I/O-Efficient Well-Separated Pair Decomposition and Its Applications

Abstract

Abstract. We present an external memory algorithm to compute a well-separated pair decomposition (WSPD) of a given point set P in

Pruning spanners and constructing well-separated pair decompositions in the presence of memory hierarchies

“…For simplicity of exposition, and following the notation used by Govindarajan et al [23], we refer to a node representing the subset A ⊆ S also as A, i.e., we use the denotation for nodes and subsets interchangeably whenever the meaning is clear from the context. A split tree of S is a binary tree associated with S and is defined recursively as follows: If |S| = 1, then T consists of a single node representing the singleton.…”

“…In the literature, general surveys [4,40] as well as more specific surveys considering, for example, I/Oefficient graph algorithms [30,39] can be found. Results related to I/O-efficiently constructing (planar) spanners for point sets, sometimes allowing Steiner points and/or respecting polygonal obstacles in the plane, have been obtained by several authors [23,33,34].…”

“…As mentioned above, we can benefit from the original I/O-algorithm as most of the building blocks can be adopted directly. To keep the exposition self-contained, yet as concise as possible, we will only sketch these parts and refer the reader to the work of Govindarajan et al [23] for a more detailed description of them. Whenever a part of the I/O-algorithm has to be changed, we will outline the I/O-efficient implementation of each part along with the modifications necessary to make it cache-oblivious.…”

“…In the following, we will describe our cache-oblivious method by outlining the original I/O-algorithm along with the necessary modifications. 4 In a nutshell, Govindarajan et al [23] compute a WSPD of a given point set S as follows: They start by constructing a fair split tree T of S. The idea is to build T recursively. First, a partial fair split tree T whose leaves have size at most |S| α for some constant 1 − 1 2d α < 1 is constructed.…”

I/O-Efficient Shortest Path Queries in Geometric Spanners

“…We include these regions as the donut cells of DB . Using the fair split tree construction of [12], and choosing γ ε 6, we obtain the following result. …”