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12
Online algorithms for path selection in a nonblocking network
 SIAM Journal on Computing
, 1996
"... This paper presents the first optimaltime algorithms for path selection in an optimalsize nonblocking network. In particular, we describe an Ninput, Noutput, nonblocking network with O(N log N) boundeddegree nodes, and an algorithm that can satisfy any request for a connection or disconnection ..."
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Cited by 67 (14 self)
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This paper presents the first optimaltime algorithms for path selection in an optimalsize nonblocking network. In particular, we describe an Ninput, Noutput, nonblocking network with O(N log N) boundeddegree nodes, and an algorithm that can satisfy any request for a connection or disconnection between an input and an output in O(log N) bit steps, even if many requests are made at once. Viewed in a telephone switching context, the algorithm can put through any set of calls among N parties in O(log N) bit steps, even if many calls are placed simultaneously. Parties can hang up and call again whenever they like; every call is still put through O(log N) bit steps after being placed. Viewed in a distributed memory machine context, our algorithm allows any processor to access any idle block of memory within O(log N) bit steps, no matter what other connections have been made previously or are being made simultaneously.
Generalized connection networks for parallel pro. cessor intercommunication
 IEEE Trans. Comput
, 1978
"... AbstractA generalized connection network (GCN) is a switching network with N inputs and N outputs that can be set to pass any of the NN mappings of inputs onto outputs. This paper demonstrates an intimate connection between the problems of GCN construction, message routing on SIMD computers, and &q ..."
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AbstractA generalized connection network (GCN) is a switching network with N inputs and N outputs that can be set to pass any of the NN mappings of inputs onto outputs. This paper demonstrates an intimate connection between the problems of GCN construction, message routing on SIMD computers, and "resource partitioning." A GCN due to Ofman [7] is here improved to use less than 7.6N log N contact pairs, making it the minimal known construction. Any GCN construction leads to a new algorithm for the broadcast of messages among processing elements of an SIMD computer when each processing element is to receive one message. Previous approaches to message broadcasting have not handled the problem in its full generality. The algorithm arising from this paper's GCN takes 8 log N (or 13N 112) routing steps on an N element processor of the perfect shuffle (or meshtype) variety. If each resource in a multiprocessing environment is assigned one output of a GCN, private buses may be provided for any number of disjoint subsets of the resources. The partitioning construction derived from this paper's GCN has 5.7N log N switches, providing an alternative to "banyan networks " with O(N log N) switches but incomplete functionality. Index TermsArray processors, connection networks, message broadcasting, parallel algorithms, parallel processing, resource partitioning, SIMD machines.
Randomized Protocols for LowCongestion Circuit Routing in Multistage Interconnection Networks
"... In this paper we study randomized algorithms for circuit switching on multistage networks related to the butterfly. We devise algorithms that route messages by constructing circuits (or paths) for the messages with small congestion, dilation, and setup time. Our algorithms are based on the idea of h ..."
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Cited by 16 (5 self)
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In this paper we study randomized algorithms for circuit switching on multistage networks related to the butterfly. We devise algorithms that route messages by constructing circuits (or paths) for the messages with small congestion, dilation, and setup time. Our algorithms are based on the idea of having each message choose a route from two possibilities, a technique that has previously proven successful in simpler load balancing settings. As an application of our techniques, we propose a novel design for a data server.
On the Benefit of Supporting Virtual Channels in Wormhole Routers
 In Proceedings of the 8th Annual ACM Symposium on Parallel Algorithms and Architectures
, 1996
"... This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We show that in any network in which each physical channel, i.e., communication link, can support up to B virtual channels, it is possible to route any set of messages with L flits each, whose paths ..."
