DFS Topological Sorting

Theory If G = (E, V) is a Directed Acyclic Graph (DAG), then a topological sorting

of V is a linear ordering of V such that for each edge (u, v) in the DAG, u appears before v in the linear ordering.

The nodes on a graph can represent a task that needs to be done It accomplishes this using the depth-first search (DFS) algorithm, which

starts at a root, and explores as far as possible down the tree before backtracking

The edges linking the nodes can represent constraints that one task has on another.

The algorithm is recursive. This algorithm can only be used when the graph is acyclic and directed

Overall theory view of Topological Graph

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Topological Graph start to finishDAG Topological Order

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Setup for this algorithm As the graph is traversed, the nodes will be marked. The first time a node is visited, it will be temporarily marked, meaning that

node’s children still need to be looked at. After all the current node’s children have been visited, the node will be permanently marked.

Marking of Vertices during Sorting

An empty list will be needed, with size equal to the number of nodes in the graph

The example graph given would need this list Any time a node is marked Permanently, it is added to the head of the list

Exhausted Vertices stored in Reversed Queue

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Step by Step Start with the lowest value vertex

o if another segment(s) is unreachable from the current, start again with the next lowest valued vertex

will notice how recursion and queue-ish data structure are used Searches for the shortest path from a starting vertex to all others Once a non-visited vertex is identified, move to that vertex and recursively

continue this until we have to stop DFS Topological uses a stack to determine the last vertex visited example

1st Topological Sort Example

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Complete the problem below: Answerb:Call Stack Topo Order Stack

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Try this one too. Answerb:

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How to read the Topological Order Stack The stack that contains the Topological Order is filled with information if you

can decipher it vertices higher in the stack must be completed before some of the items below

Understanding the Topological ResultsTopological Order As a reconnected tree

This is no coincidence that all edges are flowing in one direction2 requires 0 AND 1 to be completed first (look at the original graph to confirm)7 requires what?

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Try this one on your own. Show ALL stacks and Topological Tree. Answerb:

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Code Vector used so we can add to the HEAD of the list notice the recursion

topsort in Java

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Complexity V is the number of nodes E is the number of edges Each vertex is enqueued once at most, so all queue operations run at O(|V|)

constant time. The body of the loop is executed at most once per edge, so the running time is

O(|E|). The initialization is proportional to the size of the graph used, so it runs at O(|

E| + |V|). Add(u,v) – O(1) getDegree(u) – O( |V| ) getAdjacent(u) - O( |V| + |E| ) isAdjacentTo(u,v) – O(1) Worst Case - O(|E| + |V|)

o Graph is undirected and not connected. Might fail to visit some vertices on first DFS call. Then have to search for unvisited vertex and apply a DFS there and continue until all vertices have been visited

Best Case - O(|E|)o Graph is a tree. Start at top of tree, can get to all other vertices just by

looking at edges

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Applications a DFS based topological sort is the perfect solution for breaking down

directed acyclic graphs into dependency fulfilling sequences a good example of its application (relevant to our studies here) are the

prerequisites for certain computer science courses.

Course requirements for 1999 UMBC CS major

by using a DFS topological sort, we can construct a sequence in which to take these courses that would ensure the prerequisites are always fulfilled.

For exampleo CMSC102 has a prerequisite of CMSC101 as well as MATH151

that means in any sequence of nodes, CMSC101 and MATH151 must precede CMSC 102

o Since CMSC101 has a prerequisite of MATH150, MATH150 must also always precede CMSC101.

possible sequences for CMSC102 include the following

MATH 150, MATH 151, CMSC 101, CMSC 102MATH 151, MATH 150, CMSC 101, CMSC 102MATH 150, CMSC 101, MATH 151, CMSC 102

Note that the order the prerequisites appear in is irrelevant as long as it holds that no item appears in the list without having all of its prerequisites appear before it.

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AnswersTopological Exercise (in video)

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Exercise #1

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Exercise #2

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Sourceshttp://www.just.edu.jo/~basel/algorithms/Algo%20Slides/algo_ch22_graph_part3.pdfhttp://cs.anu.edu.au/~Alistair.Rendell/Teaching/apac_comp3600/module4/basic_graph_algorithms.xhtmlhttp://www.cs.umbc.edu/courses/undergraduate/341/Lectures/Graphs/Graphs.ppthttp://www.cs.usfca.edu/~galles/visualization/TopoSortDFS.htmlhttp://www.cse.cuhk.edu.hk/~taoyf/course/2100sum11/lec14.pdfhttp://www.cse.ust.hk/faculty/golin/COMP271Sp03/Notes/MyL08.pdfhttp://www.cs.nott.ac.uk/~nza/G5BADS03/graphs3.pdfCMSC Image SourceData Structures and Algorithm Analysis in Java 3rd Edition by Mark Allen Weisshttp://www.cs.princeton.edu/courses/archive/spr10/cos226/lectures.html (topological sort under Demos)https://www.youtube.com/watch?v=jksMzq4LgfM

Topological Graph Creator (with some edits)Dia

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