Querying Metric Spaces with Bit Operations

Abstract

Metric search techniques can be usefully characterised by the time at which distance calculations are performed during a query. Most exact search mechanisms use a "just-in-time" approach where distances are calculated as part of a navigational strategy. An alternative is to use a "one-time" approach, where distances to a fixed set of reference objects are calculated at the start of each query. These distances are typically used to re-cast data and queries into a different space where querying is more efficient, allowing an approximate solution to be obtained. In this paper we use a "one-time" approach for an exact search mechanism. A fixed set of reference objects is used to define a large set of regions within the original space, and each query is assessed with respect to the definition of these regions. Data is then accessed if, and only if, it is useful for the calculation of the query solution. As dimensionality increases, the number of defined regions must increase, but the memory required for the exclusion calculation does not. We show that the technique gives excellent performance over the SISAP benchmark data sets, and most interestingly we show how increases in dimensionality may be countered by relatively modest increases in the number of reference objects used. 1 Context To set a formal context, we are interested in searching a (large) finite set of objects S which is a subset of an infinite set U , where (U, d) is a metric space: that is, an ordered pair (U, d), where U is a domain of objects and d is a total distance function d : U ×U → R, satisfying postulates of non-negativity, identity, symmetry, and triangle inequality [20]. The general requirement is to efficiently find members of S which are similar to an arbitrary member of U given as a query, where the distance function d gives the only way by which any two objects may be compared. There are many important practical examples captured by this mathematical framework, see for example [16, 20]. The simplest type of similarity query is the range search query: for some threshold t, based on a query q ∈ U , the solution set is R = {s ∈ S| d(q, s) ≤ t}. The essence of metric search is to spend time pre-processing the finite set S so that solutions to queries can be efficiently calculated using only distances among objects. In all cases therefore, distances between the data and selected This is a post-peer-review, pre-copyedit version of a paper published in Marchand-Maillet S., Silva Y., Chávez E.