TY - GEN
T1 - Distributed strong diameter network decomposition
T2 - 35th ACM Symposium on Principles of Distributed Computing, PODC 2016
AU - Elkin, Michael
AU - Neiman, Ofer
N1 - Funding Information: Supported in part by ISF grant No. (523/12) and by the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no303809. Publisher Copyright: © 2016 ACM.
PY - 2016/7/25
Y1 - 2016/7/25
N2 - For a pair of positive parameters D,x, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,x) net- work decomposition, if the supergraph G(P), obtained by contracting each of the clusters of P, can be properly x- colored. The decomposition P is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diame- ter at most D, i.e., if for every cluster C ϵ P and every two vertices u; v ϵ C, the distance between them in the induced graph G(C) of C (resp., in G) is at most D. Network decomposition is a powerful construct, very use- ful in distributed computing and beyond. It was introduced by Awerbuch et. al. [AGLP89] in the end of the eighties. These authors showed that strong (2O(√log n log log n); 2O(√log n log log n)) network decompositions can be computed in 2O(√log n log log n) distributed time. Their result was improved at the beginning of nineties by Pan- conesi and Srinivasan [PS92], who showed that 2O(√log n log log n) in all the three expressions can be replaced by 2O(√log n). Around the same time Linial and Saks [LS93] devised an ingenious randomized algorithm that constructs weak (O(log n);O(log n)) network decompositions in O(log2 n) time. Awerbuch et. al. [ABCP96] devised a randomized al- gorithm that builds a strong (O(log n);O(log n)) network decomposition in O(log4 n) time, using very large messages and heavy local computations. It was however open till now if strong network decompositions with both parame-ters 2O(√log n) can be constructed in distributed 2O(√log n) time using short messages, or if a result of [LS93] can be strengthened to provide a strong (O(log n);O(log n)) net- work decomposition within O(log2 n) time (even using large messages). In this paper we answer these long-standing open ques- tions in the afirmative, and show that strong (O(log n);O(log n)) network decompositions can be computed in O(log2 n) time. We also present a tradeo between pa- rameters of our network decomposition. Our work is inspired by and relies on the \shifted shortest path approach", due to Blelloch et. al. [BGK+14], and Miller et. al. [MPX13]. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation.
AB - For a pair of positive parameters D,x, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,x) net- work decomposition, if the supergraph G(P), obtained by contracting each of the clusters of P, can be properly x- colored. The decomposition P is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diame- ter at most D, i.e., if for every cluster C ϵ P and every two vertices u; v ϵ C, the distance between them in the induced graph G(C) of C (resp., in G) is at most D. Network decomposition is a powerful construct, very use- ful in distributed computing and beyond. It was introduced by Awerbuch et. al. [AGLP89] in the end of the eighties. These authors showed that strong (2O(√log n log log n); 2O(√log n log log n)) network decompositions can be computed in 2O(√log n log log n) distributed time. Their result was improved at the beginning of nineties by Pan- conesi and Srinivasan [PS92], who showed that 2O(√log n log log n) in all the three expressions can be replaced by 2O(√log n). Around the same time Linial and Saks [LS93] devised an ingenious randomized algorithm that constructs weak (O(log n);O(log n)) network decompositions in O(log2 n) time. Awerbuch et. al. [ABCP96] devised a randomized al- gorithm that builds a strong (O(log n);O(log n)) network decomposition in O(log4 n) time, using very large messages and heavy local computations. It was however open till now if strong network decompositions with both parame-ters 2O(√log n) can be constructed in distributed 2O(√log n) time using short messages, or if a result of [LS93] can be strengthened to provide a strong (O(log n);O(log n)) net- work decomposition within O(log2 n) time (even using large messages). In this paper we answer these long-standing open ques- tions in the afirmative, and show that strong (O(log n);O(log n)) network decompositions can be computed in O(log2 n) time. We also present a tradeo between pa- rameters of our network decomposition. Our work is inspired by and relies on the \shifted shortest path approach", due to Blelloch et. al. [BGK+14], and Miller et. al. [MPX13]. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation.
KW - Distributed Model
KW - Network decompositions
UR - http://www.scopus.com/inward/record.url?scp=84984690287&partnerID=8YFLogxK
U2 - 10.1145/2933057.2933094
DO - 10.1145/2933057.2933094
M3 - Conference contribution
T3 - Proceedings of the Annual ACM Symposium on Principles of Distributed Computing
SP - 211
EP - 216
BT - PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing
Y2 - 25 July 2016 through 28 July 2016
ER -