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Faster algorithms for finding lowest common ancestors in directed acyclic graphs
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Czumaj, Artur, Kowaluk, Miroslaw and Lingas, Andrzej (2007) Faster algorithms for finding lowest common ancestors in directed acyclic graphs. Theoretical Computer Science, Vol.380 (No.1-2). pp. 37-46. doi:10.1016/j.tcs.2007.02.053 ISSN 0304-3975.
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Official URL: http://dx.doi.org/10.1016/j.tcs.2007.02.053
Abstract
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 0 (n m).
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 0 (n (2+lambda)), where A satisfies the equation to (1, lambda, I) = 1 + 2 lambda and w (1, lambda, 1) is the exponent of the multiplication of an n x n (lambda) matrix by an n (lambda) x n matrix. By the currently best known bounds on w 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 <= n alpha where alpha approximate to 0.294, our algorithm runs in a time that is asymptotically the same as that required for multiplying two n x n matrices, that is, O (n (w)); 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 (w) ho (0.468)). This algorithm is faster than our algorithm for arbitrary dags for all values of h <= n (0.42). (C) 2007 Elsevier B. V. All rights reserved.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||
| Journal or Publication Title: | Theoretical Computer Science | ||||
| Publisher: | Elsevier Science BV | ||||
| ISSN: | 0304-3975 | ||||
| Official Date: | 21 June 2007 | ||||
| Dates: |
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| Volume: | Vol.380 | ||||
| Number: | No.1-2 | ||||
| Number of Pages: | 10 | ||||
| Page Range: | pp. 37-46 | ||||
| DOI: | 10.1016/j.tcs.2007.02.053 | ||||
| Status: | Not Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Title of Event: | 32nd International Colloquium on Automata, Languages and Programming (ICALP 2005) | ||||
| Type of Event: | Other | ||||
| Location of Event: | Lisbon, Portugal | ||||
| Date(s) of Event: | July 11-15, 2005 |
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