Online Facility Assignment for General Layout of Servers on a Line

Abstract

In the online facility assignment on a line $\textrm{OFAL}(S,c)$ with a set S of k servers and a capacity $c:S\rightarrow \mathbb {N}$, each server $s\in S$ with a capacity c(s) is placed on a line, and a request arrives on a line one-by-one. The task of an online algorithm is to irrevocably match a current request with one of the servers with vacancies before the next request arrives. An algorithm can match up to c(s) requests to a server $s\in S$. In this paper, we propose a new online algorithm PTCP (Policy Transition at Critical Point) for OFAL(S, c) and show that PTCP is $(2\alpha (S)+1)$-competitive, where $\alpha (S)$ is informally the ratio of the diameter of S to the maximum distance between two adjacent servers in S. Depending on the layout of servers, $\alpha (S)$ ranges from O(1) to O(k). Among all of known algorithms for OFAL(S, c), this upper bound on the competitive ratio is the best when $\alpha (S)$ is small. We also show that the competitive ratio of any MPFS (Most Preferred Free Servers) algorithm [6] is at least $2\alpha (S)+1$, where MPFS is a class of algorithms whose competitive ratio does not depend on a capacity c. Recall that the class MPFS includes the natural greedy algorithm and PTCP, etc. Thus, this implies that PTCP is the best for OFAL(S, c) in the class MPFS.