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22
Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers
 Journal of Computer and System Sciences
, 1996
"... This paper presents a deterministic sorting algorithm, called Sharesort, that sorts n records on an nprocessor hypercube, shuffleexchange, or cubeconnected cycles in O(log n (log log n) 2 ) time in the worst case. The algorithm requires only a constant amount of storage at each processor. Th ..."
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Cited by 67 (10 self)
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This paper presents a deterministic sorting algorithm, called Sharesort, that sorts n records on an nprocessor hypercube, shuffleexchange, or cubeconnected cycles in O(log n (log log n) 2 ) time in the worst case. The algorithm requires only a constant amount of storage at each processor. The fastest previous deterministic algorithm for this problem was Batcher's bitonic sort, which runs in O(log 2 n) time. Supported by an NSERC postdoctoral fellowship, and DARPA contracts N0001487K825 and N00014 89J1988. 1 Introduction Given n records distributed uniformly over the n processors of some fixed interconnection network, the sorting problem is to route the record with the ith largest associated key to processor i, 0 i ! n. One of the earliest parallel sorting algorithms is Batcher's bitonic sort [3], which runs in O(log 2 n) time on the hypercube [10], shuffleexchange [17], and cubeconnected cycles [14]. More recently, Leighton [9] exhibited a boundeddegree,...
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 ..."
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Cited by 29 (3 self)
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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.
FaultTolerant Meshes with Small Degree
, 1993
"... This paper presents constructions for faulttolerant twodimensional mesh architectures. The constructions are designed to tolerate k faults while maintaining a healthy n by n mesh as a subgraph. They utilize several novel techniques for obtaining tradeoffs between the number of spare nodes and th ..."
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Cited by 17 (0 self)
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This paper presents constructions for faulttolerant twodimensional mesh architectures. The constructions are designed to tolerate k faults while maintaining a healthy n by n mesh as a subgraph. They utilize several novel techniques for obtaining tradeoffs between the number of spare nodes and the degree of the faulttolerant network. We consider both worstcase and random fault distributions. In terms of worstcase faults, we give a construction that has constant degree and O(k 3 ) spare nodes. This is the first construction known in which the degree is constant and the number of spare nodes is independent of n. In terms of random faults, we present several new degree6 and degree8 constructions and show (both analytically and through simulations) that they can tolerate large numbers of randomly placed faults. A preliminary version of this paper appeared in Proceedings of the Fifth Annual ACM Symposium on Parallel Algorithms and Architectures, 1993. y California Institute of...
Edge Isoperimetric Problems on Graphs
 Bolyai Math. Series
"... We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G ..."
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Cited by 17 (6 self)
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We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G (A) = f(u; v) 2 EG j u 2 A; v 62 Ag: We omit the subscript G if the graph is uniquely defined by the context. By edge isoperimetric problems we mean the problem of estimation of the maximum and minimum of the functions I and ` respectively, taken over all subsets of VG of the same cardinality. The subsets on which the extremal values of I (or `) are attained are called isoperimetric subsets. These problems are discrete analogies of some continuous problems, many of which can be found in the book of P'olya and Szego [99] devoted to continuous isoperimetric inequalities and their numerous applications. Although the continuous isoperimetric problems have a history of thousand years, the dis...
Parallelism and Locality in Priority Queues
 In Sixth IEEE Sypmposium on Parallel and Distributed Processing
, 1994
"... We explore two ways of incorporating parallelism into priority queues. The first is to speed up the execution of individual priority operations so that they can be performed one operation per time step, unlike sequential implementations which require O(log N ) time steps per operation for an N eleme ..."
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Cited by 15 (0 self)
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We explore two ways of incorporating parallelism into priority queues. The first is to speed up the execution of individual priority operations so that they can be performed one operation per time step, unlike sequential implementations which require O(log N ) time steps per operation for an N element heap. We give an optimal parallel implementation that uses a linear array of O(log N ) processors. Second, we consider parallel operations on the priority queue. We show that using a ddimensional array (constant d) of P processors we can insert or delete the smallest P elements from a heap in time O(P 1=d log 1\Gamma1=d P ), where the number of elements in the heap is assumed to be polynomial in P . We also show a matching lower bound, based on communication complexity arguments, for a range of deterministic implementations. Finally, using randomization, we show that the time can be reduced to the optimal O(P 1=d ) time with high probability. 1 Introduction Much of the theoret...
