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@ -20,7 +20,7 @@ Topological Sort is the sorting of the vertices in such a way that, for any two
$a,b,c,d,e$ is the topologically ordering for the graph below. $a,b,c,d,e$ is the topologically ordering for the graph below.
![enter image description here](https://lh3.googleusercontent.com/zOBG-tWmznt9QeFbW-SopxIpvyDSH7RgpmYKL9fIgth_TkRQ3sfuxXsDw7iWgmmyGdqip4cay_WS) ![enter image description here](https://github.com/KingsGambitLab/Lecture_Notes/blob/master/articles/Akash%20Articles/md/Images/Topological%20Sort/1.jpg)
## Quiz Time ## Quiz Time
@ -28,7 +28,7 @@ $Q.1$ Is it always possible to find out the topological sort?
No. Whenever there is a cycle in a graph we can not find it. See the image below: No. Whenever there is a cycle in a graph we can not find it. See the image below:
![enter image description here](https://lh3.googleusercontent.com/u-MLT9wrOoXBNQDqBqEi06jrnqudSGpBCi_oYG27-WI-zd8yEOs2PiBWyLVTimeYP42c5pk-N4AC) ![enter image description here](https://github.com/KingsGambitLab/Lecture_Notes/blob/master/articles/Akash%20Articles/md/Images/Topological%20Sort/2.jpg)
We can not find proper dependency between any two vertices, becuase each of them are interdependent. We can not find proper dependency between any two vertices, becuase each of them are interdependent.
@ -37,7 +37,7 @@ So the condition for the topological sort is, the graph must be a DAG - Directed
$Q.2$ Does topological sort give a unique ordering of the vertices? $Q.2$ Does topological sort give a unique ordering of the vertices?
No, it is not unique in every case, because we can find many orderings which satisfies the condition of the topological sort. No, it is not unique in every case, because we can find many orderings which satisfies the condition of the topological sort.
![enter image description here](https://lh3.googleusercontent.com/A-uLwWY3HLE8aKWi0AAWxCDgkEidHTxi5u_CVYhASvoFZgaXr9nzXVNObaIP1BEyAUEE5yLyY6BM) ![enter image description here](https://github.com/KingsGambitLab/Lecture_Notes/blob/master/articles/Akash%20Articles/md/Images/Topological%20Sort/3.jpg)
In the graph shown in the image above, $b,a,c,d,e$ and $a,b,c,d,e$ both are valid topological sorts. In the graph shown in the image above, $b,a,c,d,e$ and $a,b,c,d,e$ both are valid topological sorts.
@ -57,13 +57,13 @@ We will use the indegree of vertices to find the topological ordering. How? Let'
In the image below, the number in the square bracket represents indegree for a vertex near to it. In the image below, the number in the square bracket represents indegree for a vertex near to it.
![enter image description here](https://lh3.googleusercontent.com/YbZlobxN3dHtY7MkIXkZRI7u3Xsv6QTsngzoqn-hiNnMOuUs8tLweaxaR8snMkCTdgOjk7rh7W68) ![enter image description here](https://github.com/KingsGambitLab/Lecture_Notes/blob/master/articles/Akash%20Articles/md/Images/Topological%20Sort/4.jpg)
As we have seen the topological ordering for the above graph is $a,b,c,d,e$. Now, what is your observation? As we have seen the topological ordering for the above graph is $a,b,c,d,e$. Now, what is your observation?
See that the vertices having indegree $0$ are appearing first in the ordering. But what next? What if we remove both the vertices and all the edges coming out from it? See that the vertices having indegree $0$ are appearing first in the ordering. But what next? What if we remove both the vertices and all the edges coming out from it?
![enter image description here](https://lh3.googleusercontent.com/czUu-KCQJ7Y78v0z4-rI4uKiXwsHPT2cmYEISfEQU3n89sEK8uZleMB5Rx6VYnxDjREvO2A94g9b) ![enter image description here](https://github.com/KingsGambitLab/Lecture_Notes/blob/master/articles/Akash%20Articles/md/Images/Topological%20Sort/5.jpg)
See now, $c$ is the vertex having $0$ indegree, which is appearning next in the ordering. See now, $c$ is the vertex having $0$ indegree, which is appearning next in the ordering.