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Models of Computation  Exploring the Power of Computing
"... Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and oper ..."
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Cited by 76 (6 self)
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Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although
Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 50 (9 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
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 41 (2 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...
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 32 (4 self)
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We prove the first timespace lower bound tradeoffs for randomized computation of decision problems.
Communicationspace tradeoffs for unrestricted protocols
 SIAM Journal on Computing
, 1994
"... This paper introduces communicating branching programs, and develops a general technique for demonstrating communicationspace tradeoffs for pairs of communicating branching programs. This technique is then used to prove communicationspace tradeoffs for any pair of communicating branching programs ..."
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Cited by 10 (0 self)
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This paper introduces communicating branching programs, and develops a general technique for demonstrating communicationspace tradeoffs for pairs of communicating branching programs. This technique is then used to prove communicationspace tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. As an example, any pair of communicating Boolean branching programs that computes matrixvector products over GF(2) requires communicationspace product Ω(n 2), provided the space used is o(n / log n). These are the first examples of communicationspace tradeoffs on a completely general model of communicating processes.
Some Topics in Parallel Computation and Branching Programs
, 1995
"... There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition ..."
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Cited by 2 (0 self)
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There are two parts of this thesis: the first part gives two constructions of branching programs; the second part contains three results on models of parallel machines. The branching program model has turned out to be very useful for understanding the computational behavior of problems. In addition, several restrictions of branching programs, for example ordered binary decision diagrams, have proven to be successful data structures in several VLSI design and verification applications. We construct a branching program of o(n log 3 n) nodes for computing any threshold function on n variables and a branching program of o(n log 4 n) nodes for determining the sum of n variables modulo a fixed divisor. These are improvements over constructions of size 2(n 3=2 ) due to Lupanov [Lup65]. The second p...
Triangle Detection Versus Matrix Multiplication: A Study of Truly Subcubic Reducibility
"... It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that s ..."
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It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that such a reduction is impossible: for example, triangle detection returns one bit, while a BMM algorithm returns n 2 bits. Similar reasoning goes for other matrix products and their corresponding triangle problems. We show this intuition is false, and present a new generic strategy for efficiently computing matrix products over algebraic structures used in optimization. We say an algorithm on n × n matrices (or nnode graphs) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0, where M is the absolute value of the largest entry (or the largest edge weight). We prove an equivalence between the existence of truly subcubic algorithms for any (min, ⊙) matrix product, the corresponding matrix product verification problem, and a corresponding triangle detection problem. Our work simplifies and unifies prior work, and has some new consequences: • The following problems either all have truly subcubic algorithms, or none of them do:
on Time–Space Tradeoffs for Branching
, 1999
"... We obtain the first nontrivial time–space tradeoff lower bound for functions f: {0, 1}nQ {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+e) n, for some constant e> 0. We also give the fir ..."
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We obtain the first nontrivial time–space tradeoff lower bound for functions f: {0, 1}nQ {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+e) n, for some constant e> 0. We also give the first separation result between the syntactic and semantic readk models (A. Borodin et al., Comput. Complexity 3 (1993), 1–18) for k> 1 by showing that polynomialsize semantic readtwice branching programs can compute functions that require exponential size on any semantic readk branching program. We also show a time–space tradeoff result on the more general Rway branching program model (Borodin et al., 1993): for any k, we give a function that requires exponential size to be computed by length kn qway branching programs, for some q=q(k). This result gives a similar tradeoff for RAMs, and thus provides the first nontrivial time–space tradedoff for decision problems in this model. © 2001 Elsevier Science (USA) 1.
Abstract
, 2001
"... We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [30] to the quantum case. Applying this ..."
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We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [30] to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other Fourier based lower bound methods, notably showing that ¯s(f) / log n, for the average sensitivity ¯s(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x ∧ y ⊕ z), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log n). 1