Bounded Stretch Homotopic Routing Using Hyperbolic Embedding of Sensor Networks

In this paper we consider lightweight routing in a wireless sensor network deployed in a complex geometric domain Σ with holes. Our goal is to find short paths of different homotopy types, i.e., paths that go around holes in different ways. In the example of Figure 1, there are three holes in the network and there are many different ways to “thread” a route from s to t. Observe that paths α, β, γ are all different in a global sense; in that, e.g., one cannot deform α to β without “lifting” it over some hole. In contrast, paths γ and δ are only different in a local manner; one can deform γ to δ continuously through local changes, keeping δ within the domain. This difference is characterized by the homotopy type of a path. Two paths in a Euclidean domain are homotopy equivalent if one can continuously deform one to the other. A set of paths that are pairwise homotopy equivalent are said to have the same homotopy type. The number of homotopy types is infinitely many (assuming there is at least one hole), as a path can loop around a hole k times, for any integer k; however, for most routing scenarios we only care about a finite number of homotopy types, corresponding to paths in the dual graph of a triangulation of Σ that do not repeat triangles (and thus do not loop around holes).