A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential) bound are essentially optimized to perform well on such worst case graphs. However, such graphs are often either unlikely or actively avoided in practice, where benign graph instances usually admit much faster solutions. To circumnavigate these drawbacks, the concept of universal complexity analysis in the distributed setting was suggested by [Kutten and Peleg, PODC'95] and actively pursued by [Haeupler et al., STOC'21]. Here, the aim is to gauge the complexity of a distributed graph problem depending on the given graph instance. The challenge is to identify and understand the graph property that allows to accurately quantify the complexity of a distributed problem on a given graph. In the present work, we consider distributed shortest paths problems in the HYBRID model of distributed computing, where nodes have simultaneous access to two different modes of communication: one is restricted by locality and the other is restricted by congestion. We identify the graph parameter of neighborhood quality and show that it accurately describes a universal bound for the complexity of certain class of shortest paths problems in the HYBRID model.

翻译：经典分布式图问题复杂度分析方法的一个缺陷在于，其大多仅揭示所谓“最坏情况”图类别的复杂度。用于证明紧（存在性）边界的算法本质上针对此类最坏情况图进行了优化。然而，这类图在实践中往往要么出现概率极低，要么被主动规避——良性图实例通常能够支持更快的求解方案。为克服这些缺陷，[Kutten和Peleg，PODC'95]提出了分布式场景下的普遍复杂度分析概念，并由[Haeupler等人，STOC'21]持续推进。该方法的核心理念在于，依据给定图实例来度量分布式图问题的复杂度。其挑战在于识别并理解能够精准量化给定图上分布式问题复杂度的图属性。在本文中，我们考虑HYBRID分布式计算模型下的最短路径问题——该模型中节点可同时使用两种不同通信模式：一种受限于局部性，另一种受限于拥塞性。我们定义了邻域质量的图参数，并证明其能够精确描述HYBRID模型中某类最短路径问题的通用复杂度边界。