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The averagecase area of Heilbronntype triangles
 RANDOM STRUCTURES AND ALGORITHMS
, 2002
"... From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points ..."
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Cited by 6 (2 self)
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From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points are chosen independently and at random (with a uniform distribution), then there exist positive constants c and C such that c/n3 <µ n < C/n3 for all large enough values of n, where µ n is the expectation of A. Moreover, c/n3 <A<C/n3, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in
A New Approach to Formal Language Theory by Kolmogorov Complexity
 Preprint, SIAM J. Comput
, 1995
"... We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic contextfree languages and nondeterministic contextfree ..."
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We present a new approach to formal language theory using Kolmogorov complexity. The main results presented here are an alternative for pumping lemma(s), a new characterization for regular languages, and a new method to separate deterministic contextfree languages and nondeterministic contextfree languages. The use of the new `incompressibility arguments' is illustrated by many examples. The approach is also successful at the high end of the Chomsky hierarchy since one can quantify nonrecursiveness in terms of Kolmogorov complexity. (This is a preliminary uncorrected version. The final version is the one published in SIAM J. Comput., 24:2(1995), 398410.) 1 Introduction It is feasible to reconstruct parts of formal language theory using algorithmic information theory (Kolmogorov complexity). We provide theorems on how to use Kolmogorov complexity as a concrete and powerful tool. We do not just want A preliminary version of part of this work was presented at the 16th International...
On superlinear lower bounds in complexity theory
 In Proc. 10th Annual IEEE Conference on Structure in Complexity Theory
, 1995
"... This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the mode ..."
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This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows nonlocal communication with memory at unit cost. We study a model that imposes a “fair cost ” for nonlocal communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorovcomplexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.
Kolmogorov Complexity and a Triangle Problem of the Heilbronn Type
"... From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such tha ..."
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Cited by 1 (0 self)
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From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c=n 3 ! n ! C=n 3 for all large enough n, where n is the expectation of A. Moreover, with probability close to one c=n 3 ! A ! C=n 3 . Our proof uses the incompressibility method based on Kolmogorov complexity. The related Heilbronn problem asks for the maximum value assumed by A over all choices of n points. 1 Introduction From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit circle, let T be the one of least area, and let A be the area of T . Let \Delta n be the maximum assumed by A over all choices of n points. H.A. Heilbronn (19081975) 1 asked for the exact value or approximation of \Delta n . The list [1, 2, 3, 5, 8, 9, 10, 11,...
Algorithmic Arguments in Physics of Computation
"... . We show the usefulness of incompressibility arguments based on Kolmogorov complexity in physics of computation by several examples. These include analysis of energy parsimonious `adiabatic' computation, and scalability of network architectures. 1 Introduction In [Shannon, 1948] C. Shannon formula ..."
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. We show the usefulness of incompressibility arguments based on Kolmogorov complexity in physics of computation by several examples. These include analysis of energy parsimonious `adiabatic' computation, and scalability of network architectures. 1 Introduction In [Shannon, 1948] C. Shannon formulated information theory dealing with the average number of bits required to communicate a message produced by a random source from a sender to a receiver who both agree on the ensemble of possible messages. In this theory, if the universe of messages consists of a two elements, a sentence "let's go drink a beer" and Homer's Illiad, both elements equally likely, then the Illiad can be transmitted by a single bit. This illustrates that, as Shannon points out, this theory does not say anything about the information content of individual objects, but only says something about the required information exchange for communication. In [Kolmogorov, 1965] A.N. Kolmogorov formulated a theory of informat...
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"... is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is c ..."
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is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is constructed with tape alphabet or number of tapes far greater than that of the original machine. In this paper, we prove that limiting Turing machines to a fixed alphabet and a fixed number of tapes rules out linear speedup. Specifically, we describe a language that can be recognized in linear time (e. g., 1.51n), and provide a proof, based on Kolmogorov complexity, that the computation cannot be sped up (e. g., below 1.49n). Without the tape and alphabet limitation, the linear speedup theorem does hold and yields machines of time complexity of the form (1 + ε)n for arbitrarily small ε> 0. Earlier results negating linear speedup in alternative models of computation have often been based on the existence of very efficient universal machines. In the vernacular of programming language theory: These models have very efficient selfinterpreters. As the second contribution of this paper, we define a class, PICSTI, of computation models that exactly captures this property, and we disprove the Linear Speedup Theorem for every model in this class, thus generalizing all similar, modelspecific proofs.
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"... Abstract. In search of a single number like Shannon’s statesymbol product to compare the complicacy of Turing Machines including those with multiple tapes and tape heads, a number called TM index is proposed, using a generic definition for single and multi tape machines. Several examples are shown ..."
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Abstract. In search of a single number like Shannon’s statesymbol product to compare the complicacy of Turing Machines including those with multiple tapes and tape heads, a number called TM index is proposed, using a generic definition for single and multi tape machines. Several examples are shown together with their TM indices, including the recently rediscovered physical