Preemptive Online Partitioning of Sequences

TL;DR

This paper introduces preemptive online algorithms for sequence partitioning, achieving constant competitive ratios and improving previous bounds by allowing separators to be removed but not reinserted.

Contribution

It presents the first simple deterministic 2-competitive preemptive algorithm and a sophisticated scheme that improves competitiveness to 1.68 for certain cases, advancing online sequence partitioning methods.

Findings

Deterministic 2-competitive preemptive algorithm for arbitrary sequences.

Improved competitiveness to 1.68 under specific conditions with a novel scheme.

Lower bounds of 4/3 and 6/5 for deterministic and randomized algorithms, respectively.

Abstract

Online algorithms process their inputs piece by piece, taking irrevocable decisions for each data item. This model is too restrictive for most partitioning problems, since data that is yet to arrive may render it impossible to extend partial partitionings to the entire data set reasonably well. In this work, we show that preemption might be a potential remedy. We consider the problem of partitioning online sequences, where $p-1$ separators need to be inserted into a sequence of integers that arrives online so as to create $p$ contiguous partitions of similar weight. While without preemption no algorithm with non-trivial competitive ratio is possible, if preemption is allowed, i.e., inserted partition separators may be removed but not reinserted again, then we show that constant competitive algorithms can be obtained. Our contributions include: We first give a simple deterministic $2$-competitive preemptive algorithm for arbitrary $p$ and arbitrary sequences. Our main contribution is the design of a highly non-trivial partitioning scheme, which, under some natural conditions and $p$ being a power of two, allows us to improve the competitiveness to $1.68$. We also show that the competitiveness of deterministic (randomized) algorithms is at least $\frac{4}{3}$ (resp. $\frac{6}{5}$). For $p=2$, the problem corresponds to the interesting special case of preemptively guessing the center of a weighted request sequence. While deterministic algorithms fail here, we provide a randomized $1.345$-competitive algorithm for all-ones sequences and prove that this is optimal. For weighted sequences, we give a $1.628$-competitive algorithm and a lower bound of $1.5$.