Results 1 
8 of
8
SmallDepth Counting Networks
, 1992
"... Generalizing the notion of a sorting network, Aspnes, Herlihy, and Shavit recently introduced a class of socalled "counting" networks, and established an O(lg 2 n) upper bound on the depth complexity of such networks. Their work was motivated by a number of practical applications arising ..."
Abstract

Cited by 43 (2 self)
 Add to MetaCart
Generalizing the notion of a sorting network, Aspnes, Herlihy, and Shavit recently introduced a class of socalled "counting" networks, and established an O(lg 2 n) upper bound on the depth complexity of such networks. Their work was motivated by a number of practical applications arising in the domain of asynchronous shared memory machines. This paper continues the analysis of counting networks, providing a number of new upper bounds. In particular, we present an explicit construction of an O(c lg* lg n) depth counting network, a randomized construction of an O(lgn)depth network (that works with extremely high probability), and using the random con struction we present an existential proof of a de terministic O(lgn)depth network. The latter result matches the trivial (lgn)depth lower bound to within a constant factor.
Packet Routing In FixedConnection Networks: A Survey
, 1998
"... We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing ..."
Abstract

Cited by 34 (3 self)
 Add to MetaCart
We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.
Hypercubic Sorting Networks
 SIAM J. Comput
, 1998
"... . This paper provides an analysis of a natural dround tournamentover n = 2 d players, and demonstrates that the tournament possesses a surprisingly strong ranking property. The ranking property of this tournament is used to design efficient sorting algorithms for a variety of different models of ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
. This paper provides an analysis of a natural dround tournamentover n = 2 d players, and demonstrates that the tournament possesses a surprisingly strong ranking property. The ranking property of this tournament is used to design efficient sorting algorithms for a variety of different models of parallel computation: (i) a comparator network of depth c \Delta lg n, c 7:44, that sorts the vast majority of the n! possible input permutations, (ii) an O(lg n)depth hypercubic comparator network that sorts the vast majority of permutations, (iii) a hypercubic sorting network with nearly logarithmic depth, (iv) an O(lgn)time randomized sorting algorithm for any hypercubic machine (other such algorithms have been previously discovered, but this algorithm has a significantly smaller failure probability than any previously known algorithm), and (v) a randomized algorithm for sorting n O(m)bit records on an (n lg n)node omega machine in O(m + lg n) bit steps. Key words. parallel sort...
SmallDepth Counting Networks and Related Topics
, 1994
"... In [5], Aspnes, Herlihy, and Shavit generalized the notion of a sorting network by introducing a class of so called "counting" networks and establishing an O(lg 2 n) upper bound on the depth complexity of such networks. Their work was motivated by a number of practical applications arising ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
In [5], Aspnes, Herlihy, and Shavit generalized the notion of a sorting network by introducing a class of so called "counting" networks and establishing an O(lg 2 n) upper bound on the depth complexity of such networks. Their work was motivated by a number of practical applications arising in the domain of asynchronous shared llemory machines. In this thesis, we continue the analysis of counting networks and produce a number of new upper bounds on their depths. Our results are predicated on the rich combinatorial structure which counting networks possess. In particular, we present a simple explicit construction of an O(lg n lg lg n)depth counting network, a randomized construction of an O(lg n)depth network (which works with extremely high probability), and we present an existential proof of a deterministic O(lg n)depth network. The latter result matches the trivial ((lg n)depth lower bound to within a constant factor. Our main result is a uniform polynomialtime construction of an O(lg n)depth counting network which depends heavily on the existential result, but makes use of extractor functions introduced in [25]. Using the extractor, we construct regular high degree hipattire graphs with extremely strong expansion properties. We believe this result is of independent interest.
A Lower Bound for Sorting Networks Based on the Shuffle Permutation
 Mathematical Systems Theory
, 1994
"... We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of ninput sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)depth bitonic sor ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of ninput sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest. 1 Introduction A variety of different classes of sorting networks has been described in the literature. Of particular interest here are the socalled AKS network [1] discovered by Ajtai, Koml'os and Szemer'edi, and the sorting networks proposed by Batcher [2]. The AKS network is the only known sorting network with O(lg n) depth. However, the topology of the network is highly irregular, and the multiplicative constant hidden by the Onotation is impractically large [1, 11]. On the other hand, the networks proposed by Batcher have a relatively simple interconnection structure and a small constant...
A SuperLogarithmic Lower Bound for Hypercubic Sorting Networks
 in Proceedings of the 21st International Colloquium on Automata, Languages, and Programming
, 1994
"... Hypercubic sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput hypercubic sorting networks with depth 2 O( p lg lg n) lg n have been discovered. These networks are the only known sorti ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Hypercubic sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput hypercubic sorting networks with depth 2 O( p lg lg n) lg n have been discovered. These networks are the only known sorting networks of depth o(lg 2 n) that are not based on expanders, and their existence raises the question of whether a depth of O(lg n) can be achieved by any hypercubic sorting network. In this paper, we resolve this question by establishing an\Omega \Gamma lg n lg lg n lg lg lg n \Delta lower bound on the depth of any ninput hypercubic sorting network. Our lower bound can be extended to certain restricted classes of nonoblivious sorting algorithms on hypercubic machines. 1 Introduction A variety of different classes of sorting networks have been described in the literature. Of particular interest here are the socalled AKS network [1] discovered by Ajtai, Koml'os, and Szemer...
A Lower Bound for Sorting Networks Based on the Shue Permutation
"... We prove an (lg 2 n = lg lg n) lower bound for the depth of ninput sorting networks based on the shue permutation. The best previously known lower bound was the trivial (lgn) bound, while the best upper bound is given by Batcher's (lg 2 n)depth bitonic sorting network. The proof technique emp ..."
Abstract
 Add to MetaCart
(Show Context)
We prove an (lg 2 n = lg lg n) lower bound for the depth of ninput sorting networks based on the shue permutation. The best previously known lower bound was the trivial (lgn) bound, while the best upper bound is given by Batcher's (lg 2 n)depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest. 1