Geometric Minimum Spanning Trees with GeoFilterKruskal

Abstract

Abstract. Let P be a set of points in R d . We propose GEOFILTERKRUSKAL, an algorithm that computes the minimum spanning tree of P using well separated pair decomposition in combination with a simple modification of Kruskal's algorithm. When P is sampled from uniform random distribution, we show that our algorithm takes one parallel sort plus a linear number of additional steps, with high probability, to compute the minimum spanning tree. Experiments show that our algorithm works better in practice for most data distributions compared to the current state of the art [31]. Our algorithm is easy to parallelize and to our knowledge, is currently the best practical algorithm on multi-core machines for d > 2.

Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering

“…PARALLELKRUSKAL pair in the WSPD to generate a graph from which an MST can be computed to obtain the EMST. However, it is not necessary to compute the BCCP for all pairs, as observed by Chatterjee et al [17]. Our implementation only computes the BCCP between a pair if their points are not yet connected in the spanning forest generated so far.…”

“…Our implementation is inspired by a variant of Kruskal's algorithm, GeoFilterKruskal (GFK). This algorithm was used for sequential EMST by Chatterjee et al [17], and for MST in general graphs by Osipov et al [49]. It improves Kruskal's algorithm by avoiding the BCCP computation between pairs unless needed, and prioritizing BCCPs between pairs with smaller cardinalities, which are cheaper, with the goal of pruning more expensive BCCP computations.…”

“…Finally, the subsequent rounds process a single pair (𝑄 1 , 𝑖). At the end of each round, we double the value of 𝛽 to ensure that there are logarithmic number of rounds and hence better depth (in contrast, the sequential algorithm of Chatterjee et al [17] increases 𝛽 by 1 every round). Throughout the algorithm, we cache the BCCP results of pairs to avoid repeated computations.…”

“…We implemented our own sequential and parallel versions of the GFK algorithm as a baseline based on Algorithm 2, which we found to be faster than the implementation of Chatterjee et al [17] in our experiments. In addition, because the original GFK algorithm requires materializing the full WSPD, its memory consumption can be excessive, limiting the algorithm's practicality.…”

“…There has been a significant amount of theoretical work on designing fast sequential EMST algorithms (e.g., [6,8,14,55,63]). There have also been some practical implementations of EMST [10,17,43,47], although most of them are sequential (part of the algorithm by Chatterjee et al [17] is parallel). The state-of-the-art EMST implementations are either based on generating a well-separated pair decomposition (WSPD) [15] and applying Kruskal's minimum spanning tree (MST) algorithm on edges produced by the WSPD [17,47], or dual-tree traversals on 𝑘-d trees integrated into Boruvka's MST algorithm [43].…”