Page Migration Algorithms Using Work Functions

Abstract

The page migration problem occurs during management of a globally addressed shared memory in a multiprocessor system. Each physical page of memory is located at a given processor, and memory references to that page by other processors are charged a cost equal to the network distance. At times the page may migrate between processors, at a cost equal to the distance times a page size factor,D. The problem is to schedule movements online so as to minimize the total cost of memory references. Page migration can also be viewed as a restriction of the 1-server with excursions problem. This paper presents a collection of algorithms and lower bounds for the page migration problem in various settings. Competitive analysis is used. The competitiveness of an online algorithm is the worst case ratio of its cost to the optimum cost on any sequence of requests. Randomized (2+1/2D)-competitive online algorithms are given for trees and products of trees, including the mesh and the hypercube, and for uniform spaces whenD=1,2. These algorithms are shown to be optimal. A lower bound of8527on the competitiveness of any deterministic algorithm (in arbitrary spaces) withD=1 is shown, disproving a conjecture of Black and Sleator. Deterministic (2+1/2D)-competitive algorithms are given for products of continuous trees under the L1metric, such as Rnwith the Manhattan metric. A deterministic algorithm for Rnunder any norm is presented and is shown to have competitive ratio tending to 1+φ=2.618…. The paper makes extensive use of the concept of a work function, a tool which provides a systematic approach to many competitive analysis problems.