Routing in Strongly Hyperbolic Unit Disk Graphs

Greedy routing has been studied successfully on Euclidean unit disk graphs, which we interpret as a special case of hyperbolic unit disk graphs. While sparse Euclidean unit disk graphs exhibit grid-like structure, we introduce strongly hyperbolic unit disk graphs as the natural counterpart containing graphs that have hierarchical network structures. We develop and analyze a routing scheme that utilizes these hierarchies. On arbitrary graphs this scheme guarantees a worst case stretch of $\max\{3, 1+2b/a\}$ for $a > 0$ and $b > 1$. Moreover, it stores $\mathcal{O}(k(\log^2{n} + \log{k}))$ bits at each vertex and takes $\mathcal{O}(k)$ time for a routing decision, where $k = \pi e (1 + a)/(2(b - 1)) (b^2 \text{diam}(G) - 1) R + \log_b(\text{diam}(G)) + 1$, on strongly hyperbolic unit disk graphs with threshold radius $R > 0$. In particular, for hyperbolic random graphs, which have previously been used to model hierarchical networks like the internet, $k = \mathcal{O}(\log^2{n})$ holds asymptotically almost surely. Thus, we obtain a worst-case stretch of $3$, $\mathcal{O}(\log^4 n)$ bits of storage per vertex, and $\mathcal{O}(\log^2 n)$ time per routing decision on such networks. This beats existing worst-case lower bounds. Our proof of concept implementation indicates that the obtained results translate well to real-world networks.