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, 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, extending the previous 3-competitive algorithm on uniform networks.

Uniform Page Migration Problem in Euclidean Space

“…The first deterministic lower bound larger than 3 for general networks was 3.148 [3], which was improved to 3.168 by Matsubayashi [5]. Although these lower bounds, larger than 3, were proved only for D = 1, the first lower bound of 3 + Ω (1) was recently proved [9], where Ω notation is with respect to D. For three points, 3-competitive deterministic algorithms with D ∈ {1, 2} [3,10] and a (3 + 1/D)-competitive deterministic algorithm with D ≥ 3 [10] were proposed. The latter algorithm asymptotically matches the lower bound of 3 + Ω (1/D) on three points for every D ≥ 3 [10].…”

“…For three points, 3 -competitive deterministic algorithms with ∈ {1,2} [3,10] and a (3 + 1/ )-competitive deterministic algorithm with ≥ 3 [10] were proposed. The latter algorithm asymptotically matches the lower bound of 3 + Ω(1/ ) on three points for every ≥ 3 [10]. As for randomized algorithms against adaptive online adversaries, a 3-competitive algorithm for general metric spaces was proposed in [4] and the upper bound of 3 is also a lower bound on two points [11].…”