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35
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 (13 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 56 (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 Lower Bounds for Satisfiability
, 2004
"... We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c an ..."
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Cited by 28 (7 self)
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We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c and space n d, where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √ 2. Our lower bounds apply to nondeterministic linear time and almost all natural NPcomplete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c. Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove timespace lower bounds for languages higher up in the polynomialtime hierarchy.
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 25 (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.
Lower bounds for randomized read/write stream algorithms
 In Proc. STOC’07
, 2007
"... Motivated by the capabilities of modern storage architectures, we consider the following generalization of the data stream model where the algorithm has sequential access to multiple streams. Unlike the data stream model, where the stream is read only, in this new model (introduced in [8, 9]) the al ..."
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Cited by 22 (1 self)
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Motivated by the capabilities of modern storage architectures, we consider the following generalization of the data stream model where the algorithm has sequential access to multiple streams. Unlike the data stream model, where the stream is read only, in this new model (introduced in [8, 9]) the algorithms can also write onto streams. There is no limit on the size of the streams but the number of passes made on the streams is restricted. On the other hand, the amount of internal memory used by the algorithm is scarce, similar to data stream model. We resolve the main open problem in [7] of proving lower bounds in this model for algorithms that are allowed to have 2sided error. Previously, such lower bounds were shown only for deterministic and 1sided error randomized algorithms [9, 7]. We consider the classical set disjointness problem that has proved to be invaluable for deriving lower bounds for many other problems involving data streams and other randomized models of computation. For this problem, we show a nearlinear lower bound on the size of the internal memory used by a randomized algorithm with 2sided error that is allowed to have o(log N / log log N) passes over the streams. This bound is almost optimal since there is a simple algorithm that can solve this problem using logarithmic memory if the number of passes over the streams is allowed to be O(log N). Applications include nearlinear lower bounds on the internal memory for wellknown problems in the literature: (1) approximately counting the number of distinct elements in the input (F0); (2) approximating the frequency of the mode of an input sequence (F ∗ ∞); (3) computing the join of two relations; and (4) deciding if some node of an XML document matches an XQuery (or XPath) query.
On approximate majority and probabilistic time
 in Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and ..."
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Cited by 22 (8 self)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�size depth3 circuits for Approximate Majority on n bits have bottom fanin Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomialsize circuits of depth 3. Prior to our work, the only known polynomialsize depth3 circuits for Approximate Majority were nonuniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a randomaccess input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
TimeSpace Lower Bounds for the PolynomialTime Hierarchy on Randomized Machines
 SIAM JOURNAL ON COMPUTING
, 2006
"... We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machine ..."
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Cited by 16 (5 self)
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We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in time n c and space n d, where QSAT ℓ denotes the problem of deciding the validity of a quantified Boolean formula with at most ℓ − 1 quantifier alternations. Moreover, d approaches 1/2 from below as c approaches 1 from above for ℓ = 2, and d approaches 1 from below as c approaches 1 from above for ℓ ≥ 3. In fact, we establish the stronger result that for any constants a ≤ 1 and c < 1+(ℓ −1)a, there exists a positive constant d such that lineartime alternating machines using space n a and ℓ − 1 alternations cannot be simulated by randomized machines with twosided error running in time n c and space n d, where d approaches a/2 from below as c approaches 1 from above for ℓ = 2 and d approaches a from below as c approaches 1 from above for ℓ ≥ 3. Corresponding to ℓ = 1, we prove that there exists a positive constant d such that the set of Boolean tautologies cannot be decided by a randomized machine with onesided error in time n 1.759 and space n d. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result.
A Survey of Lower Bounds for Satisfiability and Related Problems
 FOUNDATIONS AND TRENDS IN THEORETICAL COMPUTER SCIENCE
, 2007
"... Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow ..."
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Cited by 15 (1 self)
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Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving nontrivial lower bounds on the computational complexity of satisfiability. In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the stateoftheart results and present the underlying arguments in a unified framework.
ComparisonBased Time–Space Lower Bounds for Selection
"... We establish the first nontrivial lower bounds on timespace tradeoffs for the selection problem. We prove that any comparisonbased randomized algorithm for finding the median requires Ω(n log logS n) expected time in the RAM model (or more generally in the comparison branching program model), if we ..."
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Cited by 10 (1 self)
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We establish the first nontrivial lower bounds on timespace tradeoffs for the selection problem. We prove that any comparisonbased randomized algorithm for finding the median requires Ω(n log logS n) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the readonly input array. This bound is tight for all S ≫ log n, and remains true even if the array is given in a random order. Our result thus answers a 16yearold question of Munro and Raman, and also complements recent lower bounds that are restricted to sequential access, as in the multipass streaming model [Chakrabarti et al., SODA 2008]. We also prove that any comparisonbased, deterministic, multipass streaming algorithm for finding the median requires Ω(n log ∗ (n/s) + n log s n) worstcase time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s ≫ log 2 n. We get deterministic lower bounds for I/Oefficient algorithms as well. All proofs in this paper involve “elementary ” techniques only. 1
Pebbles and Branching Programs for Tree Evaluation
, 2009
"... We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in P), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced dary tree of height h, whose internal nodes are la ..."
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Cited by 9 (5 self)
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We introduce the tree evaluation problem, show that it is in LogDCFL (and hence in P), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced dary tree of height h, whose internal nodes are labeled with dary functions on [k] = {1,...,k}, and whose leaves are labeled with elements of [k]. Each node obtains a value in [k] equal to its dary function applied to the values of its d children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic kway branching program with Θ(k h) states solving this problem, and we prove that this upper bound is tight for h = 2 and h = 3. We introduce a simple semantic restriction called thrifty on kway branching programs solving tree evaluation problems and show that the same state bound of Θ(k h) is tight (up to a constant factor) for all h ≥ 2 for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with Θ(k h/2+1) states solving the Boolean problem