基于中心点的次线性空间距离标注
Sublinear-Space Distance Labeling using Hubs
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。我们提出了一系列新的标签计划的框架内，所谓的枢纽标签（HL，也被称为地标标签或2跳覆盖标签），其中每个节点$u$存储其距离的所有节点从一个适当选择的一组枢纽$S(u) \subseteq V$。对于被查询的节点对$(u,v)$，从$S(u)\cap S(v)$通过中心节点的最短$u-v$路径的长度然后被用作$u$和$v$之间的距离的上限.我们提出了一个枢纽标签，使我们能够解码精确的距离稀疏图使用标签的大小次线性的节点数。对于具有最多$n$节点和平均度$Δ$的图，对于任何$T \leq n$，我们的方法的标签位大小$L$和查询解码时间$T$之间的权衡由$L = O(n \log \log_ΔT / \log_ΔT)$给出。因此，我们的简单方法是第一个子线性空间距离标记稀疏图，同时承认小的解码时间（对于恒定的$Δ$，我们可以实现任何$T=ω(1)$，同时保持$L=o(n)$），它也提供了一个改进的标签大小相对于以前较慢的方法。通过使用类似的技术，我们然后提出了一般图的$2$ -加性标记方案，即，其中解码器提供任意一对节点之间的距离的2-加法近似。对于任何$T \leq n$，我们实现了几乎相同的标签大小-时间权衡$L = O(n \log^2 \log T / \log T)$。据我们所知，这是第一个使用次线性大小标签的具有恒定绝对误差的加法方案。相应的解码时间则很小（任何$T=ω(1)$都足够）。
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. We propose a series of new labeling schemes within the framework of so-called hub labeling (HL, also known as landmark labeling or 2-hop-cover labeling), in which each node $u$ stores its distance to all nodes from an appropriately chosen set of hubs $S(u) \subseteq V$. For a queried pair of nodes $(u,v)$, the length of a shortest $u-v$-path passing through a hub node from $S(u)\cap S(v)$ is then used as an upper bound on the distance between $u$ and $v$. We present a hub labeling which allows us to decode exact distances in sparse graphs using labels of size sublinear in the number of nodes. For graphs with at most $n$ nodes and average degree $Δ$, the tradeoff between label bit size $L$ and query decoding time $T$ for our approach is given by $L = O(n \log \log_ΔT / \log_ΔT)$, for any $T \leq n$. Our simple approach is thus the first sublinear-space distance labeling for sparse graphs that simultaneously admits small decoding time (for constant $Δ$, we can achieve any $T=ω(1)$ while maintaining $L=o(n)$), and it also provides an improvement in terms of label size with respect to previous slower approaches. By using similar techniques, we then present a $2$-additive labeling scheme for general graphs, i.e., one in which the decoder provides a 2-additive-approximation of the distance between any pair of nodes. We achieve almost the same label size-time tradeoff $L = O(n \log^2 \log T / \log T)$, for any $T \leq n$. To our knowledge, this is the first additive scheme with constant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any $T=ω(1)$ is sufficient).
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