Optimal path search in small worlds: dimension matters

We consider Kleinberg's celebrated small-world model (2000). This model is based on a d-dimensional grid graph of n nodes, augmented by a constant number of "long-range links" per node. It is known that this graph has diameter O(log n), and that a simple greedy search algorithm visits an expected number of O(log2 n) nodes, which is asymptotically optimal over all decentralized search algorithms. Besides the number of nodes visited, a relevant measure is the length of the path constructed by the search algorithm. A decentralized algorithm by Lebhar and Schabanel (2003) constructs paths of expected length O(log n (loglog n)2) by visiting the same number of nodes as greedy search. A natural question, posed by Kleinberg (2006), is whether there are decentralized algorithms that construct paths of length O(log n) while visiting only a poly-logarithmic number of nodes. In this paper we resolve this question. For grid dimension d=1, we answer the question in the negative, by showing that any decentralized algorithm that visits a poly-logarithmic number of nodes constructs paths of expected length O(log n loglog n). Further we show that this bound is tight; a simple variant of the algorithm by Lebhar and Schabanel matches this bound. For dimension de2, however, we answer the question in the affirmative; the bound is achieved by essentially the same algorithm we used for d=1. This is the first time that such a dichotomy, based on the dimension d, has been observed for an aspect of this model. Our results may be applicable to the design of peer-to-peer networks, where the length of the path along which data are transferred is critical for the network's performance.