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11
A Sequential Algorithm for Generating Random Graphs
, 2006
"... Abstract. We present a nearlylinear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence (di) n i=1 with maximum degree dmax = O(m 1/4−τ), our algorithm generates almost uniform random graphs with that degree sequence ..."
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Cited by 10 (0 self)
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Abstract. We present a nearlylinear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence (di) n i=1 with maximum degree dmax = O(m 1/4−τ), our algorithm generates almost uniform random graphs with that degree sequence in time O(m dmax) where m = 1 2 i di is the number of edges in the graph and τ is any positive constant. The fastest known algorithm for uniform generation of these graphs [35] has a running time of O(m 2 d 2 max). Our method also gives an independent proof of McKay’s estimate [34] for the number of such graphs. We also use sequential importance sampling to derive fully Polynomialtime Randomized Approximation Schemes (FPRAS) for counting and uniformly generating random graphs for the same range of dmax = O(m 1/4−τ). Moreover, we show that for d = O(n 1/2−τ), our algorithm can generate an asymptotically uniform dregular graph. Our results improve the previous bound of d = O(n 1/3−τ) due to Kim and Vu [31] for regular graphs. 1
Examples comparing importance sampling and the Metropolis algorithm
 Illinois Journal of Mathematics
, 2006
"... Importance sampling, particularly sequential and adaptive importance sampling, have emerged as competitive simulation techniques to Markov–chain Monte–Carlo techniques. We compare importance sampling and the Metropolis algorithm as two ways of changing the output of a Markov chain to get a different ..."
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Cited by 9 (3 self)
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Importance sampling, particularly sequential and adaptive importance sampling, have emerged as competitive simulation techniques to Markov–chain Monte–Carlo techniques. We compare importance sampling and the Metropolis algorithm as two ways of changing the output of a Markov chain to get a different stationary distribution. 1 Introduction. Let X be a finite set and π(x) be a probability on X.Forf: X→R, we want to approximate µ = ∑ f(x)π(x). (1) x
Generating Random Graphs with Large Girth
"... We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance LowDensity ParityCheck (LDPC) codes. For any constant k, α ≤ 1/2k(k + 3) and m = O(n 1+α), our algorithm generates an asymptotically uniform r ..."
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Cited by 3 (1 self)
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We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance LowDensity ParityCheck (LDPC) codes. For any constant k, α ≤ 1/2k(k + 3) and m = O(n 1+α), our algorithm generates an asymptotically uniform random graph with n vertices, m edges, and girth larger than k in polynomial time. To the best of our knowledge this is the first polynomialalgorithm for the problem. Our algorithm generates a graph by sequentially adding m edges to an empty graph with n vertices. Recently, these types of sequential methods for counting and random generation have been very successful [35, 18, 11, 7, 5, 6]. 1
Unicode in DomainSpecific Programming Languages for Modeling & Simulation ScalaTion as a Case Study
"... As recent programming languages provide improved conciseness and flexibility of syntax, the development of embedded or internal DomainSpecific Languages has increased. The field of Modeling and Simulation has had a long history of innovation in programming languages (e.g. Simula67, GPSS). Much eff ..."
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Cited by 1 (1 self)
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As recent programming languages provide improved conciseness and flexibility of syntax, the development of embedded or internal DomainSpecific Languages has increased. The field of Modeling and Simulation has had a long history of innovation in programming languages (e.g. Simula67, GPSS). Much effort has gone into the development of Simulation Programming Languages. The ScalaTion project is working to develop an embedded or internal DomainSpecific Language for Modeling and Simulation which could streamline language innovation in this domain. One of its goals is to make the code concise, readable, and in a form familiar to experts in the domain. In some cases the code looks very similar to textbook formulas. To enhance readability by domain experts, a version of ScalaTion is provided that heavily utilizes Unicode.
RightCancellability of a Family of Operations on Binary Trees
"... Introduction The product a:b, where a and b are positive integers, can be expressed as the sum of b terms, each being equal to a. Similarly, a b can be expressed as the product of b factors, each being equal to a. This basically works well because the sum and product operations for integers are a ..."
