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Faster algorithms for finding lowest common ancestors in directed acyclic graphs. (English) Zbl 1118.68102
Summary: We present two new methods for finding a Lowest Common Ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on $$n$$ vertices and $$m$$ edges.
The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag on $$n$$ vertices and $$m$$ edges in time $$O(nm)$$.
The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem (and hence also the all-pairs LCA problem) in time $$O(n^{2+\lambda })$$, where $$\lambda$$ satisfies the equation $$\omega (1,\lambda,1) = 1 + 2\lambda$$ and $$\omega (1,\lambda ,1)$$ is the exponent of the multiplication of an $$n\times n^{\lambda }$$ matrix by an $$n^{\lambda }\times n$$ matrix. By the currently best known bounds on $$\omega (1,\lambda ,1)$$, the running time of our algorithm is $$O(n^{2.575})$$. Our algorithm improves the previously known $$O(n^{2.688})$$ time-bound for the general all-pairs LCA problem in dags by Bender et al.
Our additional contribution is a faster algorithm for solving the all-pairs lowest common ancestor problem in dags of small depth, where the depth of a dag is defined as the length of the longest path in the dag. For all dags of depth at most $$h\leq n^{\alpha }$$, where $$\alpha \approx 0.294$$, our algorithm runs in a time that is asymptotically the same as that required for multiplying two $$n\times n$$ matrices, that is, $$O(n^{\omega })$$; we also prove that this running time is optimal even for dags of depth 1. For dags with depth $$h>n^{\alpha }$$, the running time of our algorithm is at most $$O(n^{\omega}\cdot h^{0.468})$$. This algorithm is faster than our algorithm for arbitrary dags for all values of $$h\leq n^{0.42}$$.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C20 Directed graphs (digraphs), tournaments 05C38 Paths and cycles 05C85 Graph algorithms (graph-theoretic aspects) 68W40 Analysis of algorithms
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