Transposition is Nearly Optimal for IID List Update

2026-03-10

AI Summary

This study addresses the online list update problem under the i.i.d. request model without prior knowledge, focusing on the performance of the transpose rule—swapping an accessed item with its predecessor. By employing a cost decomposition technique combined with a novel combinatorial injection method termed “sign cancellation,” the authors prove that, in steady state, the expected access cost of the transpose rule exceeds that of the optimal static ordering by at most 1. This result achieves an almost lossless approximation with zero memory overhead, thereby confirming Rivest’s five-decade-old conjecture on the near-optimality of the transpose rule. Moreover, it provides the first rigorous guarantee that the rule’s performance is within an unavoidable additive constant of the theoretical lower bound.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The list update problem is one of the oldest and simplest problems in online algorithms: A set of items must be maintained in a list while requests to these items arrive over time. Whenever an item is requested, the algorithm pays a cost equal to the position of the item in the list. In the i.i.d. model, where requests are drawn independently from a fixed distribution, the static ordering by decreasing access probabilities $p_1\ge p_2\ge \dots \ge p_n$ achieves the minimal expected access cost OPT$=\sum_{i=1}^n ip_i$. However, $p$ is typically unknown, and approximating it by tracking access frequencies creates undesirable overheads. We prove that the Transposition rule (swap the requested item with its predecessor) has expected access cost at most OPT$+1$ in its stationary distribution. This confirms a 50-year-old conjecture by Rivest up to an unavoidable additive constant. More abstractly, it yields a purely memoryless procedure to approximately sort probabilities via sampling. Our proof is based on a decomposition of excess cost, and its technical core is a"sign-eliminating"combinatorial injection to witness nonnegativity of a constrained multivariate polynomial.

Problem

• list update
• i.i.d. model
• online algorithms
• access cost
• probability estimation

Innovation

• Transposition rule
• list update
• i.i.d. model
• combinatorial injection
• online algorithms