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A Brief History of Generative Models for Power Law and Lognormal Distributions
 INTERNET MATHEMATICS
"... Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying ..."
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Cited by 252 (7 self)
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Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying
The Power of Two Random Choices: A Survey of Techniques and Results
 in Handbook of Randomized Computing
, 2000
"... ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately ..."
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Cited by 99 (2 self)
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ITo motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log log n with high probability. Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log log n= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically different results in load balancing. Indeed, having just two random choices (i.e.,...
How Useful Is Old Information
 IEEE Transactions on Parallel and Distributed Systems
, 2000
"... AbstractÐWe consider the problem of load balancing in dynamic distributed systems in cases where new incoming tasks can make use of old information. For example, consider a multiprocessor system where incoming tasks with exponentially distributed service requirements arrive as a Poisson process, the ..."
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Cited by 80 (10 self)
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AbstractÐWe consider the problem of load balancing in dynamic distributed systems in cases where new incoming tasks can make use of old information. For example, consider a multiprocessor system where incoming tasks with exponentially distributed service requirements arrive as a Poisson process, the tasks must choose a processor for service, and a task knows when making this choice the processor queue lengths from T seconds ago. What is a good strategy for choosing a processor in order for tasks to minimize their expected time in the system? Such models can also be used to describe settings where there is a transfer delay between the time a task enters a system and the time it reaches a processor for service. Our models are based on considering the behavior of limiting systems where the number of processors goes to infinity. The limiting systems can be shown to accurately describe the behavior of sufficiently large systems and simulations demonstrate that they are reasonably accurate even for systems with a small number of processors. Our studies of specific models demonstrate the importance of using randomness to break symmetry in these systems and yield important rules of thumb for system design. The most significant result is that only small amounts of queue length information can be extremely useful in these settings; for example, having incoming tasks choose the least loaded of two randomly chosen processors is extremely effective over a large range of possible system parameters. In contrast, using global information can actually degrade performance unless used carefully; for example, unlike most settings where the load information is current, having tasks go to the apparently least loaded server can significantly hurt performance. Index TermsÐLoad balancing, stale information, old information, queuing theory, large deviations. æ 1
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occu ..."
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Cited by 72 (5 self)
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Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.
Lower bounds for random 3SAT via differential equations
 THEORETICAL COMPUTER SCIENCE
, 2001
"... ..."
Setting 2 variables at a time yields a new lower bound for random 3SAT (Extended Abstract)
 STOC
, 2000
"... Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly ..."
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Cited by 35 (4 self)
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Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly at random from C(X) and taking their conjunction. The satisfiability threshold conjecture asserts that there exists a constant ra such that as n+ c¢, F(n, rn) is satisfiable with probability that tends to 1 if r < ra, but unsatisfiable with probability that tends to 1 if r:> r3. Experimental evidence suggests rz ~ 4.2. We prove rz> 3.145 improving over the previous best lower bound r3> 3.003 due to Frieze and Suen. For this, we introduce a satisfiability heuristic that works iteratively, permanently setting the value of a pair of variables in each round. The framework we develop for the analysis of our heuristic allows us to also derive most previous lower bounds for random 3SAT in a uniform manner and with little effort.
The Analysis of a ListColoring Algorithm on a Random Graph (Extended Abstract)
, 1997
"... We introduce a natural kcoloring algorithm and analyze its performance on random graphs with constant expected degree c (Gn,p=c/n). For k = 3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3colorable. This improves the lower bound on the threshold for random 3col ..."
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Cited by 33 (5 self)
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We introduce a natural kcoloring algorithm and analyze its performance on random graphs with constant expected degree c (Gn,p=c/n). For k = 3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3colorable. This improves the lower bound on the threshold for random 3colorability significantly and settles the last case of Q longstanding open question of Bollobas [5]. We also provide a tight asymptotic analysis of the algorithm. We show that for all k 2 3, if c 5 klnk 3/2k then the algorithm almost surely succeeds, while for any E> 0, and k sufficiently large, if c 2 (1 + E)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.
The satisfiability threshold for random 3SAT is at least 3.52
, 2003
"... We prove that a random 3SAT instance with clausetovariable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement. 1 ..."
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Cited by 31 (1 self)
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We prove that a random 3SAT instance with clausetovariable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement. 1
Threshold Phenomena in Random Graph Colouring and Satisfiability
, 1999
"... We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k ..."
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Cited by 24 (4 self)
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We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k (n, d) = Pr[G(n, d/n) is kcolourable]. Erdos asked the following fundamental question: for k 3, is there a constant c k such that for any # > 0, #) = 1 , and lim f k (n, c k + #) = 0 ? (1) We prove that for all k 3, there exists a function t k (n) such that (1) holds upon replacing c k by t k (n), thus establishing that indeed kcolourability has a sharp threshold. Let d k = sup{d lim n## f k (n, d) = 1}. Note that if c k exists then, by definition, c k = d k . For the basic and most studied case k = 3 we prove 3.84 < d 3 < 5.05 . These are the best
Selecting Complementary Pairs of Literals
, 2003
"... We analyze an algorithm that in each free step Selects & Sets to a value a pair of complementary literals of specified corresponding degrees. Unit Clauses, if they exist, are given priority. This algorithm is proved to succeed on input random 3CNF formulas of density c3 = 3.52+, establishing th ..."
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Cited by 21 (2 self)
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We analyze an algorithm that in each free step Selects & Sets to a value a pair of complementary literals of specified corresponding degrees. Unit Clauses, if they exist, are given priority. This algorithm is proved to succeed on input random 3CNF formulas of density c3 = 3.52+, establishing that the conjectured threshold value for random 3CNF formulas is at least 3.52+.