Results 1  10
of
29
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 ..."
Abstract

Cited by 136 (11 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 68 (9 self)
 Add to MetaCart
(Show Context)
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 & ..."
Abstract

Cited by 41 (2 self)
 Add to MetaCart
We obtain the first nontrivial timespace tradeoff lower bound for functions f : {0, 1}^n &rarr; {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 + &epsilon;)n, for some constant &epsilon; > 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 ..."
Abstract

Cited by 35 (1 self)
 Add to MetaCart
(Show Context)
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. ..."
Abstract

Cited by 34 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 31 (4 self)
 Add to MetaCart
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 cir ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
(Show Context)
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
, 1993
"... 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 ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
(Show Context)
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 programs are considered "identical " even if variables are different. Strings, under this model, additionally have symbols from a variable set \Pi and occurrences of one string in the other up to a renaming of the variables are sought. In this paper we show that finding the occurrences of a mlength string in a n length string under the parameterized pattern matching paradigm can be done in time O(n log ß), where ß = min(m; j\Pij); that is, independent of j\Sigmaj. Additionally, we show that in general this dependence on j\Pij is inherent to any algorithm for this problem in the comparison model  that is, our algorithm is optimal.
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. Beame has ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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. Beame has
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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.