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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 85 (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 66 (12 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...
On Parallel Hashing and Integer Sorting
, 1991
"... The problem of sorting n integers from a restricted range [1::m], where m is superpolynomial in n, is considered. An o(n log n) randomized algorithm is given. Our algorithm takes O(n log log m) expected time and O(n) space. (Thus, for m = n polylog(n) we have an O(n log log n) algorithm.) The al ..."
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Cited by 32 (8 self)
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The problem of sorting n integers from a restricted range [1::m], where m is superpolynomial in n, is considered. An o(n log n) randomized algorithm is given. Our algorithm takes O(n log log m) expected time and O(n) space. (Thus, for m = n polylog(n) we have an O(n log log n) algorithm.) The algorithm is parallelizable. The resulting parallel algorithm achieves optimal speed up. Some features of the algorithm make us believe that it is relevant for practical applications. A result of independent interest is a parallel hashing technique. The expected construction time is logarithmic using an optimal number of processors, and searching for a value takes O(1) time in the worst case. This technique enables drastic reduction of space requirements for the price of using randomness. Applicability of the technique is demonstrated for the parallel sorting algorithm, and for some parallel string matching algorithms. The parallel sorting algorithm is designed for a strong and non standard mo...
Quantum complexities of ordered searching, sorting, and element distinctness
, 2001
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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.
Optimal TimeSpace TradeOffs for Sorting
 IN PROC. 39TH IEEE SYMPOS. FOUND. COMPUT. SCI
, 1998
"... We study the fundamental problem of sorting in a sequential model of computation and in particular consider the timespace tradeoff (product of time and space) for this problem. Beame has ..."
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Cited by 14 (0 self)
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We study the fundamental problem of sorting in a sequential model of computation and in particular consider the timespace tradeoff (product of time and space) for this problem. Beame has
Quantum and Classical CommunicationSpace Tradeoffs from Rectangle Bounds
"... We derive bounds on the product of the communication C and space S for communicating circuits. The first bound applies to quantum circuits and follows from a "bipartite product" result for the discrepancy of communication problems. If for any problem f : XY the multicolor discrepancy ..."
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Cited by 13 (5 self)
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We derive bounds on the product of the communication C and space S for communicating circuits. The first bound applies to quantum circuits and follows from a "bipartite product" result for the discrepancy of communication problems. If for any problem f : XY the multicolor discrepancy of the communication matrix of f is 1/2 then the problem in which Alice receives some l inputs, Bob r inputs, and their task is to compute f(x i , y j ) for the l r pairs of inputs (x i , y j ), has a quantum communicationspace tradeo# CS (lrd log Z).
A TimeSpace Tradeoff for Boolean Matrix Multiplication
"... A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with prob ..."
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Cited by 12 (0 self)
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A timespace tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability nll2 and to be 0 with probability 1 n1/2. Letting S and T denote expected space and time of a deterministic algorithm, the tradeoff is ST = R(n3.5) for T < cln2.5 and ST = R(n3) for T> where c1, c2> 0. The lower bounds are matched to within a logarithmic factor by upper bounds in the branching program model. Thus, the tradeoff possesses a sharp break a.t T = O(n2.5). These expected case lower bounds are also the best known lower bounds for the worst case.
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 11 (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.
Graph properties checkable in linear time in the number of vertices
, 2004
"... This paper originates from the observation that many classical NP graph problems, including some NPcomplete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertexNLIN, which collects the graph problems comput ..."
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Cited by 10 (3 self)
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This paper originates from the observation that many classical NP graph problems, including some NPcomplete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertexNLIN, which collects the graph problems computable on a nondeterministic RAM in time OðnÞ; where n is the number of vertices of the input graph G ðV; EÞ; rather than its usual size jVjþjEj: It appears that this class is robust (it is defined by a natural restrictive computational device; it is logically characterized by several simple fragments of existential secondorder logic; it is closed under various combinatorial operators, including some restrictions of transitive closure) and meaningful (it contains many natural NP problems: connectivity, hamiltonicity, nonplanarity, etc.). Furthermore, the very restrictive definition of vertexNLIN seems to have beneficial effects on our ability to answer difficult questions about complexity lower bounds or separation between determinism and nondeterminism. For instance, we prove that vertexNLIN strictly contains its deterministic counterpart, vertexDLIN, and even that it does not coincide with its complementary class, covertexNLIN. Also, we prove that several famous graph problems (e.g. planarity, 2colourability) do not belong to vertexNLIN, although they are computable in deterministic time O(V + E).