Question

Let G = (V, E) be a directed acyclic graph with n vertices and m edges. Give an O(n + m) time algorithm that determines if G contains a directed path that touches every vertex in G exactly once. The graph G is given by its adjacency list representation.

Answer 1

A directed path that touches every vertex in G exactly once is called as Hamiltonian Path

Theory --->

A path that covers every vertex only once exists if

1.There is a path between any two adjacent vertex .

2.Edges only go in increasing direction in linearized order

3. So we need to find Direct Acyclic Graph or DAG that have egde(vi,vi+1) for every pair of adjacent vertex .

Then only hamiltonian path exists

Pseudo-Code or Algorithm

FindHamiltonianPath(GraphG):

1. The graphG, and let v 1,...,vn denote the vertices of G in topologically sorted order. // Here n is no. of vertex in graph

2.For i:=1,2,...,n-1: // check for every vertex that it's adjacent vertex is having a edge or not with it

3.If (vi,vi+1) ∉ E: // check list of vi that is vi+1 present or not in it

4.Return “No.” // If any pair of adjacent vertex donot have edge then no path is possible

5.Return “Yes.” // otherwise path is possible

TIme Complexity

The topological sort in step1 takesΘ(|V|+|E|)time. beacuse it uses DFS (Depth First Search) and one extra stack

Now there is a loop iterates|V|times, examining1 edge in each iteration.

So overall, we have a running time ofΘ(|V|+|E|).

where V is no.of vertex and E is no of edges

Thank You

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