1
4/12/2004 CS216, Spring 2004 1
Topological Sort
• Given a directed acyclic graph,
construct an ordering of the vertices
such that if there is a path from vito vj,
then vjappears after viin the ordering.
•indegree of v: # of edges (u, v).
v3
v6
v1
v2
v4
v7
v5
v8
4/12/2004 CS216, Spring 2004 2
1
3
4
2
4/12/2004 CS216, Spring 2004 3
void Graph::toposort(){
Vertex v, w;
for (int counter=0; counter < NUM_VERTICES;
counter++){
v = findNewVertexOfDegreeZero();
if (v == NOT_A_VERTEX)
throw CycleFound();
v.topologicalNum = counter;
for each w adjacent to v
w.indegree--;
}
}
4/12/2004 CS216, Spring 2004 4
void Graph::topsort(){
Queue q(NUM_VERTICES);
int counter = 0;
Vertex v, w;
q.makeEmpty();
for each vertex v
if (v.indegree == 0)
q.enqueue(v);
while (!q.isEmpty()){
v = q.dequeue();
v.topologicalNum = ++counter;
for each w adjacent to v
if (--w.indegree == 0)
q.enqueue(w);
}
if (counter != NUM_VERTICES)
throw CycleFound();
}
intialize the
queue
get a vertex with
indegree 0
insert new
eligible
vertices
4/12/2004 CS216, Spring 2004 5
Shortest Path Algorithms
• (“single-source” shortest path)
• Given a graph G= (V, E) and a single
distinguished vertex s, find the shortest
weighted path from sto every other
vertex in G.
weighted path length of v1, v2, … , vN:
∑
−
=
+
1
1
1,
c
N
i
ii , where ci,j is the cost of edge (vi, vj)
4/12/2004 CS216, Spring 2004 6
Unweighted Shortest Path
• Special case of the weighted problem:
all weights are 1.
• Solution: breadth-first search. Similar to
level-order traversal for trees.
v3
v6
v1
v2
v4v7
v5
v0
s
