Brief Announcement: Papillon: Greedy Routing in Rings

We construct the first n-node degree-d ring-based network with worst-case greedy routes of length Θ(log n/ log d) hops. We study greedy routing over uni-dimensional metrics defined over n nodes lying in a ring. greedy routing in graph (V, E) with distance function δ : V × V → R entails the following decision: Given target node t, node u with neighbors N(u) forwards a message to v ∈ N(u) such that δ(v, t) = minx∈N(u) δ(x, t). The distance metric we use over n nodes placed in a circle is the clockwise-distance between pairs of nodes (the full paper contains a similar study of the absolute-distance metric): δclock(u, v) = v − u if v ≥ u, n + v − u otherwise. Let ∆clock denote the worst-case greedy route length with δclock. ∆clock denotes the average greedy route length over all-pairs. Summary of results. We construct a family of network topologies, the Papillon , in which greedy routes are asymptotically optimal. Papillon has greedy routes of length ∆clock = Θ(log n/ log d) hops in the worst-case when each node has d out-going links. Papillon is the first construction that achieves asymptotically optimal worst-case greedy routes. Upon further investigation, two properties of Papillon emerge: (a) greedy routing does not send messages along shortest paths, and (b) Edge congestion with greedy routing is not uniform – some edges are used more often than others. We exhibit the first property by identifying routing strategies that result in paths shorter than those achieved by greedy routing. In fact, one of these strategies guarantees uniform edge-congestion. The Papillon network. The intuitive idea of the Papillon construction is as follows. Nodes are arranged in m levels, where n = κm. A node at level i has links to nodes one level down (wrapping at zero). Level-i links cover nodes between 4 The principles of this work can be extended to higher dimensional spaces. We focus on one-dimension for simplicity. 5 Our constructions are variants of the well-known butterfly family, hence the name Papillon.