单调图类中的距离概要（Sketching）
Sketching Distances in Monotone Graph Classes
AI总结：
该文研究单调图类中距离判定的通信问题，刻画了不同距离概要存在性与图类结构参数的等价或蕴含关系，明确了有界膨胀等性质对常数大小距离概要的关键作用。
摘要：
我们研究判定图$G$中两个顶点$x,y$是否邻近的双人通信问题，目标是确定哪些图结构允许用常数代价的随机协议求解该问题。等价地，我们考虑为图的顶点分配常数大小随机标签（即sketch，概要）的问题，这些标签能以高概率计算出邻接关系、精确距离阈值或近似距离阈值。\n我们的主要结果是，对于单调图类：常数大小邻接概要存在当且仅当该类有有界树厚（arboricity）；精确距离阈值的常数大小概要存在当且仅当该类有有界膨胀（bounded expansion）；常数大小近似距离阈值（ADT）概要意味着该类有有界膨胀；任何常数膨胀的类（即任何真闭 minor 类）都有常数大小ADT概要；一个类可以有任意小的膨胀却不具备常数大小ADT概要。
英文摘要：
We study the two-player communication problem of determining whether two vertices $x, y$ are nearby in a graph $G$, with the goal of determining the graph structures that allow the problem to be solved with a constant-cost randomized protocol. Equivalently, we consider the problem of assigning constant-size random labels (sketches) to the vertices of a graph, which allow adjacency, exact distance thresholds, or approximate distance thresholds to be computed with high probability from the labels. Our main results are that, for monotone classes of graphs: constant-size adjacency sketches exist if and only if the class has bounded arboricity; constant-size sketches for exact distance thresholds exist if and only if the class has bounded expansion; constant-size approximate distance threshold (ADT) sketches imply that the class has bounded expansion; any class of constant expansion (i.e. any proper minor closed class) has constant-size ADT sketches; and a class may have arbitrarily small expansion without admitting constant-size ADT sketches.
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