, as the presence of a cycle violates the linear ordering condition required for topological sorting.

Can a directed acyclic graph have more than one topological ordering?

Yes, a directed acyclic graph can have more than one topological ordering. The script provides examples where different selections of vertices with in-degree 0 lead to different valid orderings.

What is the first step in the process of finding a topological ordering of a graph?

The first step is to find the in-degree of each vertex in the directed graph, which will help identify the vertices that can be placed first in the topological ordering.

What happens when you delete a vertex from the graph during topological sorting?

When a vertex is deleted from the graph, all its outgoing edges are also deleted. This may change the in-degree of other vertices, which could lead to new vertices having an in-degree of 0 and being candidates for the next position in the ordering.

How does the script illustrate the process of topological sorting?

The script uses examples of directed acyclic graphs and walks through the process of identifying vertices with in-degree 0, deleting them along with their outgoing edges, and updating the in-degrees of other vertices until a topological ordering is formed.

Why is it necessary to update the in-degrees of other vertices after deleting a vertex and its edges?

Updating the in-degrees is necessary because the deletion of a vertex and its edges changes the connectivity of the graph. This may result in other vertices having their in-degree reduced to 0, making them candidates for the next step in the topological sorting.

What is the final result of topological sorting for a directed acyclic graph?

The final result is a linear ordering of all vertices such that every directed edge from vertex U to vertex V appears in the ordering as U before V. This ordering is not unique and can vary depending on the selection process.