Online Routing in Triangulations

Abstract

We consider online routing algorithms for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms, one that works for all Delaunay triangulations and the other that works for all regular triangulations; (2) a randomized memoryless algorithm that works for all triangulations; (3) an O(1) memory algorithm that works for all convex subdivisions; (4) an O(1) memory algorithm that approximates the shortest path in Delaunay triangulations; and (5) theoretical and experimental results on the competitiveness of these algorithms.

Improvement of the robustness on geographical networks by adding shortcuts

“…Something of randomness [16,17,18,15] is expected to relax the geographical constraints that tend to make cycles locally. Since many real complex systems belong to SF networks [1,2] and are embedded in a metric space [8,9], in addition, planar networks are suitable for efficient routings [22], we consider a family of planar SF network models called RA [12,11], DT [ 20,21], and DLSF [15], a non-planar basic geographical model called LESF [10], and a real data of the Internet at the AS level [30] as an example for the virtual examination. Our numerical results show that the robustness is improved by shortcuts around 10% rate maintaining the small distance D and number of hops L on the optimal paths (with respect to the shortest and the minimum number of hops, respectively) in each network, under similar degree distributions to the original ones.…”

“…In computer science, online routing algorithms [22] that guarantee delivery of messages using only local information about positions of the source, terminal, and the adjacent nodes to a current node are well-known. As a connection to SF networks, we consider Delaunay triangulation (DT) and random Apollonian (RA) network models based on planar triangulation of a polygonal region.…”

“…For adding shortcuts, the efficient routing algorithm [22] on a planar network can be extended as follows (see Figure 9) in the ellipsoid whose chord is defined by the distance of the shortest path l s on the edges of faces that intersect the straight line between the source and terminal as the two focuses. We describe the outline of procedures .…”

On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

“…6 } in the plane such that circle C i and C j touch each other iff edge {v i , v j } ∈ E. By Koebe's Representation Theorem [13] such a circle packing is guaranteed to exist for every planar graph.…”

“…While this scheme is simple and efficient, it does not guarantee message delivery. This is because a routed message may become trapped in a cycle [6] or it may reach a local minimum; this is a node that is closer to the destination than any of its neighbors [11].…”