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46
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 174 (14 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
A nonlinear time lower bound for boolean branching programs
 In Proc. of 40th FOCS
, 1999
"... Abstract: We give an exponential lower bound for the size of any lineartime Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2w ..."
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Cited by 57 (0 self)
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Abstract: We give an exponential lower bound for the size of any lineartime Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδlogδ  −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.
Lower bounds for high dimensional nearest neighbor search and related problems
, 1999
"... In spite of extensive and continuing research, for various geometric search problems (such as nearest neighbor search), the best algorithms known have performance that degrades exponentially in the dimension. This phenomenon is sometimes called the curse of dimensionality. Recent results [38, 37, 40 ..."
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Cited by 55 (2 self)
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In spite of extensive and continuing research, for various geometric search problems (such as nearest neighbor search), the best algorithms known have performance that degrades exponentially in the dimension. This phenomenon is sometimes called the curse of dimensionality. Recent results [38, 37, 40] show that in some sense it is possible to avoid the curse of dimensionality for the approximate nearest neighbor search problem. But must the exact nearest neighbor search problem suffer this curse? We provide some evidence in support of the curse. Specifically we investigate the exact nearest neighbor search problem and the related problem of exact partial match within the asymmetric communication model first used by Miltersen [43] to study data structure problems. We derive nontrivial asymptotic lower bounds for the exact problem that stand in contrast to known algorithms for approximate nearest neighbor search. 1
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 38 (5 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems.
SuperLinear TimeSpace Tradeoff Lower Bounds for Randomized Computation
, 2000
"... We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, ..."
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Cited by 33 (2 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his timespace tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...
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 ..."
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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.
TimeSpace Tradeoffs, Multiparty Communication Complexity, and NearestNeighbor Problems
 In 34th Symp. on Theory of Computing (STOC’02
, 2002
"... We extend recent techniques for timespace tradeoff lower bounds using multiparty communication complexity ideas. Using these arguments, for inputs from large domains we prove larger tradeoff lower bounds than previously known for general branching programs, yielding time lower bounds of the form T ..."
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Cited by 24 (2 self)
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We extend recent techniques for timespace tradeoff lower bounds using multiparty communication complexity ideas. Using these arguments, for inputs from large domains we prove larger tradeoff lower bounds than previously known for general branching programs, yielding time lower bounds of the form T = n) when space S = n , up from T = n log n) for the best previous results. We also prove the first unrestricted separation of the power of general and oblivious branching programs by proving that 1GAP , which is trivial on general branching programs, has a timespace tradeoff of the form T = (n=S)) on oblivious Finally, using timespace tradeoffs for branching programs, we improve the lower bounds on query time of data structures for nearest neighbor problems in d dimensions from d= log n), proved in the cellprobe model [8, 5], to d) or log d= log log d) or even d log d) (depending on the metric space involved) in slightly less general but more reasonable data structure models.
Testing of Function that have small width Branching Programs
 Proc. of 41 th FOCS
, 2000
"... Combinatorial property testing, initiated formally by Goldreich, Goldwasser and Ron in [11], and inspired by Rubinfeld and Sudan [13], deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x has the property or is being 'far ..."
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Cited by 22 (7 self)
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Combinatorial property testing, initiated formally by Goldreich, Goldwasser and Ron in [11], and inspired by Rubinfeld and Sudan [13], deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x has the property or is being 'far' from having the property. The main result here is that if G = fg : f0; 1g n ! f0; 1gg is a family of Boolean functions that have readonce branching programs of width w, then for every n and > 0 there is a randomized algorithm that always accepts every x 2 f0; 1g n if g(x) = 1, and rejects it with hight probability if at least n bits of x should be modified in order for it to be in g 1 (1). The algorithm queries ( 2 w ) O(w) many queries. In particular, for constant and w, the query complexity is O(1). This generalizes the results of Alon et. al. [2] asserting that regular languages are efficiently (; O(1))testable. 1. Introduction Combinatorial property testing, initia...
Testing membership in languages that have small width branching programs
 SIAM Journal on Computing
"... Abstract. Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and ..."
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Cited by 22 (5 self)
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Abstract. Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and
Tighter lower bounds for nearest neighbor search and related problems in the cell probe model
 In Proc. 32nd Annu. ACM Symp. Theory Comput
, 2000
"... We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, twosided error, algorithms in Yao’s cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we ..."
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Cited by 16 (0 self)
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We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, twosided error, algorithms in Yao’s cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n, d) cells, each containing poly(d, log n) bits. We get a lower bound of Ω(d / log n) on the search time, a significant improvement over the recent bound of Ω(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.