  Distance computation is one of the most fundamental primitives used in
communication networks. The cost of effectively and accurately computing
pairwise network distances can become prohibitive in large-scale networks such
as the Internet and Peer-to-Peer (P2P) networks. To negotiate the rising need
for very efficient distance computation, approximation techniques for numerous
variants of this question have recently received significant attention in the
literature. The goal is to preprocess the graph and store a small amount of
information such that whenever a query for any pairwise distance is issued, the
distance can be well approximated (i.e., with small stretch) very quickly in an
online fashion. Specifically, the pre-processing (usually) involves storing a
small sketch with each node, such that at query time only the sketches of the
concerned nodes need to be looked up to compute the approximate distance. In
this paper, we present the first theoretical study of distance sketches derived
from distance oracles in a distributed network. We first present a fast
distributed algorithm for computing approximate distance sketches, based on a
distributed implementation of the distance oracle scheme of [Thorup-Zwick, JACM
2005]. We also show how to modify this basic construction to achieve different
tradeoffs between the number of pairs for which the distance estimate is
accurate and other parameters. These tradeoffs can then be combined to give an
efficient construction of small sketches with provable average-case as well as
worst-case performance. Our algorithms use only small-sized messages and hence
are suitable for bandwidth-constrained networks, and can be used in various
networking applications such as topology discovery and construction, token
management, load balancing, monitoring overlays, and several other problems in
distributed algorithms.
