Competitive Analysis of Online Facility Assignment for General Layout of Servers on a Line

TL;DR

This paper introduces a new online algorithm PTCP for facility assignment on a line, achieving the best known competitive ratio for layouts with small diameter-to-adjacency ratios, and establishes its optimality within a certain class.

Contribution

The paper proposes the PTCP algorithm for OFAL, providing a tight competitive ratio bound and demonstrating its optimality among MPFS algorithms for various server layouts.

Findings

PTCP has a competitive ratio of (2α(S)+1).

The ratio is optimal within the MPFS class.

Performance varies with server layout, from constant to linear in k.

Abstract

In the online facility assignment on a line ${\rm OFAL}(S,c)$ with a set $S$ of $k$ servers and a capacity $c:S\to\mathbb{N}$, 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 $\mathrm{OFAL}(S,c)$ 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 constant (independent of $k$) to $k-1$. Among all of known algorithms for $\mathrm{OFAL}(S,c)$, 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 is at least $2\alpha(S)+1$. For $\mathrm{OFAL}(S,c)$, recall that MPFS is a class of algorithms whose competitive ratio does not depend on a capacity $c$ and it includes the natural greedy algorithm and PTCP, etc. Thus, this implies that PTCP is the best for $\mathrm{OFAL}(S,c)$ in the class MPFS.