Topological Sort
• Introduction.
• Definition of Topological Sort.
• Topological Sort is Not Unique.
• Topological Sort Algorithms.
• An Example.
• Implementation of Source Removal Algorithm.
• Review Questions.
1
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.
2
What is a DAG?
•
A directed acyclic graph (DAG) is a directed graph without cycles.
Example:
•
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 outdegree 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
3
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:
4
Definition of Topological Sort
• Example3: The graph in (a) can be topologically sorted as in (b)
(a)
(b)
5
Topological Sort is not unique
• Topological sort is not unique.
• The following are all topological sort of the graph below:
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.
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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
7
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;
}
8
Topological Sort: Source Removal Example
•
The number beside each vertex is the in-degree of the vertex at the start of the
algorithm.
1
2
3
0
2
A
B
C
D
E
F
G
H
I
J
1
0
2
2
0
D
G
A
B
F
H
J
E
I
C
9
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)]++;
}
10
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;
}
11
Review Questions
1. 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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