Exact and Approximate Map-Reduce Algorithms for Convex Hull

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 subset of 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 Theta(vertical bar P vertical bar(epsilon)), for a small constant epsilon > 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 vertical bar P vertical bar. 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.