Exact and approximate map-reduce algorithms for convex hull

Document Type

Conference Proceeding

Publication Date

1-1-2018

Abstract

Given a set of points P in the Euclidean plane, the classic problem of convex hull in computational geometry asks to compute the smallest convex polygon C with the vertex set X ⊆ P, such that every point in P belongs to C. In our knowledge, only two map-reduce convex hull algorithms have been designed so far. The exact map-reduce algorithm designed by Goodrich et al. (2011) is intricate and runs in constant number of rounds when the mappers and reducers have a memory of Θ(|P|ε), for a small constant ε >0. Otherwise, their algorithm runs in logarithmic number of rounds with high probability. In Big Data, easy-to-implement constant-round map-reduce algorithms are highly preferred. The other exact map-reduce algorithm, designed by Eldawy et al. (2011), does not perform efficiently when X contains sufficiently high number of points from P. In this paper, we design two new simple constant-round map-reduce algorithms along with map-reduce implementable pruning heuristics to address the above shortcomings. Our first algorithm CH-MR is exact and outperforms Eldawy et al.’s algorithm when reasonable computing resources are available, and the heuristics are able to prune away sufficient number of points. The second algorithm, named APXCH-MR, can run efficiently on any point set to return an approximate convex hull, when the input parameters are sub-linear in |P|. The designed algorithms are theoretically analyzed in the light of the popular MRC model. Our algorithms are easy to implement and do not use any complicated data structure.

Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volume

11346 LNCS

First Page

480

Last Page

494

Digital Object Identifier (DOI)

10.1007/978-3-030-04651-4_32

ISSN

03029743

E-ISSN

16113349

ISBN

9783030046507

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