[Paper Review] List Factoring and Relative Worst Order Analysis

Martin R. Ehmsen, Jens S. Kohrt+1 more|arXiv (Cornell University)|Sep 29, 2010

Optimization and Search Problems23 references4 citations

TL;DR

This paper introduces list factoring as a novel proof technique for relative worst order analysis (RWOA), applying it to the static list accessing problem to resolve discrepancies between competitive analysis and experimental results. It demonstrates that Move-To-Front, Time-Stamp, and Frequency-Count are equally good under RWOA—contrary to competitive analysis—while Transpose is significantly worse, aligning with empirical observations and establishing list factoring as a powerful tool for RWOA.

ABSTRACT

Relative worst order analysis is a supplement or alternative to competitive analysis which has been shown to give results more in accordance with observed behavior of online algorithms for a range of different online problems. The contribution of this paper is twofold. First, it adds the static list accessing problem to the collection of online problems where relative worst order analysis gives better results. Second, and maybe more interesting, it adds the non-trivial supplementary proof technique of list factoring to the theoretical toolbox for relative worst order analysis.

Motivation & Objective

• To address the inconsistency between competitive analysis, which deems Move-To-Front optimal, and experimental results, which show Frequency-Count and Transpose perform similarly and better than expected.
• To extend the relative worst order analysis (RWOA) framework to the static list accessing problem, where it yields results more aligned with empirical behavior.
• To introduce and validate list factoring as a new, non-trivial proof technique for RWOA, enabling finer-grained algorithm comparison.
• To compare randomized algorithms BIT and Randomized-Move-To-Front under RWOA, showing they are incomparable despite competitive analysis favoring BIT.
• To demonstrate that RWOA can provide more nuanced and realistic performance evaluations than competitive analysis in online algorithms.

Proposed method

• Adapts list factoring—a technique from competitive analysis—into the relative worst order analysis framework for the first time.
• Uses phase-based analysis to compare request sequences under worst permutations for different algorithms.
• Applies worst-order projection and pairwise properties to model algorithm behavior across different request patterns.
• Employs cost analysis with partial and full cost models to evaluate performance under RWOA.
• Constructs specific request sequences (e.g., repetitions of ⟨b, a^m⟩) to test worst-case behavior and compare algorithms.
• Uses asymptotic cost bounds and inequalities to show that RMTF’s cost is strictly less than BIT’s in worst permutations, proving incomparability.

Experimental results

Research questions

• RQ1 Can list factoring be successfully adapted to relative worst order analysis to improve algorithm comparison in online problems?
• RQ2 Why does competitive analysis fail to reflect experimental results in list accessing, and can RWOA resolve this?
• RQ3 Are Move-To-Front, Frequency-Count, and Time-Stamp equally good under RWOA, despite differing competitive ratios?
• RQ4 Is Randomized-Move-To-Front comparable to BIT under RWOA, or are they incomparable as suggested by the results?
• RQ5 Can the list factoring technique be generalized beyond competitive analysis and RWOA to other performance measures?

Key findings

• List factoring is successfully extended to relative worst order analysis, marking the first use of this technique in that framework.
• Under RWOA, Move-To-Front, Time-Stamp, and Frequency-Count are all equally good, resolving the discrepancy with competitive analysis.
• Transpose is shown to be significantly worse than the other three algorithms under RWOA, consistent with experimental findings.
• Randomized-Move-To-Front and BIT are incomparable under RWOA, despite BIT being 7/4-competitive and RMTF being 2-competitive under competitive analysis.
• The worst-case request sequence for RMTF is constructed using repeated patterns like ⟨b, a^m⟩, and its cost is proven strictly less than that of BIT on its worst permutation.