New Competitiveness Bounds for the Shared Memory Switch

TL;DR

This paper advances the theoretical understanding of buffer management in shared memory switches by tightening competitiveness bounds for the Longest Queue Drop policy and establishing new lower bounds for all deterministic algorithms.

Contribution

It provides simplified proofs for existing bounds, improves the general lower bound from 4/3 to , and introduces novel methods including linear programming and computer simulations to establish tighter bounds for LQD.

Findings

Lower bound for all deterministic algorithms improved to 

LQD is at least 1.44546086-competitive

Linear programming approach yields a lower bound of 1.4427902

Abstract

We consider one of the simplest and best known buffer management architectures: the shared memory switch with multiple output queues and uniform packets. It was one of the first models studied by competitive analysis, with the Longest Queue Drop (LQD) buffer management policy shown to be at least $\sqrt{2}$- and at most $2$-competitive; a general lower bound of $4/3$ has been proven for all deterministic online algorithms. Closing the gap between $\sqrt{2}$ and $2$ has remained an open problem in competitive analysis for more than a decade, with only marginal success in reducing the upper bound of $2$. In this work, we first present a simplified proof for the $\sqrt{2}$ lower bound for LQD and then, using a reduction to the continuous case, improve the general lower bound for all deterministic online algorithms from $\frac 43$ to $\sqrt{2}$. Then, we proceed to improve the lower bound of $\sqrt{2}$ specifically for LQD, showing that LQD is at least $1.44546086$-competitive. We are able to prove the bound by presenting an explicit construction of the optimal clairvoyant algorithm which then allows for two different ways to prove lower bounds: by direct computer simulations and by proving lower bounds via linear programming. The linear programming approach yields a lower bound for LQD of $1.4427902$ (still larger than $\sqrt{2}$).