Parallel Construction of k-d Trees and Related Problems

k-d tree (or Multidimensional binary search tree) is often used as a data structure for organizing and manipulating spatial data. We present several algorithms for the parallel construction of k-d trees on coarse-grained distributed memory parallel machines. We analyze them both theoretically and experimentally. Experimental results on the CM-5 are reported. 1 Multidimensional Binary Search Trees Consider a set of n points in k dimensional space. Let d1; d2; : : : ; dk denote the k dimensions. The root of the k-d tree 3] corresponds to the set of all points. A dimension dl is chosen to nd the median coordinate of all the points along dl. The points are partitioned into two approximately equal sized sets-one set containing all the points whose coordinates along dimension dl are less than or equal to this median and a second set containing all the remaining points. The two subpartitions are represented by the children of the root node. The tree is built recursively until each leaf corresponds to one point. In homogeneous trees, internal nodes are used to store the median points. Non-homogeneous trees store points only at the leaves. Partial construction of k-d trees is used in several applications. In parallel graph partitioning, p partitions are created to distribute the graph to p processors, requiring the construction of only the rst log p levels of the k-d tree. In hierarchical applications like the n-body simulation, clustering of physically proximate objects is essential and the k-d tree ooers such a clustering scheme. In databases, records can be treated as points in an appropriate space by mapping each key to a coordinate and the resulting point set can be organized using a k-d tree. In constructing the tree, a node is partitioned only if all its records do not t in one disk sector. Our focus is on algorithms for eecient parallel construction of balanced, non-homogeneous k-d trees on coarse-grained distributed memory parallel computers. For all nodes at level i of the tree (deening the root to be at level 0), we use dimension d (i mod k)+1. Other variations can be easily implemented with minor changes in our algorithms without signiicantly aaecting their running time. 2 Model of Parallel Computation A coarse-grained parallel computer consists of several relatively powerful processors connected by an interconnection network. Most of the commercially available parallel computers including SGI Challenge, Intel Paragon and Delta, Cray T3D, …