When to Give Up on a Parallel Implementation

TL;DR

This paper investigates online scheduling strategies in the Serial Parallel Decision Problem, demonstrating that different commitment models have distinct optimal competitive ratios in a massively parallel setting.

Contribution

It establishes the separation of power among instantly, eventually, and never-committing schedulers in the massively parallel regime, providing tight bounds for their competitive ratios.

Findings

Instantly-committing schedulers have a competitive ratio of 2.

Eventually-committing schedulers have a ratio between 1.618 and 1.678.

Never-committing schedulers have a ratio between 1.366 and 1.500.

Abstract

In the Serial Parallel Decision Problem (SPDP), introduced by Kuszmaul and Westover [SPAA'24], an algorithm receives a series of tasks online, and must choose for each between a serial implementation and a parallelizable (but less efficient) implementation. Kuszmaul and Westover describe three decision models: (1) \defn{Instantly-committing} schedulers must decide on arrival, irrevocably, which implementation of the task to run. (2) \defn{Eventually-committing} schedulers can delay their decision beyond a task's arrival time, but cannot revoke their decision once made. (3) \defn{Never-committing} schedulers are always free to abandon their progress on the task and start over using a different implementation. Kuszmaul and Westover gave a simple instantly-committing scheduler whose total completion time is $3$-competitive with the offline optimal schedule. They conjectured that the three decision models should admit different competitive ratios, but left upper bounds below $3$ in any model as an open problem. In this paper, we show that the powers of instantly, eventually, and never committing schedulers are distinct, at least in the ``massively parallel regime''. The massively parallel regime of the SPDP is the special case where the number of available processors is asymptotically larger than the number of tasks to process, meaning that the \emph{work} associated with running a task in serial is negligible compared to its \emph{runtime}. In this regime, we show (1) The optimal competitive ratio for instantly-committing schedulers is $2$, (2) The optimal competitive ratio for eventually-committing schedulers lies in $[1.618, 1.678]$, (3) The optimal competitive ratio for never-committing schedulers lies in $[1.366, 1.500]$.