AI 中文总结
本研究分析随机在线欧几里得匹配中贪心算法的竞争比，证明除二维外均为常数，二维为$\Theta(\sqrt{\log n})$，通过环面分析转移至立方体。
AI 中文摘要
我们研究在线度量匹配中的贪心算法，其中$n$个服务器和$n$个请求独立且均匀地从$[0,1]^d$中采样。服务器初始可用，贪心算法将每个到达的请求不可撤销地匹配到其最近的可用服务器，产生它们之间距离的成本。我们证明，对于每个固定的$d\ne2$，贪心算法具有$O(1)$的竞争比，而对于$d=2$，竞争比为$\Theta(\sqrt{\log n})$。此前，对于$d=1$的情况已证明常数竞争性[BFP23]，而对于更高维度，此设置没有已知的非平凡结果。我们的证明首先在平坦环面上分析贪心算法，然后将估计结果转移回立方体。
英文摘要
We study Greedy for online metric matching with $n$ servers and $n$ requests sampled independently and uniformly from $[0,1]^d$. Servers are available initially, and Greedy irrevocably matches each arriving request to its closest available server, incurring a cost of their distance. We prove that Greedy has competitive ratio $O(1)$ for every fixed $d\ne2$, and $Θ(\sqrt{\log n})$ for $d=2$. Previously, constant competitiveness was shown for $d = 1$ [BFP23], and no non-trivial results for this setting were known for higher dimensions. Our proof first analyzes Greedy on the flat torus and then transfers the estimates back to the cube.
