We show via two different algorithms that finding the length of the longest
path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and
also in the complement class co-UL. The result extends to toroidal DAGs as
well

Limaye, Nutan

Mahajan, Meena

Nimbhorkar, Prajakta

English

arXiv

We present a transformation from longest paths to shortest paths in sub-classes of directed acyclic graphs (DAGs). The transformation needs log-space and oracle access to reachability in the same class of graphs. As a corollary, we obtain our main result: Longest Paths in planar DAGs is in UL ∩ co-UL. The result also extends to toroidal DAGs. Further, we show that Longest Paths in max-unique DAGs where the target node is the unique sink is in UL ∩ co-UL. We show that for planar DAGs with the promise that the number of distinct paths is bounded by a polynomial, counting paths can be done by an unambiguous pushdown automaton equipped with an auxiliary log-space worktape and running in polynomial time. This bound also holds if we want to compute the number of longest paths, or shortest paths, and this number is bounded by a polynomial (irrespective of the total number of paths). Along the way, we show that counting paths in general DAGs can be done by a deterministic pushdown automaton with an auxiliary log-space worktape and running in polynomial time, and hence is in the complexity class LogDCFL, provided the number of paths is bounded by a polynomial and the target node is the only sink

Nutan Limaye

Meena Mahajan

Prajakta Nimbhorkar

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Longest paths in planar DAGs  in Unambiguous Log-space

Longest Paths in Planar DAGs in Unambiguous Log-Space

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.243.335

Longest paths in Planar DAGs in Unambiguous Logspace

Longest paths in Planar DAGs in Unambiguous Logspace

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