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36
Randomization and Derandomization in SpaceBounded Computation
 In Proceedings of the 11th Annual IEEE Conference on Computational Complexity
, 1996
"... This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators tha ..."
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Cited by 36 (0 self)
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This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic spacebounded computations, and the application of such generators to obtain deterministic simulations.
On Recycling the Randomness of States in Space Bounded Computation
 In Proceedings of the ThirtyFirst Annual ACM Symposium on the Theory of Computing
, 1999
"... Let M be a logarithmic space Turing machine (or a polynomial width branching program) that uses up to k 2 p log n (read once) random bits. For a fixed input, let P i (S) be the probability (over the random string) that at time i the machine M is in state S, and assume that some weak estimation of ..."
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Cited by 34 (14 self)
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Let M be a logarithmic space Turing machine (or a polynomial width branching program) that uses up to k 2 p log n (read once) random bits. For a fixed input, let P i (S) be the probability (over the random string) that at time i the machine M is in state S, and assume that some weak estimation of the probabilities P i (S) is known or given or can be easily computed. We construct a logarithmic space pseudorandom generator that uses only logarithmic number of truly random bits and outputs a sequence of k bits that looks random to M . This means that a very weak estimation of the state probabilities of M is sufficient for a full derandomization of M and for constructing pseudorandom sequences for M . We have several applications of the main theorem, as stated within. To prove our theorem, we introduce the idea of recycling the state S of the machine M at time i as part of the random string for the same machine at later time. That is, we use the entropy of the random variable S in o...
Short Random Walks On Graphs
 in Proceedings of the TwentyFifth Annual ACM Symposium on Theory of Computing
, 1993
"... . The short term behavior of random walks on graphs is studied, in particular, the rate at which a random walk discovers new vertices and edges. A conjecture by Linial that the expected time to find N distinct vertices is O(N 3 ) is proved. In addition, an upper bound of O(M 2 ) on the expected ..."
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Cited by 28 (2 self)
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. The short term behavior of random walks on graphs is studied, in particular, the rate at which a random walk discovers new vertices and edges. A conjecture by Linial that the expected time to find N distinct vertices is O(N 3 ) is proved. In addition, an upper bound of O(M 2 ) on the expected time to traverse M edges, and O(MN ) on the expected time to either visit N vertices or traverse M edges (whichever comes first) is proved. Key words. random walk, graph, Markov chain AMS subject classification. 60J15 1. Introduction. Consider a simple random walk on G, an undirected graph with n vertices and m edges. At each time step, if the walk is at vertex v, it moves to a vertex chosen uniformly at random from the neighbors of v. Random walks have been studied extensively, and have numerous applications in theoretical computer science, including spaceefficient algorithms for undirected connectivity [4, 8], derandomization [1], recycling of random bits [10, 15], approximation algori...
Fast Connected Components Algorithms For The EREW PRAM
 SIAM J. COMPUT
, 1999
"... We present fast and e#cient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusiveread, exclusivewrite (EREW) PRAM. On a graph with n vertices and m edges, our randomized algorithm runs ..."
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Cited by 26 (3 self)
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We present fast and e#cient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusiveread, exclusivewrite (EREW) PRAM. On a graph with<F3.492e+05> n<F3.822e+05> vertices and<F3.492e+05> m<F3.822e+05> edges, our randomized algorithm runs in<F3.492e+05><F3.822e+05> O(log<F3.492e+05><F3.822e+05> n) time using<F3.492e+05> (m<F3.822e+05> +<F3.492e+05> n<F2.77e+05><F2.072e+05> 1+#<F3.822e+05><F3.492e+05> )/<F3.822e+05> log<F3.492e+05> n<F3.822e+05> EREW processors (for any fixed<F3.492e+05> # ><F3.822e+05> 0). A variant uses<F3.492e+05> (m<F3.822e+05> +<F3.492e+05><F3.822e+05><F3.492e+05> n)/<F3.822e+05> log<F3.492e+05> n<F3.822e+05> processors and runs in<F3.492e+05><F3.822e+05> O(log<F3.492e+05> n<F3.822e+05> log log<F3.492e+05><F3.822e+05> n) time. A deterministic version of the algorithm runs in<F3.492e+05><F3.822e+05> O(log<F2.77e+05><F2.072e+05><F2.77e+05> 1.5<F3.492e+05><F3.822e+05> n) time using<F3.492e+...
Symmetric Logspace is Closed Under Complement
 CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE
, 1994
"... We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL. ..."
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Cited by 26 (1 self)
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We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL.
Algorithmic Derandomization via Complexity Theory
 In Proceedings of the 34th annual ACM Symposium on Theory of Computing (STOC
, 2002
"... We point out how the methods of Nisan [Nis90, Nis92], originally developed for derandomizing spacebounded computations, may be applied to obtain polynomialtime and NC derandomizations of several probabilistic algorithms. Our list includes the randomized rounding steps of linear and semidefinit ..."
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Cited by 25 (1 self)
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We point out how the methods of Nisan [Nis90, Nis92], originally developed for derandomizing spacebounded computations, may be applied to obtain polynomialtime and NC derandomizations of several probabilistic algorithms. Our list includes the randomized rounding steps of linear and semidefinite programming relaxations of optimization problems, parallel derandomization of discrepancytype problems, and the JohnsonLindenstrauss lemma, to name a few.
Improved Algorithms via Approximations of Probability Distributions
 Journal of Computer and System Sciences
, 1997
"... We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC alg ..."
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Cited by 24 (2 self)
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We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
Many Random Walks Are Faster Than One
"... We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time–the expected time required to visit every node in a ..."
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Cited by 21 (3 self)
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We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time–the expected time required to visit every node in a graph at least once–and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speedup in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speedup is sometimes possible, but that some natural graphs allow only a logarithmic speedup. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected stconnectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.
The Complexity of Reconfiguring Network Models
, 1992
"... This paper concerns some of the theoretical complexity aspects of the reconfigurable network model. The computational power of the model is investigated under several variants, depending on the type of switches (or switch operations) assumed by the network nodes. Computational power is evaluated by ..."
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Cited by 19 (5 self)
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This paper concerns some of the theoretical complexity aspects of the reconfigurable network model. The computational power of the model is investigated under several variants, depending on the type of switches (or switch operations) assumed by the network nodes. Computational power is evaluated by focusing on the set of problems computable in constant time in each variant. A hierarchy of such problem classes corresponding to different variants is shown to exist and is placed relative to traditional classes of complexity theory. Department of Mathematics and Computer Science, The Haifa University, Haifa, Israel. Email: yosi@mathcs2.haifa.ac.il y Department of Computer Science, Technische Universitat Munchen, 80290 Munchen, Germany. Email: lange@informatik.tumuenchen.de z Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel. Email: peleg@wisdom.weizmann.ac.il. Supported in part by an Allon Fellowship, by a Bantrell Fellowship an...