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Lossless condensers, unbalanced expanders, and extractors
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bi ..."
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Cited by 101 (20 self)
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Abstract Trevisan showed that many pseudorandom generator constructions give rise to constructionsof explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(log n) additional random bitsthat maps n bits strings to poly(log K) bit strings, such that any source with support size Kis mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date.By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough minentropies our extractors can output all of the randomness with only O(log n) bits. We also obtain a new disperser that works for every entropy loss, uses an O(log n)bit seed, and has only O(log n) entropy loss. This is the best disperser construction to date,and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced
Efficient Routing and Scheduling Algorithms for Optical Networks
"... This paper studies the problems of dedicating routes and scheduling transmissions in optical networks. In optical networks, the vast bandwidth available in an optical fiber is utilized by partitioning it into several channels, each at a different optical wavelength. A connection between two nodes is ..."
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Cited by 81 (4 self)
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This paper studies the problems of dedicating routes and scheduling transmissions in optical networks. In optical networks, the vast bandwidth available in an optical fiber is utilized by partitioning it into several channels, each at a different optical wavelength. A connection between two nodes is assigned a specific wavelength, with the constraint that no two connections sharing a link in the network can be assigned the same wavelength. This paper classifies several models related to optical networks and presents optimal or nearoptimal algorithms for permutation routing and/or scheduling problems in many of these models. some scheduling problems in one specific model.
Approximation algorithms for disjoint paths and related routing and packing problems
 Mathematics of Operations Research
, 2000
"... Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems consi ..."
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Cited by 60 (1 self)
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Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multicommodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented. Key words and phrases. Disjoint paths, approximation algorithms, unsplittable flow, routing, packing, integer programming, multicommodity flow, randomized algorithms, rounding, linear programming. 1
On the Fault Tolerance of Some Popular BoundedDegree Networks
 SIAM Journal on Computing
, 1992
"... In this paper, we analyze the ability of several boundeddegree networks that are commonly used for parallel computation to tolerate faults. Among other things, we show that an Nnode butterfly containing N 1\Gammaffl worstcase faults (for any constant ffl ? 0) can emulate a faultfree butterfly ..."
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Cited by 50 (9 self)
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In this paper, we analyze the ability of several boundeddegree networks that are commonly used for parallel computation to tolerate faults. Among other things, we show that an Nnode butterfly containing N 1\Gammaffl worstcase faults (for any constant ffl ? 0) can emulate a faultfree butterfly of the same size with only constant slowdown. Similar results are proved for the shuffleexchange graph. Hence, these networks become the first connected boundeddegree networks known to be able to sustain more than a constant number of worstcase faults without suffering more than a constantfactor slowdown in performance. We also show that an Nnode butterfly whose nodes fail with some constant probability p can emulate a faultfree version of itself with a slowdown of 2 O(log N) , which is a very slowly increasing function of N . The proofs of these results combine the technique of redundant computation with new algorithms for (packet) routing around faults in hypercubic networks. Tech...
Eigenvalues and Expansion of Regular Graphs
 Journal of the ACM
, 1995
"... The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kr ..."
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Cited by 50 (1 self)
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The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best known explicit expanders. The spectral method yielded a lower bound of k=4 on the expansion of linear sized subsets of kregular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k=2. Moreover, we construct a family of kregular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k=2. This shows that k=2 is the best bound one can obtain using the second eigenvalue method. We also show an upper bound of roughly 1 + p k \Gamma 1 on the average degree of linearsized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2 p k \Gamma 1. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) th...
Short Paths in Expander Graphs
 In Proceedings of the 37th Annual Symposium on Foundations of Computer Science
, 1996
"... Graph expansion has proved to be a powerful general tool for analyzing the behavior of routing algorithms and the interconnection networks on which they run. We develop new routing algorithms and structural results for boundeddegree expander graphs. Our results are unified by the fact that they ..."
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Cited by 46 (1 self)
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Graph expansion has proved to be a powerful general tool for analyzing the behavior of routing algorithms and the interconnection networks on which they run. We develop new routing algorithms and structural results for boundeddegree expander graphs. Our results are unified by the fact that they are all based upon, and extend, a body of work asserting that expanders are rich in short, disjoint paths. In particular, our work has consequences for the disjoint paths problem, multicommodity flow, and graph minor containment. We show: (i) A greedy algorithm for approximating the maximum disjoint paths problem achieves a polylogarithmic approximation ratio in boundeddegree expanders. Although our algorithm is both deterministic and online, its performance guarantee is an improvement over previous bounds in expanders. (ii) For a multicommodity flow problem with arbitrary demands on a boundeddegree expander, there is a (1+ ")optimal solution using only flow paths of polylogarithmi...
Fast Algorithms for BitSerial Routing on a Hypercube
, 1991
"... In this paper, we describe an O(log N)bitstep randomized algorithm for bitserial message routing on a hypercube. The result is asymptotically optimal, and improves upon the best previously known algorithms by a logarithmic factor. The result also solves the problem of online circuit switching in ..."
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Cited by 38 (10 self)
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In this paper, we describe an O(log N)bitstep randomized algorithm for bitserial message routing on a hypercube. The result is asymptotically optimal, and improves upon the best previously known algorithms by a logarithmic factor. The result also solves the problem of online circuit switching in an O(1)dilated hypercube (i.e., the problem of establishing edgedisjoint paths between the nodes of the dilated hypercube for any onetoone mapping). Our algorithm is adaptive and we show that this is necessary to achieve the logarithmic speedup. We generalize the BorodinHopcroft lower bound on oblivious routing by proving that any randomized oblivious algorithm on a polylogarithmic degree network requires at least \Omega\Gammaast 2 N= log log N) bit steps with high probability for almost all permutations. 1 Introduction Substantial effort has been devoted to the study of storeandforward packet routing algorithms for hypercubic networks. The fastest algorithms are randomized, and c...
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 35 (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.
Efficient Routing in Optical Networks
, 1996
"... This paper studies the problem of dedicating routes to connections in optical networks. In optical networks, the vast bandwidth available in an optical fiber is utilized by partitioning it into several channels, each at a different optical wavelength. A connection between two nodes is assigned a spe ..."
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Cited by 32 (0 self)
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This paper studies the problem of dedicating routes to connections in optical networks. In optical networks, the vast bandwidth available in an optical fiber is utilized by partitioning it into several channels, each at a different optical wavelength. A connection between two nodes is assigned a specific wavelength, with the constraint that no two connections sharing a link in the network can be assigned the same wavelength. This paper considers optical networks with and without switches, and different types of routing in these networks. It presents optimal or nearoptimal constructions of optical networks in these cases and algorithms for routing connections, specifically permutation routing for the networks constructed here.