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12
The Computational Complexity of Universal Hashing
 Theoretical Computer Science
, 2002
"... Any implementation of CarterWegman universal hashing from nbit strings to mbit strings requires a timespace tradeoff of TS = Ω(nm). The bound holds in the general boolean branching program model, and thus in essentially any model of computation. As a corollary, computing a+b*c in any f ..."
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Cited by 74 (2 self)
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Any implementation of CarterWegman universal hashing from nbit strings to mbit strings requires a timespace tradeoff of TS = &Omega;(nm). The bound holds in the general boolean branching program model, and thus in essentially any model of computation. As a corollary, computing a+b*c in any field F requires a quadratic timespace tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the...
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.
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 & ..."
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Cited by 47 (4 self)
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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...
Determinism versus NonDeterminism for Linear Time RAMs with Memory Restrictions
 In Proc. of 31st STOC
, 1998
"... Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is suffici ..."
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Cited by 41 (2 self)
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Our computational model is a random access machine with n read only input registers each containing c log n bits of information and a read and write memory. We measure the time by the number of accesses to the input registers. We show that for all k there is an epsilon > 0 so that if n is sufficiently large then the elements distinctness problem cannot be solved in time kn with epsilon n bits of read and write memory, that is, there is no machine with this values of the parameters which decides whether there are two different input registers whose contents are identical. We also show that there is a simple decision problem that can be solved in constant time (actually in two steps) using nondeterministic computation, while there is no deterministic linear time algorithm with epsilon n log n bits read and write memory which solves the problem. More precisely if we allow kn time for some fixed constant k, then there is an epsilon > 0 so that the problem cannot be solved with epsilon n log n bits of read and write memory if n is sufficiently large. The decision problem is the following: "Find two different input registers, so that the Hamming distance of their contents is at most c log n".
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...
Optimal and Efficient Clock Synchronization Under Drifting Clocks (Extended Abstract)
 IN PROCEEDINGS OF THE 18TH ANNUAL ACM SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING
, 1999
"... We consider the classical problem of clock synchronization in distributed systems. Previously, this problem was solved optimally and efficiently only in the case when all individual clocks are nondrifting, i.e., only for systems where all clocks advance at the rate of real time. In this paper, we ..."
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Cited by 31 (1 self)
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We consider the classical problem of clock synchronization in distributed systems. Previously, this problem was solved optimally and efficiently only in the case when all individual clocks are nondrifting, i.e., only for systems where all clocks advance at the rate of real time. In this paper, we present a new algorithm for systems with drifting clocks, which is the first optimal algorithm to solve the problem efficiently: clock drift bounds and message latency bounds may be arbitrary; the computational complexity depends on the communication pattern of the system in a way which is bounded by a polynomial in the network size for most systems. More specifically, the complexity is polynomial in the maximal number of messages known to be sent but not received, the relative system speed, and timestamp s...
Tight lower bounds for stconnectivity on the NNJAG model
 SIAM J. on Computing
, 1999
"... Abstract. Directed stconnectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Compu ..."
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Cited by 8 (1 self)
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Abstract. Directed stconnectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, 1993, pp. 218–227]. Let n be the number of nodes in the input graph and S and T be the space and time used by the NNJAG, respectively. We show that, for any δ>0, if an NNJAG uses space S ∈ O(n1−δ), then T ∈ 2Ω(log2 (n/S)) ; otherwise n log n) / log log n) S
Quantum timespace tradeoffs for sorting
 Proceedings of 35th ACM STOC
, 2003
"... We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We o ..."
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Cited by 8 (2 self)
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We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds S, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T = O(n
TimeSpace Tradeoffs in Algebraic Complexity
"... We exhibit a new method for showing lower bounds for timespace tradeoffs of polynomial evaluation procedures given by straightline programs. From the tradeoff results obtained by this method we deduce lower space bounds for polynomial evaluation procedures running in optimal nonscalar time. Time, ..."
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Cited by 2 (2 self)
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We exhibit a new method for showing lower bounds for timespace tradeoffs of polynomial evaluation procedures given by straightline programs. From the tradeoff results obtained by this method we deduce lower space bounds for polynomial evaluation procedures running in optimal nonscalar time. Time, denoted by L, is measured in terms of nonscalar arithmetic operations and space, denoted by S, is measured by the maximal number of pebbles (registers) used during the given evaluation procedure. The timespace tradeoff function considered in this paper is LS². We show that for "almost all"...