Greedy Routing with Bounded Stretch

Abstract

Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomial-time algorithm that embeds combinatorial unit disk graphs (CUDGs -a CUDG is a UDG without any geometric information) into O(log 2 n)-dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated Lipton-Tarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)-stretch greedy embedding into O(log 2 n)-dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constant-stretch compact routing scheme in which each node is assigned an O(log 3 n)-bit label. To the best of our knowledge, this result yields the best known stretch-space trade-off for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10%; only few routes have a stretch above 1.5.

Efficient Geometric Routing in Large-Scale Complex Networks with Low-Cost Node Design

“…In [14] authors proposed another embedding in the hyperbolic plane which requires O(log n) bits (with n being the network size). [10] proposed an embedding in a Euclidean O(log 2 n)-dimensional space and a poly-logarithmic dimension embedding into Euclidean space was proposed in [11].…”

“…A greedy embedding for a given graph G(V, E) into a metric space X, is a function from V(G) to X such that for all graph nodes s t, s has a neighbor u which decreases the distance toward t in metric space X. Several works proposed greedy embeddings in different metric spaces such as hyperbolic plane [8], [9] and multidimensional Euclidean spaces [10], [11]. Greedy forwarding based on these embeddings never gives rise to local minima and thus 100% successful delivery of the packets is guaranteed.…”

“…In this embedding, instead of relying on complex geometry, network nodes coordinates are derived from a spanning tree of the network. This decreases the overall overhead/complexity of the scheme compared to approaches based on hyperbolic [8], [9] or high-dimensional Euclidean spaces [10], [11]. The less complex the approach, the more likely it would be used in practice.…”

Greedy Zone Routing: Robust and Scalable Routing in Wireless Ad-hoc Networks

“…There are several attempts to overcome this by introducing multiple trees embedded instead of a single spanning tree [15], [5]. This suggests a node to hold multiple coordinates, corresponding to different spanning trees.…”

“…Nevertheless, neither of the two proposals do not provide guarantees on routing stretch. In contrast [15], propose the use of multiple tree embeddings, where a constant factor stretch bound is proven for combinatorial unit disk graphs by computing a structure of trees which covers the graph with constant stretch. This is the only case for which a stretch bound is known.…”