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Topological Sort

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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Topological Sort

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

  2. Introduction • 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.

  3. What is a DAG? 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

  4. Definition of Topological Sort • 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:

  5. Definition of Topological Sort • Example3: The graph in (a) can be topologically sorted as in (b) (a) (b)

  6. s1 = {a, b, c, d, e, f, g, h, i} 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:

  7. Topological Sort Algorithms: DFS based algorithm • 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

  8. Topological Sort Algorithms: Source Removal Algorithm • 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; • }

  9. 1 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.

  10. Implementation of Topological Sort • 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)]++; }

  11. Implementation of Topological Sort 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; }

  12. Review Questions • 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.

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