Estimating Graph Parameters from Random Order Streams

TL;DR

This paper introduces a novel technique for converting constant-time approximation algorithms into random order streaming algorithms, enabling efficient estimation of graph parameters with provable guarantees.

Contribution

The authors present a new algorithmic approach that transforms existing approximation algorithms into streaming algorithms for graphs in random order models.

Findings

Approximate number of connected components with additive error using specific space bounds.

Estimate minimum spanning tree weight within a multiplicative factor with defined space complexity.

Approximate maximum independent set size in planar graphs with particular space requirements.

Abstract

We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with probability at least $2/3$, $\bullet$ the number of connected components of $G$ can be approximated up to an additive error of $\varepsilon n$ using $(\frac{1}{\varepsilon})^{O(1/\varepsilon^3)}$ space, $\bullet$ the weight of a minimum spanning tree of a connected input graph with integer edges weights from $\{1,\dots,W\}$ can be approximated within a multiplicative factor of $1+\varepsilon$ using $\big(\frac{1}{\varepsilon}\big)^{\tilde O(W^3/\varepsilon^3)}$ space, $\bullet$ the size of a maximum independent set in planar graphs can be approximated within a multiplicative factor of $1+\varepsilon$ using space $2^{(1/\varepsilon)^{(1/\varepsilon)^{\log^{O(1)} (1/\varepsilon)}}}$.