Question

Write an algorithm to find the longest path in a DAG, where the length of the path is measured
by the number of edges that it contains. What is the asymptotic complexity of your algorithm?

Answer 1

Begin
initially mark all nodes as unvisited
for each node i in the graph,do
if i is not visited, then
topological sort(i,visited,stack)
done

make distances of all vertices as -infinite
dist[start] :=0
while stack is not empty,do
pop stack and take into nextVert
if dist[nextVert] is not equal to -infinite,then
for each vertices v which is adjacent to nextVert,do
if cost[nextVert,v] is not equal to -infinite,then
if dist[v]<dist[nextVert]+cost[nextVert,v],then
dist[v] :=dist[nextVert]+cost[nextVert,v]
done
done

for all vertices i in the graph do
if dist[i]=-infinite,then
display infinty
else display dist[i]
done
End

FOR TOPOLOGICALSORT(U,VISITED,STACK)

Begin
mark u as visited
for all vertex v,which is connected with u,do
if v is not visited then
topologicalsort(v,visited,stack)
done
push u into stack
End

Asymptotic complexity for topological sorting is O(V+E) so the overall asymptotic complexity is also considered as O(V+E) where V represents Vertices and E represents Edges