In this article, we present a construction of a spanner on a set of $n$ points in $\mathbf{R}^d$ that we call a heavy path WSPD spanner. The construction is parameterized by a constant $s > 2$ called the separation ratio. The size of the graph is $O(s^dn)$ and the spanning ratio is at most $1 + 2/s + 2/(s - 1)$. We also show that this graph has a hop spanning ratio of at most $2\lg n + 1$. We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex $v$ of the graph to store $O(\mathrm{deg}(v)\log n)$ bits of information, where $\mathrm{deg}(v)$ is the degree of $v$. The routing ratio is at most $1 + 4/s + 1/(s - 1)$ and at least $1 + 4/s$ in the worst case. The number of edges on the routing path is bounded by $2\lg n + 1$. We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension $\lambda$, the heavy path WSPD spanner has size $O(s^\lambda n)$ where $s$ is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case. Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most $1 + (2 + \frac{\tau}{\tau-1})/s + 1/(s - 1)$ in the worst case, where $\tau \ge 11$ is a constant related to the doubling dimension.

翻译：本文提出了一种在$\mathbf{R}^d$中$n$个点集上构造扳手图的方法，我们将其称为重型路径WSPD扳手。该构造由常数$s > 2$（称为分隔比）参数化。图的规模为$O(s^d n)$，展宽比至多为$1 + 2/s + 2/(s - 1)$。我们还证明该图的跳展宽比至多为$2\lg n + 1$。针对重型路径WSPD扳手，我们提出了一种无记忆的局部路由算法。该路由算法要求图的顶点$v$存储$O(\mathrm{deg}(v)\log n)$比特信息，其中$\mathrm{deg}(v)$为$v$的度数。路由比至多为$1 + 4/s + 1/(s - 1)$，在最坏情况下至少为$1 + 4/s$。路由路径上的边数被限定在$2\lg n + 1$以内。随后我们证明重型路径WSPD扳手可在有界倍增维度的度量空间中进行构造。这类度量空间作为欧氏空间的推广，已在计算几何领域得到广泛研究。我们证明，在倍增维度为$\lambda$的度量空间中，重型路径WSPD扳手的规模为$O(s^\lambda n)$，其中$s$为分隔比。其展宽比和跳展宽比与欧氏情形相同。最后，我们证明该局部路由算法在有界倍增维度情形下依然有效。顶点所需存储量不变，但路由比在最坏情况下变为至多$1 + (2 + \frac{\tau}{\tau-1})/s + 1/(s - 1)$，其中$\tau \ge 11$为与倍增维度相关的常数。