DistributionSensitive Construction of the Greedy Spanner
Abstract
The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it take Omega(n^2) time, limiting its applicability on large data sets. We observe that for many point sets, the greedy spanner has many `short' edges that can be determined locally and usually quickly, and few or no `long' edges that can usually be determined quickly using local information and the wellseparated pair decomposition. We give experimental results showing large to massive performance increases over the stateoftheart on nearly all tests and reallife data sets. On the theoretical side we prove a nearlinear expected time bound on uniform point sets and a nearquadratic worstcase bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of tspanners we give on such point sets: we give a geometric property that holds with high probability on such point sets. This property implies that if an edge set on these points has tpaths between pairs of points `close' to each other, then it has tpaths between all pairs of points. This characterization gives a O(n log^2 n log^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give a O((n + E) log^2 n log log n) expected time algorithm on uniformly distributed points that determines if E is a tspanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.
 Publication:

arXiv eprints
 Pub Date:
 January 2014
 arXiv:
 arXiv:1401.1085
 Bibcode:
 2014arXiv1401.1085A
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms
 EPrint:
 16 pages,22 figures. Full version of the ESA 2014 publication with the same title