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Cited by 10 (4 self)
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This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We show that in any network in which each physical channel, i.e., communication link, can support up to B virtual channels, it is possible to route any set of messages with L flits each, whose paths have congestion C and dilation D in (L + D)C(D log D) 1=B 2 O(log (C=D)) =B flit steps, where a flit step is the time taken to transmit a single flit across a link. We also prove a nearly matching lower bound, i.e., for any values of C, D, B, and L, where C; D B + 1 and L = (1 +\Omega\Gamma302 D, we show how to construct a network and a set of Lflit messages whose paths have congestion C and dilation D that require\Omega\Gamma LCD 1=B =B) flit steps to route. These upper and lower bounds imply that increasing the buffering capacity and the bandwidth of each physical channel by a factor of B can speed up a wormhole routing algorithm by a superlinear factor, i.e., a factor signi...
Improved Bounds on Nonblocking 3Stage Clos Networks
 SIAM JOURNAL OF COMPUTING
, 2007
"... We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in [0, 1] and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König’s theorem says that the number ..."
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Cited by 6 (0 self)
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We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in [0, 1] and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König’s theorem says that the number of colors needed is exactly the maximum degree. For this generalization, we show that 2.557n + o(n) colors are sufficient, where n is the maximum total weight adjacent to any vertex, improving the previously best bound of 2.833n + O(1) due to Du et al. Our analysis is interesting on its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous onedimensional bin packing instance which we can prove allows perfect packing. This question is motivated by the question of the rearrangeability of 3stage Clos networks. In that context, the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unitsized bins needed to pack the weights of the incident edges. In that setting, we are able to improve the bound to 2.5480n + o(n), also improving a bound of 2.5625n + O(1) of Du et al. We also consider the online version of this problem in which edges have to be colored as soon as they are revealed. In this context, we can show that 5n colors are enough. This contrasts with the best known lower bound of 3n − 2 by Tsai, Wang, and Hwang but improves upon the previous best upper bound of 5.75n obtained by Gao and Hwang. Additionally, we show several improved bounds for more restricted versions of the problem. These online bounds are achieved by simple and easytoimplement algorithms, inspired by the first fit heuristic for bin packing.
TimeSpace Tradeoffs for BacktoBack FFT Algorithms
 IEEE Trans. Computing C32
, 1983
"... [15] N. Wirth, "Modula: A language for modular multiprogramming," ..."
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[15] N. Wirth, &quot;Modula: A language for modular multiprogramming,&quot;
On Nonblocking Properties of the Benes Network
 In Proceedings of the 6th Annual European Symposium on Algorithms
, 1998
"... A network is called nonblocking if every unused input can be connected by a path through unused edges to any unused output, regardless of which inputs and outputs have been already connected. The Benes network of dimension n is shown to be strictly nonblocking if only a suitable chosen fraction ..."
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A network is called nonblocking if every unused input can be connected by a path through unused edges to any unused output, regardless of which inputs and outputs have been already connected. The Benes network of dimension n is shown to be strictly nonblocking if only a suitable chosen fraction of 1=n of inputs and outputs is used. This has several consequences. First, there is a very simple strict sense nonblocking network with N = 2 n inputs and outputs, namely a (n + log n + 1){ dimensional Benes network. Its depth is O(log N ), it has O(N log 2 N) edges and it is not constructed of expanders. Secondly it leads to a (3 log N){competitive randomized algorithm for a (log N){dimensional Benes network and a O(log 2 N){competitive randomized algorithm for a (log N){dimensional hypercube, for routing permanent calls.
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"... Abstract This paper analyzes the effect of virtual channels on the performance of wormhole routing algorithms. We show that in a network in which each edge can emulate up to q virtual channels, it is possible to route any set of bbit messages whose paths have congestion c and dilation d in (b + d)c ..."
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Abstract This paper analyzes the effect of virtual channels on the performance of wormhole routing algorithms. We show that in a network in which each edge can emulate up to q virtual channels, it is possible to route any set of bbit messages whose paths have congestion c and dilation d in (b + d)c(d log d) 1=q 2 O(log \Lambda (c=d)) bitsteps. We also prove a nearly matching lower bound, i.e., for any values of c, d, q, and b, where c ss d and b * d, we show how to construct a network and a set of bbit messages whose paths have congestion c and dilation d that require \Omega (bcd