Routing in polygonal domains

Abstract

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex of P . Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.For any fixed ε > 0, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + ε. The labels have O(log n) bits, and the routing tables are of size O((ε −1 + h) log n). The preprocessing time is O(n 2 log n). It can be improved to O(n 2 ) for simple polygons.

Bounded-Degree Spanners in the Presence of Polygonal Obstacles

“…There has been extensive research into routing algorithms, including routing on the visibility graph in the presence of line segment obstacles [5,9,10]. For polygonal obstacles, Banyassady et al [1] developed a routing algorithm on the visibility graph with polygonal obstacles, though this work requires some additional information to be stored at the vertices.…”

Routing in Unit Disk Graphs without Dynamic Headers

“…Again, the number of bits for the labels is poly-logarithmic [21]. The same holds for visibility graphs of simple polygons [4]. Moreover, see [2] for different routing compact routing schemes in networks with low doubling dimension.…”

“…More precisely, Chan and Skrepetos show that given V , one can compute in O(n log n + (1/ε)n) time a decomposition tree T for DG(V ) with the following properties. 4 • Every node µ of T is assigned two sets: port(µ) ⊆ V (µ) ⊆ V . The subgraph of DG(V ) induced by V (µ) is connected and the vertices in port(µ) are called portals.…”

“…For notational ease, we restrict the explicit constants in our stretch bounds to powers of two 4. The reader familiar with the work of Chan and Skrepetos may notice that we have slightly extended the notion of portals: while Chan and Skrepetos define portals only for inner nodes, we also define portals for the leaves.…”

Routing in Histograms

“…We consider routing in a particularly interesting class of geometric graphs, namely visibility graphs of polygons. Banyassady et al [3] presented a routing scheme for polygonal domains with n vertices and h holes that uses O(log n) bits for the label, O((ε −1 + h) log n) bits for the routing tables, and achieves a stretch of 1 + ε, for any fixed ε > 0. However, their approach is efficient only if the edges of the visibility graph are weighted with their Euclidean lengths.…”