Asymptotically Optimal Online Page Migration on Three Points

Abstract

This paper addresses the page migration problem: given online requests from nodes on a network for accessing a page stored in a node, to output online migrations of the page. Serving a request costs the distance between the request and the page, and migrating the page costs the migration distance multiplied by the page size D ≥ 1. The objective is to minimize the total sum of service costs and migration costs. Black and Sleator conjectured that there exists a 3-competitive deterministic algorithm for every graph. Although the conjecture was disproved for the case D = 1, whether or not an asymptotically (with respect to D) 3-competitive deterministic algorithm exists for every graph is still open. In fact, we did not know if there exists a 3-competitive deterministic algorithm for an extreme case of three nodes with D ≥ 2. As the first step toward an asymptotic version of the Black and Sleator conjecture, we present 3- and (3 + 1/D)-competitive algorithms on three nodes with D = 2 and D ≥ 3, respectively, and a lower bound of 3 + Ω(1/D) that is greater than 3 for every D ≥ 3. In addition to the results on three nodes, we also derive ρ-competitiveness on complete graphs with edge-weights between 1 and 2 − 2/ρ for any ρ ≥ 3, improving the previous 3-competitive algorithm on uniform networks.