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We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected. We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication.In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and s − t cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree, shortest paths, and minimum cut. Many of these results are the first non-trivial lower bounds for both exact and approximate distributed computation and they resolve previous open questions. Moreover, our unconditional lower bound of approximating minimum spanning tree (MST) subsumes and improves upon the previous hardness of approximation bound of Elkin [STOC 2004] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [FOCS 1999]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm, for any approximation factor.Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.

A radio network is a synchronous network of processors that communicate by transmitting messages to their neighbors, where a processor receives a message in a given step if and only if it is silent in this step and precisely one of its neighbors transmits. In this paper we prove the existence of a family of radius-2 networks on n vertices for which any broadcast schedule requires at least sZ(log* n) rounds of transmissions. This matches an upper bound of O(log* n) rounds for networks of radius 2 proved earlier by Bar-Yehuda, Goldreich, and Itai, in "Proceedings of the 4th ACM Symposium on Principles of Distributed Computing, 1986," pp. 98-107.

Given a graph G = (V E), a subgraph G' = (V E ' ) is a t-spanner of G if for every u, u E V the distance from u to u in G' is at most t times longer than that distance in G. This paper presents some results concerning the existence and efficient constructability of sparse spanners for various classes of graphs, including general undirected graphs, undirected chordal graphs, and general directed graphs.