On triangulation of simple networks

Network triangulation is a method for estimating distances betweennodes in the network, by letting every node measure its distanceto a few beacon nodes, and deducing the distance between every two nodes x,y by using their measurements to their common beaconsand applying the triangle inequality. Kleinberg, Slivkins and Wexler [FOCS 2004] initiated a theoretical study of triangulation in metric spaces, and Slivkins [PODC 2005] subsequently showed that metrics of bounded doubling dimension admit a triangulation thatapproximates arbitrarily well all pairwise distances using only O(log n) beacons per point, where n is the number of points in the network. He then asked whether this term is necessary (for doubling metrics). We provide the first lower bounds on the number of beacons required for a triangulation in some specific simple networks. In particular, these bounds (i) answer Slivkins' question positively, even for one-dimensional metrics, and (ii) prove that, up to constants, Slivkins' triangulation achieves an optimal number of beacons (as a function of the approximation guarantee and the doubling dimension).