Topological Sorting Source Removal Method | Dec & Conq Tech.| L 119 | Design & Analysis of Algorithm

Summary

This session of CSC Guru introduces the Source Removal Method for topological sorting, an algorithmic technique used to order vertices in a directed acyclic graph (DAG). The process involves identifying vertices with no incoming edges (in-degree zero) and removing them along with their outgoing edges, breaking ties alphabetically. The session demonstrates the method with a step-by-step example, emphasizing the importance of checking for cycles as a prerequisite. The final topological order is derived from the sequence in which vertices are removed, showcasing a simple and effective approach to DAG ordering.

Takeaways

• 📚 Topological sorting is a method used in graph theory to order vertices in a directed graph such that all directed edges go from left to right.
• 🔍 There are two main methods for topological sorting: DFS (Depth-First Search) and the Source Removal method, which is discussed in the script.
• 🔄 Topological sorting is not possible if the graph contains a cycle; the absence of cycles is a prerequisite for the sorting process.
• 🔑 The Source Removal method involves identifying vertices with no incoming edges (in-degree of zero) and removing them along with their outgoing edges.
• 🔤 In case of a tie, where multiple vertices have an in-degree of zero, the vertex to be removed is chosen based on alphabetical order.
• 🔄 The process involves iteratively identifying and removing vertices with an in-degree of zero until the graph is empty.
• 📈 The order in which vertices are removed during the Source Removal method is the topological order or sequence of the graph.
• 📝 The script provides a step-by-step guide on how to perform topological sorting using the Source Removal method, starting with identifying vertices with in-degree zero.
• 📉 The example in the script demonstrates the process of topological sorting by sequentially removing vertices and their edges until the graph is empty.
• 🔍 The final topological order is the sequence in which vertices were removed from the graph during the sorting process.
• 👍 The Source Removal method is presented as a simple and logical approach to achieve topological sorting of a directed acyclic graph (DAG).

Q & A

What is topological sorting?

Topological sorting is a method for ordering vertices in a directed graph such that for every directed edge from vertex A to vertex B, vertex A comes before vertex B in the ordering.

What are the two methods for implementing topological sorting?

The two methods for implementing topological sorting are the Depth-First Search (DFS) method and the Source removal method.

What is the purpose of topological sorting in algorithmic design?

Topological sorting is used in algorithmic design to arrange vertices in a way that respects the dependencies between tasks, ensuring that all prerequisites are completed before subsequent tasks are started.

What is the first step in the Source removal method for topological sorting?

The first step in the Source removal method is to identify the vertex with no incoming edges, also known as the vertex with an in-degree of zero.

How is the tie broken if there are multiple vertices with in-degree zero?

The tie is broken by selecting the vertex that comes first in alphabetical order.

What is the main constraint for applying topological sorting to a graph?

The main constraint is that the graph must not contain any cycles. Topological sorting is only possible on Directed Acyclic Graphs (DAGs).

What happens when a vertex with in-degree zero is identified in the Source removal method?

When a vertex with in-degree zero is identified, it is removed from the graph along with all its outgoing edges.

Can topological sorting be applied to any directed graph?

No, topological sorting can only be applied to directed graphs that are acyclic, meaning they do not contain any cycles.

What is the final result of the Source removal method after all vertices have been removed?

The final result is a topological ordering of the vertices, which is a sequence that represents the order in which the vertices were removed.

How does the Source removal method differ from the DFS method for topological sorting?

The Source removal method focuses on removing vertices with no incoming edges and their outgoing edges, while the DFS method explores the graph by visiting vertices and marking them as visited in a depth-first manner.

Outlines

00:00

📚 Introduction to Topological Sorting with Source Removal Method

This paragraph introduces the concept of topological sorting, a method used in graph theory to order vertices such that all directed edges go from left to right. The session focuses on the Source Removal Method, an algorithmic design technique that relies on the in-degree and out-degree of vertices. The speaker explains that topological sorting is not possible if the graph contains a cycle and emphasizes the importance of checking for cycles first. The process involves identifying vertices with no incoming edges (in-degree of zero), removing them along with their outgoing edges, and continuing this process until the graph is empty. The order in which vertices are removed represents the topological sort.

05:02

🔄 Step-by-Step Guide to Implementing Source Removal Method

This paragraph provides a step-by-step guide on how to implement the Source Removal Method for topological sorting. It reiterates the importance of ensuring the graph does not contain a cycle before proceeding. The method involves identifying vertices with an in-degree of zero and removing them along with their outgoing edges. If multiple vertices have an in-degree of zero, the tie is broken by selecting the vertex that appears first in alphabetical order. The process is repeated until the graph is empty, and the sequence in which vertices are removed constitutes the topological sort. The speaker assures that the Source Removal Method is a simple logic and thanks the viewers for watching the video.

