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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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