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20
Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
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Cited by 93 (9 self)
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We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
Quantum Algorithms for Element Distinctness
 SIAM Journal of Computing
, 2001
"... We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison c ..."
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Cited by 58 (11 self)
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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1
TimeSpace Tradeoffs for Branching Programs
, 1999
"... We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0 ..."
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Cited by 44 (2 self)
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We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n → {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + ε)n, for some constant ε > 0. We also give the first separation result between the syntactic and semantic readk models [BRS93] for k > 1 by showing that polynomialsize semantic readtwice branching programs can compute functions that require exponential size on any syntactic readk branching program. We also show...
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 29 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
TimeSpace Tradeoff Lower Bounds for Randomized Computation of Decision Problems
 In Proc. of 41st FOCS
, 2000
"... We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. ..."
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Cited by 28 (2 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems.
A General Sequential TimeSpace Tradeoff for Finding Unique Elements
 SIAM Journal on Computing
, 1991
"... An optimal R(n2) lower bound is shown for the timespace product of any Rway branching program that determines those values which occur exactly once in a list of n integers in the range [l, R] where R 1 n. This Q(n2) tradeoff also applies to the sorting problem and thus improves the previous times ..."
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Cited by 27 (2 self)
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An optimal R(n2) lower bound is shown for the timespace product of any Rway branching program that determines those values which occur exactly once in a list of n integers in the range [l, R] where R 1 n. This Q(n2) tradeoff also applies to the sorting problem and thus improves the previous timespace tradeoffs for sorting. Because the Rway branching program is a such a powerful model these timespace product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as offline multitape Turing machines and offline logcost RAMS. 1
On the Complexity of SAT
, 1999
"... We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform circui ..."
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Cited by 25 (1 self)
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We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform circuits.
Alphabet Dependence in Parameterized Matching
 Information Processing Letters
, 1994
"... The classical pattern matching paradigm is that of seeking occurrences of one string in another, where both strings are drawn from an alphabet set \Sigma. A recently introduced model is that of parameterized pattern matching. The main motivation for this scheme lies in software maintenance where pro ..."
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Cited by 20 (4 self)
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The classical pattern matching paradigm is that of seeking occurrences of one string in another, where both strings are drawn from an alphabet set \Sigma. A recently introduced model is that of parameterized pattern matching. The main motivation for this scheme lies in software maintenance where program fragments are considered "identical" even if variables names are different. Besides the fixed symbols from \Sigma, strings under this model have additional symbols from a variable set \Pi and occurrences of one string in the other are sought, where renaming of the variables from \Pi is allowed in a match. In this paper we provide an algorithm to find all occurrences of a pattern string of length m in a text string of length n under the parameterized pattern matching model. Our algorithm takes time O(n log ß), where ß = min(m; j\Pij), independent of j\Sigmaj. Our algorithm is optimal since we show that this dependence on j\Pij is inherent to any algorithm for this problem in the compari...
Optimal TimeSpace TradeOffs for Sorting
 In Proc. 39th IEEE Sympos. Found. Comput. Sci
, 1998
"... We study the fundamental problem of sorting in a sequential model of computation and in particular consider the timespace tradeoff (product of time and space) for this problem. ..."
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Cited by 10 (0 self)
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We study the fundamental problem of sorting in a sequential model of computation and in particular consider the timespace tradeoff (product of time and space) for this problem.
A TimeSpace Tradeoff for Boolean Matrix Multiplication
"... A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with prob ..."
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Cited by 7 (0 self)
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A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with probability 1 n1/2. Letting S and T denote expected space and time of a deterministic algorithm, the tradeoff is ST = R(n3.5) for T < cln2.5 and ST = R(n3) for T> where c1, c2> 0. The lower bounds are matched to within a logarithmic factor by upper bounds in the branching program model. Thus, the tradeoff possesses a sharp break a.t T = O(n2.5). These expected case lower bounds are also the best known lower bounds for the worst case.