Routing in Unit Disk Graphs without Dynamic Headers
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Let V⊂ℝ^2 be a set of n sites in the plane. The unit disk graph DG(V) of V is the graph with vertex set V in which two sites v and w are adjacent if and only if their Euclidean distance is at most 1. We develop a compact routing scheme for DG(V). The routing scheme preprocesses DG(V) by assigning a label l(v) to every site v in V. After that, for any two sites s and t, the scheme must be able to route a packet from s to t as follows: given the label of a current vertex r (initially, r=s) and the label of the target vertex t, the scheme determines a neighbor r' of r. Then, the packet is forwarded to r', and the process continues until the packet reaches its desired target t. The resulting path between the source s and the target t is called the routing path of s and t. The stretch of the routing scheme is the maximum ratio of the total Euclidean length of the routing path and of the shortest path in DG(V), between any two sites s, t ∈ V. We show that for any given ε>0, we can construct a routing scheme for DG(V) with diameter D that achieves stretch 1+ε and label size O(log Dlog^3n/loglog n) (the constant in the O-Notation depends on ε). In the past, several routing schemes for unit disk graphs have been proposed. Our scheme is the first one to achieve poly-logarithmic label size and arbitrarily small stretch without storing any additional information in the packet.
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