Topological Sorting Source Removal Method | Dec & Conq Tech.| L 119 | Design & Analysis of Algorithm
Summary
TLDRThis 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
- ๐ 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.
- ๐ There are two main methods for topological sorting: DFS (Depth-First Search) and the Source Removal method, which is discussed in the script.
- ๐ Topological sorting is not possible if the graph contains a cycle; the absence of cycles is a prerequisite for the sorting process.
- ๐ The Source Removal method involves identifying vertices with no incoming edges (in-degree of zero) and removing them along with their outgoing edges.
- ๐ค 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.
- ๐ The process involves iteratively identifying and removing vertices with an in-degree of zero until the graph is empty.
- ๐ The order in which vertices are removed during the Source Removal method is the topological order or sequence of the graph.
- ๐ 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.
- ๐ The example in the script demonstrates the process of topological sorting by sequentially removing vertices and their edges until the graph is empty.
- ๐ The final topological order is the sequence in which vertices were removed from the graph during the sorting process.
- ๐ 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
๐ 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.
๐ 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.
Mindmap
Keywords
๐กTopological Sorting
๐กDFS (Depth-First Search)
๐กSource Removal Method
๐กIn-Degree
๐กOutgoing Edges
๐กCycle
๐กAlgorithmic Design
๐กDecrease and Confer Degree
๐กAlphabetical Order
๐กTie-breaking
๐กGraph
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
e e
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