Keywords

💡Topological Sorting

Topological Sorting is a linear ordering of vertices in a directed graph such that for every directed edge UV from vertex U to vertex V, U comes before V in the ordering. It is a key concept in the video, as it is the main objective of the discussed algorithms. The script describes topological sorting as ordering vertices along a horizontal line so that all directed edges go from left to right, which is essential for understanding the process and its applications.

💡DFS (Depth-First Search)

DFS is an algorithm for traversing or searching tree or graph data structures. In the context of the video, it is one of the two methods mentioned for implementing topological sorting, although the script focuses on the Source Removal method. The mention of DFS establishes a comparison point for the method that is being discussed in detail.

💡Source Removal Method

The Source Removal Method is the primary focus of the video and is a specific algorithm for performing topological sorting. It involves iteratively identifying vertices with no incoming edges (in-degree of zero) and removing them along with their outgoing edges. The script outlines the steps of this method, emphasizing its simplicity and logical progression in achieving topological sorting.

💡In-Degree

In a directed graph, the in-degree of a vertex is the number of incoming edges to that vertex. The script explains that the Source Removal Method requires identifying vertices with an in-degree of zero, which is a crucial step in the topological sorting process. It is used to determine which vertices can be safely removed from the graph during sorting.

💡Outgoing Edges

Outgoing edges are the edges that originate from a vertex and point to other vertices in a directed graph. The script mentions that when a vertex with in-degree zero is identified, it must be removed along with its outgoing edges. This action is part of the iterative process of the Source Removal Method for topological sorting.

💡Cycle

A cycle in a graph is a path that starts and ends at the same vertex, passing through a sequence of edges without repeating any vertex more than once. The script points out that topological sorting is only possible if the graph does not contain a cycle, making the detection of cycles a critical preliminary step before attempting to sort.

💡Algorithmic Design

Algorithmic Design refers to the process of designing algorithms to solve specific problems. The script mentions topological sorting as coming under this category, indicating that it is a methodical approach to ordering vertices in a directed graph. The video's theme revolves around the application of algorithmic design techniques to achieve topological sorting.

💡Decrease and Confer Degree

While the term 'confer degree' seems to be a misinterpretation in the script, the related concept of 'decrease' in the context of topological sorting likely refers to reducing the in-degree of vertices by removing them and their outgoing edges. This process is integral to the Source Removal Method, as it progressively simplifies the graph towards a state where topological sorting can be achieved.

💡Alphabetical Order

In the event of a tie, where multiple vertices have an in-degree of zero, the script suggests breaking the tie by removing vertices in alphabetical order. This rule ensures a consistent and fair method for choosing which vertex to remove when there is more than one option, thus maintaining the integrity of the topological sorting process.

💡Tie-breaking

Tie-breaking is the process of resolving situations where there is more than one option to choose from, as in the case of multiple vertices having an in-degree of zero. The script uses the term in the context of the Source Removal Method, where alphabetical order is used as the criterion for breaking ties, ensuring a clear path forward in the sorting process.

💡Graph

A graph is a mathematical structure consisting of vertices, or nodes, and edges that connect these vertices. The script discusses a graph as the fundamental structure on which topological sorting is performed. The entire process of topological sorting, as described in the video, is based on the properties and connections within a given graph.

Highlights

Introduction to the topic of topological sorting and its implementation methods.

Explanation of two methods for topological sorting: DFS and Source removal method.

DFS method has been previously discussed, now focusing on the Source removal method.

Topological sorting is an algorithmic design technique based on degrees.

The concept of topological sorting as ordering vertices along a horizontal line.

The importance of identifying a vertex with no incoming edges (in-degree zero) in the Source removal method.

Process of removing a vertex and its outgoing edges when its in-degree is zero.

Tie-breaking rule in case of multiple vertices with in-degree zero: alphabetical order.

The main constraint for topological sorting: the graph must not contain any cycles.

Step-by-step guide to implementing topological sorting using the Source removal method.

Demonstration of removing vertices and edges step by step to achieve topological sorting.

Final result of topological sorting is an empty graph with vertices ordered.

Topological sorting as a sequence of vertex removals from the graph.

The simplicity of the Source removal method in topological sorting.

The necessity to check for cycles before implementing topological sorting.

