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89
Two applications of inductive counting for complementation problems
 SIAM Journal of Computing
, 1989
"... nondeterministic spacebounded complexity classes are closed under complementation, two further applications of the inductive counting technique are developed. First, an errorless probabilistic algorithm for the undirected graph st connectivity problem that runs in O(log n) space and polynomial exp ..."
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Cited by 53 (4 self)
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nondeterministic spacebounded complexity classes are closed under complementation, two further applications of the inductive counting technique are developed. First, an errorless probabilistic algorithm for the undirected graph st connectivity problem that runs in O(log n) space and polynomial expected time is given. Then it is shown that the class LOGCFL is closed under complementation. The latter is a special case of a general result that shows closure under complementation of classes defined by semiunbounded fanin circuits (or, equivalently, nondeterministic auxiliary pushdown automata or treesize bounded alternating Turing machines). As one consequence, it is shown that small numbers of "role switches " in twoperson pebbling can be eliminated.
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 52 (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...
TIGHT ANALYSES OF TWO LOCAL LOAD BALANCING ALGORITHMS
 SIAM J. COMPUT.
, 1999
"... This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We ..."
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Cited by 51 (5 self)
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This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We show that within O(∆/α) steps, the algorithm reduces the maximum difference in tokens between any two nodes to at most O((d 2 log n)/α), where ∆ is the global imbalance in tokens (i.e., the maximum difference between the number of tokens at any node initially and the average number of tokens), n is the number of nodes in the network, and α is the edge expansion of the network. The time bound is tight in the sense that for any graph with edge expansion α, and for any value ∆, there exists an initial distribution of tokens with imbalance ∆ for which the time to reduce the imbalance to even ∆/2 is at least Ω(∆/α). The bound on the final imbalance is tight in the sense that there exists a class of networks that can be locally balanced everywhere (i.e., the maximum difference in tokens between any two neighbors is at most 2d), while the global imbalance remains Ω((d 2 log n)/α). Furthermore, we show that upon reaching a state with a global imbalance of O((d 2 log n)/α), the time for this algorithm to locally balance the network can be as large as Ω(n 1/2). We extend our analysis to a variant of this algorithm for dynamic and asynchronous
Randomized Routing on FatTrees
 Advances in Computing Research
, 1996
"... Fattrees are a class of routing networks for hardwareefficient parallel computation. This paper presents a randomized algorithm for routing messages on a fattree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the ..."
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Cited by 51 (11 self)
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Fattrees are a class of routing networks for hardwareefficient parallel computation. This paper presents a randomized algorithm for routing messages on a fattree. The quality of the algorithm is measured in terms of the load factor of a set of messages to be routed, which is a lower bound on the time required to deliver the messages. We show that if a set of messages has load factor on a fattree with n processors, the number of delivery cycles (routing attempts) that the algorithm requires is O(+lg n lg lg n) with probability 1 \Gamma O(1=n). The best previous bound was O( lg n) for the offline problem in which the set of messages is known in advance. In the context of a VLSI model that equates hardware cost with physical volume, the routing algorithm can be used to demonstrate that fattrees are universal routing networks. Specifically, we prove that any routing network can be efficiently simulated by a fattree of comparable hardware cost. 1 Introduction Fattrees constitute...
Scalable Network Architectures Using The Optical Transpose Interconnection System (OTIS)
, 1996
"... The Optical Transpose Interconnection System (OTIS) proposed in [14] makes use of freespace optical interconnects to augment an electronic system with nonlocal interconnections. In this paper, we show how these connections can be used to implement a largescale system with a given network topology ..."
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Cited by 45 (0 self)
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The Optical Transpose Interconnection System (OTIS) proposed in [14] makes use of freespace optical interconnects to augment an electronic system with nonlocal interconnections. In this paper, we show how these connections can be used to implement a largescale system with a given network topology using small copies of a similar topology. In particular, we show that, using OTIS, an N 2 node 4D mesh can be constructed from N copies of the Nnode 2D mesh, an N 2 node hypercube can be constructed from N copies of the Nnode hypercube, and an (N 2 ; ff 2 ; c=2) expander can be constructed from N copies of an (N; ff; c) expanders, all with small slowdown. We also show how this expander construction can be used to build multibutterfly networks in a scalable fashion. Finally, we demonstrate how the OTIS connections can be used to produce a bitparallel crossbar using many copies of bitserial crossbars with minimal overhead. 1 Introduction In principle, optical interconnect tec...
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 36 (9 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...
BSP vs LogP
, 1996
"... A quantitative comparison of the BSP and LogP models of parallel computation is developed. We concentrate on a variant of LogP that disallows the socalled stalling behavior, although issues surrounding the stalling phenomenon are also explored. Very efficient cross simulations between the two model ..."
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Cited by 30 (4 self)
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A quantitative comparison of the BSP and LogP models of parallel computation is developed. We concentrate on a variant of LogP that disallows the socalled stalling behavior, although issues surrounding the stalling phenomenon are also explored. Very efficient cross simulations between the two models are derived, showing their substantial equivalence for algorithmic design guided by asymptotic analysis. It is also shown that the two models can be implemented with similar performance on most pointtopoint networks. In conclusion, within the limits of our analysis that is mainly of an asymptotic nature, BSP and (stallfree) LogP can be viewed as closely related variants within the bandwidthlatency framework for modeling parallel computation. BSP seems somewhat preferable due to its greater simplicity and portability, and slightly greater power. LogP lends itself more naturally to multiuser mode.
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.
Fast Algorithms for Routing Around Faults in Multibutterflies and RandomlyWired Splitter Networks
 IEEE Transactions on Computers
, 1992
"... This paper describes simple deterministic O(log N)step algorithms for routing permutations of packets in multibutterflies and randomlywired splitter networks. The algorithms are robust against faults (even in the worst case), and are efficient from a practical point of view. As a consequence, we fi ..."
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Cited by 27 (8 self)
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This paper describes simple deterministic O(log N)step algorithms for routing permutations of packets in multibutterflies and randomlywired splitter networks. The algorithms are robust against faults (even in the worst case), and are efficient from a practical point of view. As a consequence, we find that the multibutterfly is an excellent candidate for a highbandwidth lowdiameter switching network underlying a sharedmemory machine. Index TermsFault tolerance, interconnection network, multibutterfly, multistage network, routing algorithm. 1 Introduction Networks derived from hypercubes form the architectural basis of most parallel computers, including machines such as the BBN Butterfly, the Connection Machine, the IBM RP3 and GF11, the Intel iPSC, and the NCUBE. The butterfly, in particular, is quite popular, and has been demonstrated to perform reasonably well in practice. An example of an 8input butterfly is illustrated in Figure 1. The nodes in this graph represent switches,...
Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
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Cited by 26 (0 self)
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This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;