Longest path in a DAG

Question

To find the longest path in a DAG, I\'m aware of 2 algorithms: algo 1: do a topological sort + use dynamic programming on the result of the sort ~ or ~ algo 2: enumerate all

Answer 1

No DFS needed. Algorithm : takes a DAG G. Each arc holds one variable E

for each node with no predecessor :
    for each of his leaving arcs, E=1.
for each node whose predecessors have all been visited :
    for each of his leaving arcs, E=max(E(entering arcs))+1.

max_path is the highest E within edges when all nodes have been processed.

Answer 2

Your second option is incorrect: DFS does not explore all possible paths, unless your graph is a tree or a forest, and you start from the roots. The second algorithm that I know is negating the weights and finding the shortest path, but it is somewhat slower than the top sort + DP algorithm that you listed as #1.

Answer 3

Enumerate all paths in a DAG using "DFS":

import numpy as np

def enumerate_dag(g):

    def enumerate_r(n, paths, visited, a_path = []):
        a_path += [n]
        visited[n] = 1
        if not g[n]:
            paths += [list(a_path)]
        else:
            for nn in g[n]:
                enumerate_r(nn, paths, visited, list(a_path))

    paths, N = [], len(g)
    visited = np.zeros((N), dtype='int32')

    for root in range(N):
        if visited[root]: continue
        enumerate_r(root, paths, visited, [])

    return paths