Webbläsaren som du använder stöds inte av denna webbplats. Alla versioner av Internet Explorer stöds inte längre, av oss eller Microsoft (läs mer här: * https://www.microsoft.com/en-us/microsoft-365/windows/end-of-ie-support).

Var god och använd en modern webbläsare för att ta del av denna webbplats, som t.ex. nyaste versioner av Edge, Chrome, Firefox eller Safari osv.

Faster algorithms for finding lowest common ancestors in directed acyclic graphs

In English

Författare

Summary, in English

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.

Avdelning/ar

  • Computer Science

Publiceringsår

2007

Språk

Engelska

Sidor

37-46

Publikation/Tidskrift/Serie

Theoretical Computer Science

Volym

380

Issue

1-2

Dokumenttyp

Artikel i tidskrift

Förlag

Elsevier

Ämne

  • Computer Science

Nyckelord

  • directed acyclic graphs
  • lowest common ancestors
  • matrix
  • multiplication
  • time complexity

Status

Published

Projekt

  • VR 2005-4085

ISBN/ISSN/Övrigt

  • ISSN: 0304-3975