The iterative process of identifying and removing vertices with in-degree zero.

Conclusion summarizing the simplicity and logic of the Source removal method.

Transcripts

welcome to CSC Guru in this session we

will discuss the topic topological

shorting topological shorting can be

implemented with the help of two methods

one is DFS method and another one is

Source removal method DFS method already

we have discussed with example so now we

will discuss Source removal method so

this topological shorting is a method

based on decrees and confer degree that

is it comes under the algorithmic design

technique decrease and cont topological

shorting is nothing but ordering of

vertices along a horizontal line so that

all directed edges go from left to right

so in this session we will discuss

Source removal method so the design

steps for Source removal method is in

the given graph identify the vertex with

no incoming age the particular vertex

with in degree zero so you need to

identify the vertex in the given graph

such that its in degree should be zero

okay and delete along with the outgoing

ages that particular Vortex you need to

delete along with this outgoing ages

okay if there are several vertices with

no incoming edges break the tie

orbitally obviously we will break the

tie in alphabetical order so

alphabetical order which vertex comes

first that we will take it first and

remove first okay so the order in which

vertices are visited and deleted one by

one results in topological short

very simple logic identify the edge with

integre zero remove it one by one in the

given graph identify the edge with

ingree zero and remove it okay and

finally the order of the deleted edges

is nothing but the topological sh so now

we will discuss with one example so this

is the given graph so first thing we

need to consider in this graph is check

whether this graph forms a cycle if this

graph forms a cycle in the sense

topological shorting is not possible

okay that is a main constraint so for

any graph given first you need to check

whether it is having any cycle see here

there is no cycle the direction you have

to consider it should not form a cycle

so this graph does not form any cycle so

we will Implement topological shorting

so first thing is step one you need to

identify the vortex with ingree 0o so

here one also ingree 0 two also ingree 0

so what we will do we will break the tie

and remove the vertex one so if you are

removing vertex one along with its

outgoing Edge you have to remove the

vortex also you have to remove and the

outgoing Edge also you have to remove so

if you are removing in the sense you

will get the graph like

this so one you will remove and the

remaining vertices and its edges

everything you have to include okay so

this is the graph you will get it after

after removing one next step in this

step check the next Vortex with degree Z

which is nothing but Vortex 2 in degree

zero so next step remove the vortex 2 so

if you are removing Vortex 2 along with

its outgoing Edge in the Sun the graph

will be like

this 3 4 and five only will be there

along with it

edges okay next step identify the next

not with the IND degree zero that is

nothing but Vortex 3 so remove the

vortex 3 along with its outgoing Hedges

okay so here if you are removing three

you will get Vortex four and five okay

next step identify the vertex with IND

degree 0o the next Vortex with IND

degree 0o that is nothing but four

remove vertex four along with its

outgoing Edge so next node with integre

0 is four and remove four if you are

removing four you will get vertex five

only and the next step remove five also

okay so now the graph is empty okay see

now if you are ordering the vertices if

you are ordering the removed vertices in

the S you will get topological shorting

so topological shorting or otherwise

topological sequence also they will tell

it both is same only so topological

shorting is nothing but first we have

removed vortex one okay and then we have

removed Vortex two and then we have

removed Vortex three and then we have

removed Vortex four and then we have

removed Vortex 5 so this is nothing but

topological ordering of

vertices so this is nothing but

topological shorting and this is nothing

but the ordering of vertices so Source

removal method is nothing but in the

given graph first you have to check

whether the graph forms a cycle so if

the graph forms a cycle in the sense you

cannot able to implement any method even

DFS method also not possible Source

removal method also not possible okay so

first thing you have to check this

constraint so if the given graph does

not form a cycle in the sense we can

Implement topological shorting so how we

can Implement in the sense step by step

identify one vertex with IND degree Z if

more than one vertex is having ingree

zero in

you need to break the tie in

alphabetical order so alphabetical order

which Vortex comes first that you have

to remove first okay so identify the

vortex with integre zero remove that

Vortex along with its outgoing age both

you have to remove it okay so in each

step identify one vertex with integre

zero if the graph does not forms a cycle

in the sense this is possible if it

forms a cycle in the sense you cannot

able to sort it out okay so every step

identify the vertex with inte degree

zero and remove that Vortex along with

its outgoing Edge so step by step if you

are implementing in the Sun finally the

graph will be empty okay so the order in

which you have deleted the vertices if

you are implementing in the sense that

is nothing but topological shorting so

topological shorting is simply ordering

of

vertices so this Source removal method

is very simple logic

thank you for watching this

video

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