Maintenance of multidimensional histograms

Abstract

We present a space- and time- efficient algorithm for maintaining multidimensional histograms for data that is dynamic, i.e., subject to updates that may be increments or decrements. Both space used as well as per-update and computing times are polylogarithmic in the input data size; this is the first known algorithm in the data stream model for this problem with this property. One of the powerful motivation for studying data stream algorithms is in analyzing traffic log from IP networks where d-dimensional data (for small d) is common. Hence, our results are of great interest. The result itself is achieved by generalizing methods known for maintenance of unidimensional histograms under updates - finding significant tensor generalizations of one dimensional wavelets, approximating distributions by robust representations - and relationships amongst histograms such as those between tensor wavelets or hierarchical histograms and general histograms. © Springer-Verlag Berlin Heidelberg 2003.