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12
Timings for Associative Operations on the MASC Model
, 2001
"... The MASC (Multiple Associative Computing) model is a generalized associativestyle computational model that naturally supports massive dataparallelism and also controlparallelism. A wide range of applications has been developed on this model. Recent research has compared its power to the power of ..."
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Cited by 17 (8 self)
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The MASC (Multiple Associative Computing) model is a generalized associativestyle computational model that naturally supports massive dataparallelism and also controlparallelism. A wide range of applications has been developed on this model. Recent research has compared its power to the power of other popular parallel models such as the PRAM and MMB models using simulations. However, the simulation of MMB has identified some important issues regarding the cost of certain basic MASC operations required for associative computing such as broadcasts, reductions, and associative searches. This paper investigates these issues and gives background information and an analysis of timings for these operations, based on implementation techniques and comparison fairness with respect to other models. It aims to provide justification and clarify arguments on the timings for these constanttime or nearly constanttime basic MASC operations.
Circuit and Decision Tree Complexity of Some Number Theoretic Problems
, 1998
"... We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems. In particular, we prove that deciding if a given integer is squarefree and testing coprimality of two int ..."
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Cited by 12 (10 self)
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We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems. In particular, we prove that deciding if a given integer is squarefree and testing coprimality of two integers by unbounded fanin circuits of bounded depth requires superpolynomial size. 1 Introduction In recent years spectral techniques based on the Abstract Harmonic Analysis on the hypercube have been shown to represent a very useful tool for obtaining lower complexity bounds. Various links between Fourier coefficients of Boolean functions and their complexity characteristics have been studied in a number of works, see [1, 2, 3, 4, 6, 8, 13, 19, 20, 22, 23]. In particular, these spectral techniques have been successfully applied to the parity function and to threshold functions. Institut fur Informatik, Technische Universitat Munchen, D80290 Munchen, Germany. bernasco@informatik.tumue...
On the average sensitivity of testing squarefree numbers
 in "Proc. 5th Intern. Computing and Combin. Conf.", Lect. Notes in Comp. Sci
, 1627
"... Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain a linear lower bound on the average sensitivity of the Boolean function deciding whether a given integer is squarefree. This result allows us to de ..."
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Cited by 7 (7 self)
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Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain a linear lower bound on the average sensitivity of the Boolean function deciding whether a given integer is squarefree. This result allows us to derive a quadratic lower bound for the formula size complexity of testing squarefree numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. \Lambda Supported by DFG grant Me 1077/141.
On polynomial representations of Boolean functions related to some number theoretic problems
 Electronic Colloq. on Comp. Compl
, 1998
"... Abstract. We say a polynomial P over ZM strongly Mrepresents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P onesidedly Mrepresents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of pol ..."
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Cited by 6 (4 self)
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Abstract. We say a polynomial P over ZM strongly Mrepresents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P onesidedly Mrepresents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of polynomials over Z M, which strongly or onesidedly Mrepresent the Boolean function deciding if a given nbit integer is squarefree. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function. 1
The average sensitivity of squarefreeness
 Comp. Compl
, 1999
"... Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymtotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is squarefree ..."
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Cited by 5 (3 self)
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Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymtotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is squarefree. This result allows us to derive a quadratic lower bound for the formula size complexity of testing squarefree numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. *Supported by DFG grant Me 1077/141.#
Complexity of Boolean Functions on PRAMs  Lower Bound Techniques
"... Determining time necessary for computing important functions on parallel machines is one of the most important problems in complexity theory for parallel algorithms. Recently, a substantial progress has been made in this area. In this survey paper, we discuss the results that have been obtained for ..."
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Determining time necessary for computing important functions on parallel machines is one of the most important problems in complexity theory for parallel algorithms. Recently, a substantial progress has been made in this area. In this survey paper, we discuss the results that have been obtained for three types of parallel random access machines (PRAMs): CREW, ROBUST and EREW. 1 Introduction Parallel random access machine (PRAM) is a most abstract model of parallel computers, where interprocessor communication is realized using a shared memory. Each processor of a PRAM can access any cell of a shared memory in one computation step. This is certainly an unrealistic assumption, but it makes the analysis of the parallel algorithms much easier, and we can can concentrate ourselves on inherent complexity of a given problem. This is a reason, why most parallel algorithms have been described in terms of PRAMs. Each PRAM consists of a collection of processors and common memory cells. Each comp...
On the Complexity of Some Arithmetic Problems over F2[T]
"... In this paper, we study various combinatorial complexity characteristics of Boolean functions related to some natural arithmetic problems about polynomials over IF 2 . In particular, we consider the Boolean function deciding whether a given polynomial over IF 2 is squarefree. We obtain an exponentia ..."
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In this paper, we study various combinatorial complexity characteristics of Boolean functions related to some natural arithmetic problems about polynomials over IF 2 . In particular, we consider the Boolean function deciding whether a given polynomial over IF 2 is squarefree. We obtain an exponential lower bound on the size of a decision tree for this function, and derive an asymptotic formula, having a linear main term, for its average sensitivity. This allows us to estimate other complexity characteristics such as the formula size, the average decision tree depth and the degrees of exact and approximative polynomial representations of this function. Finally, using a di#erent method, we show that squarefree testing and deciding irreducibility of polynomials over IF 2 are not in AC
Approximate compaction and paddedsorting on CREW PRAMs
"... Paddedsorting is a task of placing input items in an array in a nondecreasing order, but with free space between consecutive elements allowed. For many applications, paddedsorting is as useful as sorting. Approximate compaction and compression are closely related problems. It is known that time c ..."
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Paddedsorting is a task of placing input items in an array in a nondecreasing order, but with free space between consecutive elements allowed. For many applications, paddedsorting is as useful as sorting. Approximate compaction and compression are closely related problems. It is known that time complexity of paddedsorting on randomized CRCW PRAMs is dramatically lower than time complexity of sorting. We analyze time complexity of these problems on CREW PRAMs (deterministic and randomized) and get tight lower und upper bounds depending on the size of free space. We extend our lower bounds to approximate compaction and compression.