Bounds on series-parallel slowdown

TL;DR

This paper investigates how adding precedence constraints in activity networks affects execution time, disproving a previous conjecture and proposing new bounds on slowdown ratios in parallel program models.

Contribution

It disproves a conjecture that the slowdown ratio is bounded by two without workload considerations and introduces a conjecture of a 4/3 bound when workload is known, supported by algebraic proofs.

Findings

Disproved the conjecture that the slowdown ratio is at most two.

Proposed that a 4/3 slowdown ratio is achievable with workload knowledge.

Analyzed a polynomial-time algorithm related to achieving the 4/3 bound.

Abstract

We use activity networks (task graphs) to model parallel programs and consider series-parallel extensions of these networks. Our motivation is two-fold: the benefits of series-parallel activity networks and the modelling of programming constructs, such as those imposed by current parallel computing environments. Series-parallelisation adds precedence constraints to an activity network, usually increasing its makespan (execution time). The slowdown ratio describes how additional constraints affect the makespan. We disprove an existing conjecture positing a bound of two on the slowdown when workload is not considered. Where workload is known, we conjecture that 4/3 slowdown is always achievable, and prove our conjecture for small networks using max-plus algebra. We analyse a polynomial-time algorithm showing that achieving 4/3 slowdown is in exp-APX. Finally, we discuss the implications of our results.