Transposition is Nearly Optimal for IID List Update

TL;DR

This paper proves that the simple transposition rule nearly optimally solves the IID list update problem, confirming a long-standing conjecture and providing a memoryless method to approximate sorting probabilities.

Contribution

It establishes that the transposition rule is nearly optimal in expectation, confirming Rivest's conjecture and offering a memoryless approach to approximate probability sorting.

Findings

Transposition rule achieves expected cost at most OPT+1.

Confirms Rivest's 50-year-old conjecture.

Provides a new combinatorial proof technique.

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.