Paid exchanges are worth the price

Marc P. Renault,Alejandro López-Ortiz,Adi Rosén

doi:10.1016/j.tcs.2020.01.031

Marc P. Renault, Alejandro López-Ortiz + Show 1 more

Open Access

https://doi.org/10.1016/j.tcs.2020.01.031

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Abstract

We consider the list update problem as defined in the seminal work on competitive analysis by Sleator and Tarjan [13]. An instance of the problem consists of a sequence of requests to access items in a linked list. After an item is accessed, that item can be moved to any position forward in the list at no cost (a move called free exchange), and, at any time, any two adjacent items can be swapped at a cost of 1 (a move called paid exchange). The cost to access an item is equal to its current position in the list. The goal is to dynamically rearrange the list so as to minimize the total cost (accrued from accesses and exchanges) over the request sequence.We show a lower bound of 12/11 on the worst-case ratio between the performance of an (offline) optimal algorithm that can only perform free exchanges and that of an (offline) optimal algorithm that can perform both paid and free exchanges. This answers the question of the asymptotic relative power of the two models which has been open since Reingold and Westbrook [11] showed in 1996 that Sleator and Tarjan erred in [13] when they claimed that the two models are equivalent.

Highlights

The list update problem consists of a linked list of items and a finite request sequence
We show a lower bound of 12/11 on the worst-case ratio between the performance of an optimal algorithm that can only perform free exchanges and that of an optimal algorithm that can perform both paid and free exchanges
We showed that the difference in the performance between an offline optimal algorithm restricted to free exchanges and an unrestricted offline optimal algorithm is at least a multiplicative factor of 12/11, answering a question that has been open since 1996 [10]

Summary

Introduction

The list update problem consists of a linked list of items and a finite request sequence. In [12], Sleator and Tarjan claim that an algorithm that uses paid exchanges and free exchanges can be converted to an algorithm that uses only free exchanges without increasing the cost This claim turns out not to be true as Reingold and Westbrook gave the counterexample of the request sequence 3, 2, 2, 3 for a list of length 3 with a starting configuration of 1, 2, 3 [10]. The competitive ratio of 1.6 for the comb algorithm [1] (as described above) implies an upper bound of 1.6 on the worst case ratio between the cost of an optimal algorithm restricted to free exchanges and the cost of an unrestricted optimal algorithm, over all finite request sequences. As all online algorithms with currently best known competitive ratios use only free exchanges [8], our result suggests that, in order to achieve better upper bounds, it may be useful to consider online algorithms that make use of paid exchanges

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