# LCA Queries in Directed Acyclic Graphs

## Abstract

We present two 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*). As a corollary, we obtain an *O*(*n* ^{2})-time algorithm for finding genealogical distances considerably improving the previously known *O*(*n* ^{2.575}) time-bound for this problem.

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 \(O(n^{{2}+\frac{1}{4-\omega}})\), where *ω* =2.376 is the exponent of the fastest known matrix multiplication algorithm. This improves the previously known \(O(n^{\frac{\omega+3}{2}})\) time-bound for the general all-pairs LCA problem in dags.

## Keywords

Common Ancestor Direct Acyclic Graph Transitive Closure Source Vertex Lower Common Ancestor## Preview

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