Simple Space-Time Trade-Offs for AESA

Abstract

We consider indexing and range searching in metric spaces. The best method known is AESA, in practice requiring the fewest number of distance evaluations to answer range queries. The problem with AESA is its space complexity, requiring storage for Θ(n ²) distance values to index n objects. We give several methods to reduce this cost. The main observation is that exact distance values are not needed, but lower and upper bounds suffice. The simplest of our methods need only Θ(n ²) bits (as opposed to words) of storage, but the price to pay is more distance evaluations, the exact cost depending on the dimension, as compared to AESA. To reduce this efficiency gap we extend our method to use b distance bounds, requiring $\Theta(n^2\log_2(b))$ bits of storage. The scheme uses also Θ(b) or Θ(bn) words of auxiliary space. We experimentally show that using b ∈ {1,...,16} (depending on the problem instance) gives good results. Our preprocessing and side computation costs are the same as for AESA. We propose several improvements, achieving e.g. O(n ^1 + α) construction cost for some 0 < α< 1, and a variant using even less space.