t-Spanners for metric space searching

Abstract

The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G ′ (V, E) as the representation of the metric database. A t-spanner is a subgraph G ′ (V, E) of a graph G(V, A), such that E ⊆ A and G ′ approximates the shortest path costs over G within a precision factor t.Our key idea is to regard the t-spanner as an approximation to the complete graph of distances among the objects, and to use it as a compact device to simulate the large matrix of distances required by successful search algorithms such as AESA. The t-spanner properties imply that we can use shortest paths over G ′ to estimate any distance with bounded error factor t.For this sake, several t-spanner construction, updating, and search algorithms are proposed and experimentally evaluated. We show that our technique is competitive against current approaches. For example, in a metric space of documents our search time is only 9% over AESA, yet we need just 4% of its space requirement. Similar results are obtained in other metric spaces.Finally, we conjecture that the essential metric space property to obtain good tspanner performance is the existence of clusters of elements, and enough empirical evidence is given to support this claim. This property holds in most real-world metric spaces, so we expect that t-spanners will display good behavior in most practical applications. Furthermore, we show that t-spanners have a great potential for improvements.

Parallel query processing on distributed clustering indexes

“…Pivoting techniques select some objects as pivots, calculate the distance among all objects and the pivots, and use them in the triangle inequality to discard objects during search. Many algorithms are based on this idea [1,3,5,8,9,14,23,24,27,28,30,31,33,[35][36][37]. Clustering techniques divide the collection of data into groups called clusters such that similar objects fall into the same group.…”

Solving similarity joins and range queries in metric spaces with the list of twin clusters

“…In metric spaces, a natural approach to solve this problem consists in indexing one or both sets independently (by using any from the plethora of metric indices [3,4,7,8,[10][11][12]14,15,17,21,27,[36][37][38]40,41,44,[47][48][49][50], most of them compiled in [13,26,46,51]), and then solving range queries for all the involved elements over the indexed sets. In fact, this is the strategy proposed in [19], where the authors use the D-index [18] in order to solve similarity self joins.…”

Effective use of space for pivot-based metric indexing structures

“…The t-spanners structure [12] approximates the original distance matrix by introducing an error rate that can be bounded. The setup time for the t-spanners is even worse than the AESA, limiting its use for only small cardinalities, but its query performance is close to AESA while consuming less memory.…”