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19
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
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...
A General Sequential TimeSpace Tradeoff for Finding Unique Elements
 SIAM Journal on Computing
, 1991
"... An optimal R(n2) lower bound is shown for the timespace product of any Rway branching program that determines those values which occur exactly once in a list of n integers in the range [l, R] where R 1 n. This Q(n2) tradeoff also applies to the sorting problem and thus improves the previous times ..."
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Cited by 32 (3 self)
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An optimal R(n2) lower bound is shown for the timespace product of any Rway branching program that determines those values which occur exactly once in a list of n integers in the range [l, R] where R 1 n. This Q(n2) tradeoff also applies to the sorting problem and thus improves the previous timespace tradeoffs for sorting. Because the Rway branching program is a such a powerful model these timespace product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as offline multitape Turing machines and offline logcost RAMS. 1
Alphabet Dependence in Parameterized Matching
, 1993
"... The classical pattern matching paradigm is that of seeking occurrences of one string in another, where both strings are drawn from an alphabet set \Sigma. A recently introduced model is that of parameterized pattern matching; the main motivation for this scheme lies in software maintenance where pro ..."
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Cited by 24 (5 self)
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The classical pattern matching paradigm is that of seeking occurrences of one string in another, where both strings are drawn from an alphabet set \Sigma. A recently introduced model is that of parameterized pattern matching; the main motivation for this scheme lies in software maintenance where programs are considered "identical " even if variables are different. Strings, under this model, additionally have symbols from a variable set \Pi and occurrences of one string in the other up to a renaming of the variables are sought. In this paper we show that finding the occurrences of a mlength string in a n length string under the parameterized pattern matching paradigm can be done in time O(n log ß), where ß = min(m; j\Pij); that is, independent of j\Sigmaj. Additionally, we show that in general this dependence on j\Pij is inherent to any algorithm for this problem in the comparison model  that is, our algorithm is optimal.
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
Inverse Pattern Matching
, 1996
"... Let a textstring T of n symbols from some alphabet \Sigma and an integer m ! n be given. A pattern P of length m over \Sigma is sought such that P minimizes (alternatively, maximizes) the total number of pairwise character mismatches generated when P is compared with all m character substrings of ..."
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Cited by 12 (2 self)
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Let a textstring T of n symbols from some alphabet \Sigma and an integer m ! n be given. A pattern P of length m over \Sigma is sought such that P minimizes (alternatively, maximizes) the total number of pairwise character mismatches generated when P is compared with all m character substrings of T . Two additional variants of the problem are obtained by adding the constraint that P be (respectively, not be) a substring of T . Efficient sequential algorithms are proposed in this paper for the problem and its variants. Key Words: Design and analysis of algorithms, combinatorial algorithms on words, pattern matching, inverse pattern matching, Hamming distance, digital signature. 1 Introduction Inverse pattern matching refers to the task of inferring from a given textstring T a short pattern string P such that P is, by some measure, most typical (or, alternatively, most anomalous) in the context of T . This problem arises in a wide variety of applications and takes up numerous flavors...
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
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
An Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX
, 1995
"... We prove an exponential lower bound on the size of any fixeddegree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n \Gamma 1 lower bound of Rabin [R72] on the depth of algebraic decision trees for this problem. The proof in fac ..."
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Cited by 7 (6 self)
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We prove an exponential lower bound on the size of any fixeddegree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n \Gamma 1 lower bound of Rabin [R72] on the depth of algebraic decision trees for this problem. The proof in fact gives an exponential lower bound on size for the polyhedral decision problem MAX= of testing whether the jth number is the maximum among a list of n real numbers. Previously, except for linear decision trees, no nontrivial lower bounds on the size of algebraic decision trees for any familiar problems are known. We also establish an interesting connection between our lower bound and the maximum number of minimal cutsets for any rankd hypergraphs on n vertices.
TimeSpace Lower Bounds for Undirected and Directed STConnectivity on JAG Models
, 1993
"... Directed and undirected stconnectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graph ..."
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Cited by 5 (2 self)
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Directed and undirected stconnectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graphs), these upper bounds are not simultaneously achievable. This uses new entropy techniques to prove tight bounds on a game involving a helper and a player that models a computation having precomputed information about the input stored in its bounded space. The second result proves that a JAG requires a timespace tradeoff of T \Theta S 1 2 2\Omega i mn 1 2 j to compute directed stconnectivity. The third result proves a timespace tradeoff of T \Theta S 1 3 2\Omega i m 2 3 n 2 3 j on a version of the...