Summary
Imagine trying to organize pairs of points in space so that each pair is nicely separated but still covers all points. The paper studies methods to create these groupings, called well-separated pair decompositions (WSPDs), which make calculations on points easier and faster. The authors introduce faster, more accurate ways to build these groups with proven bounds, especially in two dimensions, and provide experimental evidence that their techniques are close to the best possible. They also show how to convert one type of grouping (cover WSPD) into another (partition WSPD) while keeping the size small. These improvements help in practical scenarios where organizing point data efficiently matters.
What this means in practice
- •For mapping software engineers: Speed up spatial queries by efficiently grouping points with guaranteed small overhead, improving offline and online geometric computations.
- •For network designers: Use smaller, certified approximations of point pair groupings to optimize distance-based communication structures in wireless and sensor networks.
Abstract
Given n points and $ε>0$, a well-separated pair decomposition (WSPD) is a collection of dumbbells {X,Y}, where X and Y have diameter at most $ε$ times their distance, such that every pair of points has one point in X and the other in Y for some dumbbell. A partition WSPD covers every pair exactly once, and a cover WSPD at least once. Callahan and Kosaraju give a partition WSPD with $O(n/ε^d)$ dumbbells in $R^d$, but many point sets have far smaller WSPDs of both kinds. For $d\ge2$, the known $O(1)$-approximations of the minimum size WSPD that preserve the separation $ε$ have unspecified factors, above $10^7$ for $d=2$ by our estimates. For cover WSPDs, we give an $O(n^2\log n)$-time sweep approximation algorithm with a factor 40 in the plane for $ε\le1/16$. It extends to $R^d$ with a factor $2^{O(d)}$. The sweep is also a lower-bound tool: its number of steps is a lower bound on the WSPD size. For partition WSPDs, we analyze quadtree and fair split tree constructions: in the plane and for $ε\le1/5$, they give constant approximation factors, the smallest of which is 252 for a randomly shifted and scaled quadtree. In experiments with $ε=1/20$, the number of sweep steps is at least 0.91 OPT for n = 500, where the optimum OPT is known, and a local search raises it to a lower bound of 0.99 OPT. A greedy variant of the sweep, which is still a certified 40-approximation, is at most 1.28 times the lower bound for n = 10000. Followed by a new local-search heuristic that shrinks the WSPD, it is within 0.02% of the optimum on every instance with n = 500, and at most 1.10 times the lower bound for n = 10000. We then show how to convert the cover WSPDs into partition WSPDs. For n = 10000, the conversion gives on average 1.13 times the lower bound, against 1.05 for the cover WSPD itself and 1.55 for the Callahan-Kosaraju partition WSPD.