Hierarchical Bin Buffering: Online Local Moments for Dynamic External Memory Arrays

TL;DR

This paper introduces hierarchical bin buffering algorithms that efficiently compute local moments over large external memory arrays, significantly reducing query times with minimal storage, applicable to various data analysis tasks.

Contribution

It presents novel hierarchical buffering techniques that improve query efficiency for local moments in large external memory arrays, using less storage than wavelet-based methods.

Findings

Query time reduced to O(sqrt n) with simple bin partitioning.

Hierarchical buffering achieves logarithmic query time.

Overlapped Bin Buffering uses minimal storage while maintaining efficiency.

Abstract

Local moments are used for local regression, to compute statistical measures such as sums, averages, and standard deviations, and to approximate probability distributions. We consider the case where the data source is a very large I/O array of size n and we want to compute the first N local moments, for some constant N. Without precomputation, this requires O(n) time. We develop a sequence of algorithms of increasing sophistication that use precomputation and additional buffer space to speed up queries. The simpler algorithms partition the I/O array into consecutive ranges called bins, and they are applicable not only to local-moment queries, but also to algebraic queries (MAX, AVERAGE, SUM, etc.). With N buffers of size sqrt{n}, time complexity drops to O(sqrt n). A more sophisticated approach uses hierarchical buffering and has a logarithmic time complexity (O(b log_b n)), when using N hierarchical buffers of size n/b. Using Overlapped Bin Buffering, we show that only a single buffer is needed, as with wavelet-based algorithms, but using much less storage. Applications exist in multidimensional and statistical databases over massive data sets, interactive image processing, and visualization.