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Longest Path in a weighted DAG
By peoplePleaser, history, 4 years ago, In English

Does there exist a way an efficient method to find the longest path in a DAG among all possible paths? I know how to solve this in $$$O(V*(V+E))$$$ by considering each vertex as the source, and then using topo sort, and iterating over the vertices in that order and update the distances accordingly. Any suggestions/help would be appreciated or maybe we can prove there doesn't exist a faster method. Thanks!

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4 years ago, # |
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4 years ago, # |
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4 years ago, # |
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The algorithm you described should be $$$O(V+E)$$$, because you can just relax each edge once in your topo-sort order.

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    4 years ago, # ^ |
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    It's $$$O(V+E)$$$ if we fix the source vertex, right? but we need to find the longest path over all possible paths.

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      4 years ago, # ^ |
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      Just initialize all vertices to have value 0 instead of -INF; alternatively, make a super-source that connects to all vertices.

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        4 years ago, # ^ |
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        Thanks, I thought of this solution. But how to handle the case when all weights are negative? We will always end up with a value of zero, right?

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          4 years ago, # ^ |
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          You can store longest non-empty path, in which case you relax $$$val[v] = \max(val[v], \max(val[u], 0) + e)$$$

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            4 years ago, # ^ |
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            Thanks got it!

            My bad! I am such a fool, if all weights are negative, the answer is the largest (least negative) weight edge.

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