Graph embedding is a fundamental problem of mapping nodes of a guest graph into a host graph while minimizing the distance distortion, with broad applications, including virtual network embeddings into physical topologies, VLSI design, or community detection in social networks. However, in many real-world applications the guest graph changes over time and the embedding can adapt to these changes (e.g. virtual machine migration in network embeddings). Static embeddings are inherently inefficient in comparison to adaptive embeddings, but it remains an unresolved algorithmic challenge to design efficient embedding algorithms that adapt to the demand on-the-fly, i.e., that are online. In this paper, we derive optimal deterministic and randomized online algorithms for the online graph embedding problem in star host graphs. This is an essential building block on the way to design algorithms for more complex host graphs, representing a single node and its neighborhood. We start by presenting a $1.5$-competitive deterministic algorithm and showing that no deterministic algorithm can perform better. Our main contribution is a randomized algorithm that achieves a significantly better competitive ratio of $11/9 \approx 1.222$. Both the deterministic and the randomized algorithms are optimal, which we prove by deriving tight lower bounds for the competitiveness of any algorithm.

翻译：图嵌入是一个基本问题，旨在将客户图的节点映射到宿主图中，同时最小化距离失真，其在虚拟网络嵌入物理拓扑、超大规模集成电路设计或社交网络社区检测等领域有广泛应用。然而，在许多实际应用中，客户图随时间变化，嵌入方式需适应这些变化（例如，网络嵌入中的虚拟机迁移）。静态嵌入本质上不如自适应嵌入高效，但设计能实时适应需求（即在线）的高效嵌入算法仍是一个未解决的计算挑战。本文针对星形宿主图中的在线图嵌入问题，推导出了最优的确定性与随机在线算法。这是为更复杂宿主图（代表单个节点及其邻域）设计算法的基础模块。我们首先提出一种竞争比为1.5的确定性算法，并证明任何确定性算法都无法表现更优。我们的主要贡献在于一种随机算法，其竞争比显著提升至11/9（约1.222）。我们通过导出任何算法竞争比的下界，证明确定性和随机算法均为最优。