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Introduction The product a:b, where a and b are positive integers, can be expressed as the sum of b terms, each being equal to a. Similarly, a b can be expressed as the product of b factors, each being equal to a. This basically works well because the sum and product operations for integers are associative; to push this process one level further (i.e. define a new operation by iterating exponentiation), one needs to decides on how to order the operations in the expression a " a " : : : " a (where " is the exponentiation operation). One solution is to always perform the operations in a fixed order, usually righttoleft (see Blackley and Borosh [1] or Knuth [2]). Another, richer solution is to use the structure of a binary tree to set the order, and use binary trees
Discrete Mathematics and Theoretical Computer Science 2, 1998, 2733
"... Introduction The product a:b, where a and b are positive integers, can be expressed as the sum of b terms, each being equal to a. Similarly, a b can be expressed as the product of b factors, each being equal to a. This basically works well because the sum and product operations for integers are a ..."
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Introduction The product a:b, where a and b are positive integers, can be expressed as the sum of b terms, each being equal to a. Similarly, a b can be expressed as the product of b factors, each being equal to a. This basically works well because the sum and product operations for integers are associative; to push this process one level further (i.e. define a new operation by iterating exponentiation), one needs to decides on how to order the operations in the expression a " a " : : : " a (where " is the exponentiation operation). One solution is to always perform the operations in a fixed order, usually righttoleft (see Blackley and Borosh [1] or Knuth [2]). Another, richer solution is to use the structure of a binary tree to set the order, and use binary trees instead of integers. Blondel [3, 4] defines a family of operations on binary trees. Each new operation is defined in terms of the preceding one. The first three operations are generalizations of addition, multiplicati
Effectiveness ∗
, 2011
"... We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker ..."
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We describe axiomatizations of several aspects of effectiveness: effectiveness of transitions; effectiveness relative to oracles; and absolute effectiveness, as posited by the ChurchTuring Thesis. Efficiency is doing things right; effectiveness is doing the right things. —Peter F. Drucker
and biological creativity
, 2010
"... We present an informationtheoretic analysis of Darwin’s theory of evolution, modeled as a hillclimbing algorithm on a fitness landscape. Our space of possible organisms consists of computer programs, which are subjected to random mutations. We study the random walk of increasing fitness made by a ..."
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We present an informationtheoretic analysis of Darwin’s theory of evolution, modeled as a hillclimbing algorithm on a fitness landscape. Our space of possible organisms consists of computer programs, which are subjected to random mutations. We study the random walk of increasing fitness made by a single mutating organism. In two different models we are able to show that evolution will occur and to characterize the rate of evolutionary progress, i.e., the rate of biological creativity. Key words and phrases: metabiology, evolution of mutating software, random walks in software space, algorithmic information theory 1
Five Dimensional Dynamical
, 1911
"... The dynamical triangulations approach to quantum gravity is investigated in detail for the first time in five dimensions. In this case, the most general action that is linear in components of the fvector has three terms. It was suspected that the corresponding space of couplings would yield a rich ..."
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The dynamical triangulations approach to quantum gravity is investigated in detail for the first time in five dimensions. In this case, the most general action that is linear in components of the fvector has three terms. It was suspected that the corresponding space of couplings would yield a rich phase structure. This work is primarily motivated by the hope that this new viewpoint will lead to a deeper understanding of dynamical triangulations in general. Ultimately, this research programme may give a better insight into the potential application of dynamical triangulations to quantum gravity. This thesis serves as an exploratory study of this uncharted territory. The five dimensional (k, l) moves used in the Monte Carlo algorithm are proven to be ergodic in the space of combinatorially equivalent simplicial 5manifolds. A statement is reached regarding the possible existence of an exponential upper bound on the number of combinatorially equivalent triangulations of the 5sphere. Monte Carlo simulations reveal nontrivial phase structure which is analysed in some detail. Further investigations deal with the geometric and fractal nature of triangulations. This is followed by a characterisation of the weak coupling limit in terms of stacked spheres. Simple graph theory