Longest acyclic path in a directed unweighted graph

Question

What algorithm can be used to find the longest path in an unweighted directed acyclic graph?

Answer 1

Wikipedia has an algorithm: http://en.wikipedia.org/wiki/Longest_path_problem

Looks like they use weightings, but should work with weightings all set to 1.

Answer 2

As long as the graph is acyclic, all you need to do is negate the edge weights and run any shortest-path algorithm.

EDIT: Obviously, you need a shortest-path algorithm that supports negative weights. Also, the algorithm from Wikipedia seems to have better time complexity, but I'll leave my answer here for reference.

Answer 3

Dynamic programming. It is also referenced in Longest path problem, given that it is a DAG.

The following code from Wikipedia:

algorithm dag-longest-path is
    input: 
         Directed acyclic graph G
    output: 
         Length of the longest path

    length_to = array with |V(G)| elements of type int with default value 0

    for each vertex v in topOrder(G) do
        for each edge (v, w) in E(G) do
            if length_to[w] <= length_to[v] + weight(G,(v,w)) then
                length_to[w] = length_to[v] + weight(G, (v,w))

    return max(length_to[v] for v in V(G))

Answer 4

Can be solved by critical path method:
1. find a topological ordering
2. find the critical path
see [Horowitz 1995], Fundamentals of Data Structures in C++, Computer Science Press, New York.

Greedy strategy(e.g. Dijkstra) will not work, no matter:1. use "max" instead of "min" 2. convert positive weights to negative 3. give a very large number M and use M-w as weight.