Minimum Linear Gossip Graphs and Maximal Linear (\Delta; k)Gossip Graphs
, 1994
"... Gossiping is an information dissemination problem in which each node of a communication network has a unique piece of information that must be transmitted to all other nodes using twoway communications between pairs of nodes along the communication links of the network. In this paper, we study goss ..."
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Cited by 14 (4 self)
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Gossiping is an information dissemination problem in which each node of a communication network has a unique piece of information that must be transmitted to all other nodes using twoway communications between pairs of nodes along the communication links of the network. In this paper, we study gossiping using a linear cost model of communication which includes a startup time and a propagation time which is proportional to the amount of information transmitted. A minimum linear gossip graph is a graph (modelling a network), with the minimum possible number of links, in which gossiping can be completed in minimum time under the linear cost model. For networks with an even number of nodes, we prove that the structure of minimum linear gossip graphs is independent of the relative values of the startup and unit propagation times. We prove that this is not true when the number of nodes is odd. We present four infinite families of minimum linear gossip graphs. We also present minimum linea...
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 ..."
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Cited by 14 (2 self)
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. 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...
A Hypercubic Sorting Network with Nearly Logarithmic Depth
 In Proceedings of the 24th Annual ACM Symposium on Theory of Computing
, 1992
"... A natural class of "hypercubic" sorting networks is defined. The regular structure of these sorting networks allows for elegant and efficient implementations on any of the socalled hypercubic networks (e.g., the hypercube, shuffleexchange, butterfly, and cubeconnected cycles). This class of so ..."
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Cited by 7 (4 self)
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A natural class of "hypercubic" sorting networks is defined. The regular structure of these sorting networks allows for elegant and efficient implementations on any of the socalled hypercubic networks (e.g., the hypercube, shuffleexchange, butterfly, and cubeconnected cycles). This class of sorting networks contains Batcher's O(lg 2 n)depth bitonic sort, but not the O(lg n)depth sorting network of Ajtai, Koml'os, and Szemer'edi. In fact, no o(lg 2 n) depth compareinterchange sort was previously known for any of the hypercubic networks. In this paper, we prove the existence of a family of 2 O( p lg lg n) lg ndepth hypercubic sorting networks. Note that this depth is o(lg 1+ffl n) for any constant ffl ? 0. 1 Introduction A comparator network is an ninput, noutput acyclic circuit made up of wires and 2input, 2output comparator gates. The input wires of the network are numbered from 0 to n \Gamma 1, as are the output wires. The inputs to the network may be tho...
On Bandwidth, Cutwidth, and Quotient Graphs
"... . The bandwidth and the cutwidth are fundamental parameters which can give indications on the complexity of many problems described in terms of graphs. In this paper, we present a method for finding general upper bounds for the bandwidth and the cutwidth of a given graph from those of any of its quo ..."
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Cited by 6 (2 self)
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. The bandwidth and the cutwidth are fundamental parameters which can give indications on the complexity of many problems described in terms of graphs. In this paper, we present a method for finding general upper bounds for the bandwidth and the cutwidth of a given graph from those of any of its quotient graphs. Moreover, general lower bounds are obtained by using vertexand edgebisection notions. These results are used, in a second time, to study various interconnection networks: by choosing convenient vertex partitions and judicious internal numberings for the vertices of the partition subsets, we show that bounds previously known for hypercubes can be easily reproven, and we give original bounds for 2Dmesh, binary de Bruijn, ShuffleExchange, FFT, Butterfly, and CCC graphs. 1 Introduction In all this paper, we will denote by V (G) and E(G) the vertex and edgesets of a nvertex graph G. When studying problems described in terms of graphs, it is often useful to have a good knowl...
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 sorting ..."
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Cited by 6 (5 self)
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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...