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# Topological Sort - PowerPoint PPT Presentation

Topological Sort. Introduction. Definition of Topological Sort. Topological Sort is Not Unique. Topological Sort Algorithms. An Example. Implementation of Source Removal Algorithm. Review Questions. Introduction.

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Presentation Transcript

• Introduction.

• Definition of Topological Sort.

• Topological Sort is Not Unique.

• Topological Sort Algorithms.

• An Example.

• Implementation of Source Removal Algorithm.

• Review Questions.

• There are many problems involving a set of tasks in which some of the tasks must be done before others.

• For example, consider the problem of taking a course only after taking its prerequisites.

• Is there any systematic way of linearly arranging the courses in the order that they should be taken?

Yes! - Topological sort.

Example:

• A directed acyclic graph (DAG) is a directed graph without cycles.

• DAGs arrise in modeling many problems that involve prerequisite constraints (construction projects, course prerequisites, document version control, compilers, etc.)

• Some properties of DAGs:

• Every DAG must have at least one vertex with in-degree zero and at least one vertex with out-degree zero

• A vertex with in-degree zero is called a source vertex, a vertex with out-degree zero is called a sink vertex.

• G is a DAG iff each vertex in G is in its own strongly connected component

• Every edge (v, w) in a DAG G has finishingTime[w] < finishingTime[v] in a DFS traversal of G

• Given a directed graph G = (V, E) a topological sort of G is an

ordering of V such that for any edge (u, v), u comes before v in

the ordering.

• Example1:

• Example2:

• Example3: The graph in (a) can be topologically sorted as in (b)

(a)

(b)

s2 = {a, c, b, f, e, d, h, g, i}

s3 = {a, b, d, c, e, g, f, h, i}

s4 = {a, c, f, b, e, h, d, g, i}

etc.

Topological Sort is not unique

• Topological sort is not unique.

• The following are all topological sort of the graph below:

• Topological-Sort(G)

• {

• 1. Call dfsAllVertices on G to compute f[v] for each vertex v

• 2. If G contains a back edge (v, w) (i.e., if f[w] > f[v]) , report error ;

• 3. else, as each vertex is finished prepend it to a list;// or push in stack

• 4. Return the list; // list is a valid topological sort

• }

• Running time is O(V+E), which is the running time for DFS.

Topological order: A C D B E H F G

• The Source Removal Topological sort algorithm is:

• Pick a source u [vertex with in-degree zero], output it.

• Remove u and all edges out of u.

• Repeat until graph is empty.

• int topologicalOrderTraversal( ){

• int numVisitedVertices = 0;

• while(there are more vertices to be visited){

• if(there is no vertex with in-degree 0)

• break;

• else{

• select a vertex v that has in-degree 0;

• visit v;

• numVisitedVertices++;

• delete v and all its emanating edges;

• }

• }

• return numVisitedVertices;

• }

• 3

2

0

2

A

B

C

D

E

F

G

H

I

J

1

0

2

0

2

D G A B F H J E I C

Topological Sort: Source Removal Example

• The number beside each vertex is the in-degree of the vertex at the start of the algorithm.

• The algorithm is implemented as a traversal method that visits the vertices in a topological sort order.

• An array of length |V| is used to record the in-degrees of the vertices. Hence no need to remove vertices or edges.

• A priority queue is used to keep track of vertices with in-degree zero that are not yet visited.

public int topologicalOrderTraversal(Visitor visitor){

int numVerticesVisited = 0;

int[] inDegree = new int[numberOfVertices];

for(int i = 0; i < numberOfVertices; i++)

inDegree[i] = 0;

Iterator p = getEdges();

while (p.hasNext()) {

Edge edge = (Edge) p.next();

Vertex to = edge.getToVertex();

inDegree[getIndex(to)]++;

}

BinaryHeap queue = new BinaryHeap(numberOfVertices);

p = getVertices();

while(p.hasNext()){

Vertex v = (Vertex)p.next();

if(inDegree[getIndex(v)] == 0)

queue.enqueue(v);

}

while(!queue.isEmpty() && !visitor.isDone()){

Vertex v = (Vertex)queue.dequeueMin();

visitor.visit(v);

numVerticesVisited++;

p = v.getSuccessors();

while (p.hasNext()){

Vertex to = (Vertex) p.next();

if(--inDegree[getIndex(to)] == 0)

queue.enqueue(to);

}

}

return numVerticesVisited;

}

• List the order in which the nodes of the directed graph GB are visited by

• topological order traversal that starts from vertex a. Use both DFS-based and Source Removal algorithm

• 2. What kind of DAG has a unique topological sort?

• 3. Generate a directed graph using the required courses for your major. Now apply topological sort on the directed graph you obtained.