We consider a single-facility location problem, where agents are positioned on the real line and are partitioned into multiple disjoint districts. The goal is to choose a location (where a public facility is to be built) so as to minimize the total distance of the agents from it. This process is distributed: the positions of the agents in each district are first aggregated into a representative location for the district, and then one of the district representatives is chosen as the facility location. This indirect access to the positions of the agents inevitably leads to inefficiency, which is captured by the notion of distortion. We study the discrete version of the problem, where the set of alternative locations is finite, as well as the continuous one, where every point of the line is an alternative, and paint an almost complete picture of the distortion landscape of both general and strategyproof distributed mechanisms.

翻译：我们考虑的是单一设施地点问题,即代理人位于实际线路上,并被分成多个不连接区;目标是选择一个地点(建造公共设施的地方),以便尽可能减少代理人与该地点之间的全部距离;这一过程是分散的:每个地区代理人的位置首先汇总到一个有代表性的地区,然后选择一个地区代表作为设施地点;这种间接接触代理人的位置不可避免地导致效率低下,这被扭曲的概念所掩盖;我们研究问题的离散版本,其中替代地点是有限的,以及连续版本,其中每一条线的每个点都是替代的,并描绘出一个几乎完整的一般和有战略防护的分布机制的扭曲面貌。