List Factoring and Relative Worst Order Analysis

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.

💡 Research Summary

The paper addresses a notable gap in the evaluation of online algorithms by applying Relative Worst‑Order Analysis (RWOA) to the static list‑accessing problem, a classic online problem that has traditionally been studied using competitive analysis. Competitive analysis, while providing worst‑case guarantees, often yields bounds that are too pessimistic and do not reflect observed algorithmic behavior. RWOA, introduced as a supplement to competitive analysis, compares two algorithms on the same multiset of requests by pairing each algorithm’s most disadvantageous request order with the other’s most advantageous order. This pairing yields a relative worst‑order ratio that can be significantly tighter and more indicative of practical performance.

The authors first review the state of the art: RWOA has been successfully applied to paging, matching, and scheduling problems, consistently delivering results that align better with empirical data than traditional competitive ratios. However, no prior work had extended RWOA to the list‑accessing domain, where the Move‑to‑Front (MTF), Transpose (TRANS), and Free‑Cache (FC) heuristics are the canonical algorithms.

A central contribution of the paper is the introduction of the list‑factoring technique into the RWOA framework. List factoring, originally devised for competitive analysis, decomposes a list of length n into two sub‑lists (L₁ and L₂) and analyzes the cost contributed by each sub‑list independently. The technique relies on two key properties of the cost function: linearity (the total cost is the sum of per‑access costs) and exchange invariance (swapping two items changes the cost in a predictable, additive way). By proving that these properties hold for the static list‑access model, the authors are able to lift the factorization argument to RWOA.

Using list factoring, the paper derives precise relative worst‑order ratios for the three algorithms under study:

1. MTF vs. TRANS – The relative worst‑order ratio is strictly less than 1, establishing that MTF dominates TRANS even when each algorithm is evaluated on its own worst‑order permutation. This result improves upon the classic competitive analysis, where both algorithms share the same competitive bound of 2.

2. MTF vs. FC – While FC can outperform MTF on certain request multisets (particularly those with high‑frequency items clustered toward the end of the list), the overall relative worst‑order ratio still favors MTF. The analysis shows that the advantage of FC is limited to specific patterns and does not survive the worst‑order pairing.

These findings demonstrate that RWOA captures nuanced performance differences that competitive analysis masks. Moreover, the successful adaptation of list factoring to RWOA expands the analytical toolbox for researchers: the technique can now be employed to simplify cost calculations in any online problem where the cost function satisfies linearity and exchange invariance.

The paper also discusses the limitations of the approach. List factoring requires a fixed list size and a cost model that is additive across accesses; extending the method to dynamic lists or non‑additive cost functions would require new theoretical developments. Nevertheless, the authors argue that many practical online problems—such as cache replacement policies, job scheduling, and even certain network routing scenarios—exhibit the necessary structure, suggesting a broad applicability of the combined RWOA‑list‑factoring methodology.

In conclusion, the work makes two substantive contributions: (i) it enriches the set of problems where RWOA provides more realistic performance guarantees by adding the static list‑accessing problem, and (ii) it introduces list factoring as a robust proof technique within the RWOA paradigm. The results not only reaffirm the superiority of MTF over its classic rivals in a more discriminating analytical setting but also open avenues for future research on other online problems where fine‑grained performance analysis is essential.