Single pass sparsification in the streaming model with edge deletions

TL;DR

This paper introduces a method for constructing cut sparsifiers in a single pass over streaming data with edge deletions, using sketching and sampling techniques to achieve efficient updates and sparsifier construction.

Contribution

It presents the first single-pass streaming algorithm for dynamic cut sparsifiers that handles edge deletions efficiently.

Findings

Achieves $ ilde{O}(1/ ext{e}^2)$ update time per stream element.

Constructs sparsifiers with $O(n ext{log}^3 n/ ext{e}^2)$ edges.

Handles edge deletions in a single pass, improving over previous multi-pass or deletion-incompatible methods.

Abstract

In this paper we give a construction of cut sparsifiers of Benczur and Karger in the {\em dynamic} streaming setting in a single pass over the data stream. Previous constructions either required multiple passes or were unable to handle edge deletions. We use $\tilde{O}(1/\e^2)$ time for each stream update and $\tilde{O}(n/\e^2)$ time to construct a sparsifier. Our $\e$-sparsifiers have $O(n\log^3 n/\e^2)$ edges. The main tools behind our result are an application of sketching techniques of Ahn et al.[SODA'12] to estimate edge connectivity together with a novel application of sampling with limited independence and sparse recovery to produce the edges of the sparsifier.