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.

翻译：在直线上在线设施分配问题 ${\rm OFAL}(S,c)$ 中，给定一个包含 $k$ 台服务器的集合 $S$ 以及容量函数 $c:S\to\mathbb{N}$，每台服务器 $s\in S$ 具有容量 $c(s)$ 并放置在直线上，请求按顺序逐个到达。在线算法的任务是在下一个请求到达前，将当前请求不可撤销地匹配到一台有空位的服务器。一台服务器 $s\in S$ 最多可匹配 $c(s)$ 个请求。本文针对 $\mathrm{OFAL}(S,c)$ 提出了一种新的在线算法 PTCP（临界点策略转换），并证明 PTCP 是 $(2\alpha(S)+1)$-竞争的，其中 $\alpha(S)$ 非正式地定义为 $S$ 的直径与 $S$ 中相邻服务器间最大距离之比。根据服务器的布局，$\alpha(S)$ 的取值范围从常数（与 $k$ 无关）到 $k-1$。在已知的 $\mathrm{OFAL}(S,c)$ 算法中，当 $\alpha(S)$ 较小时，该竞争比上界是最优的。我们还证明了任意 MPFS（最优先空闲服务器）算法的竞争比至少为 $2\alpha(S)+1$。对于 $\mathrm{OFAL}(S,c)$，MPFS 是一类竞争比不依赖于容量 $c$ 的算法，其中包括自然贪心算法和 PTCP 等。因此，这表明在 MPFS 类中，PTCP 是 $\mathrm{OFAL}(S,c)$ 问题的最优算法。