Space lower bounds for low-stretch greedy embeddings

Greedy (geometric) routing is an important paradigm for routing in communication networks. It uses an embedding of the nodes of a network into points of a space (e.g., RdRd) equipped with a distance function (e.g., the Euclidean distance ℓ2ℓ2) and uses as address of each node the coordinates of the corresponding point. The embedding has particular properties so that the routing decision for a packet is taken greedily based only on the addresses of the current node, its neighbors, and the destination node and the distances of the corresponding points. In this way, there is no need to keep extensive routing information (e.g., routing tables) at the nodes. Embeddings that allow for this functionality are called greedy embeddings. Even though greedy embeddings do exist for several spaces and distance functions, they usually result in paths of high stretch, i.e., significantly longer than the corresponding shortest paths.In this paper, we show that greedy embeddings in low-dimensional (Euclidean) spaces necessarily have high stretch. In particular, greedy embeddings of n  -node graphs with optimal stretch require at least Ω(n)Ω(n) dimensions for distance ℓ2ℓ2. This result disproves a conjecture by Maymounkov (2006) [14] stating that greedy embeddings of optimal stretch are possible in Euclidean spaces with polylogarithmic number of dimensions. Furthermore, we present trade-offs between the stretch and the number of dimensions of the host space. Our results imply that every greedy embedding into a space with polylogarithmic number of dimensions (and Euclidean distance) has stretch Ω(log⁡nlog⁡log⁡n). We extend this result for a distance function used by an O(log⁡n)O(log⁡n)-stretch greedy embedding presented by Flury et al. (2009) [9]. Our lower bound implies that their embedding has almost best possible stretch.