稀疏图中通过Hub标号精确计算距离的困难性
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
摘要：
距离标记方案是将位标记分配给无向、无权图的顶点，使得任何顶点对之间的距离可以仅从它们的标记解码。一类重要的距离标记方案是枢纽标记方案，其中节点$v \in G$存储其到所谓的枢纽$S_v \subseteq V$的距离，选择该距离使得对于任何$u,v \in V$，存在属于某个最短$uv$路径的$w \in S_u \cap S_v$。注意，对于大多数现有的图类，现有的最佳距离标记构造在某点至少使用枢纽标记方案作为关键构建块。我们的兴趣在于稀疏图的中心标记，即，具有$|E(G)| = O(n)$的那些，对于这些，我们示出了轮毂组的平均尺寸的$\frac{n}{2^{O(\sqrt{\log n})}}$的下限。此外，我们展示了一个枢纽标记的建设稀疏图的平均大小$O(\frac{n}{RS(n)^{c}})$的一些$0 < c < 1$，其中$RS(n)$是所谓的Ruzsa-Szemer{é}di函数，连接到结构的诱导匹配稠密图。这意味着进一步将管座标记尺寸的下限提高到$\frac{n}{2^{(\log n)^{o(1)}}}$将需要在$RS(n)$下限研究中取得突破，在过去70年中，$RS(n)$的下限一直抵制实质性改进。对于一般的距离标记的稀疏图，我们证明了一个下界的$\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$，其中$SumIndex(n)$是通信复杂性的总和指数问题$Z_n$。我们的研究结果表明，最好的可实现的枢纽标签的大小和距离标签的大小在稀疏图可能是$Θ(\frac{n}{2^{(\log n)^c}})$的一些$0<c < 1$。
英文摘要：
A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{é}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $Θ(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$.
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