Results 1  10
of
22
Universal Limit Laws for Depths in Random Trees
 SIAM Journal on Computing
, 1998
"... Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a ..."
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Cited by 51 (8 self)
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Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.
On the variance of the height of random binary search trees
 SIAM J
, 1995
"... Abstract. Let Hn be the height of a random binary search tree on n nodes. We show that there exist constants α = 4.311 ·· · and β = 1.953 ·· · such that E(Hn) = αln n − βln ln n + O(1), We also show that Var(Hn) = O(1). ..."
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Cited by 40 (3 self)
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Abstract. Let Hn be the height of a random binary search tree on n nodes. We show that there exist constants α = 4.311 ·· · and β = 1.953 ·· · such that E(Hn) = αln n − βln ln n + O(1), We also show that Var(Hn) = O(1).
An Analysis of the (µ+1) EA on Simple PseudoBoolean Functions (Extended Abstract)
"... Carsten Witt FB Informatik, LS 2 Univ. Dortmund 44221 Dortmund, Germany carsten.witt@cs.unidortmund.de Abstract. Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first r ..."
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Cited by 33 (9 self)
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Carsten Witt FB Informatik, LS 2 Univ. Dortmund 44221 Dortmund, Germany carsten.witt@cs.unidortmund.de Abstract. Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first rigorous analysis of the (+1) EA on pseudoBoolean functions is presented. For three example functions wellknown from the analysis of the (1+1) EA, bounds on the expected runtime and success probability are derived. For two of these functions, upper and lower bounds on the expected runtime are tight, and the (+1) EA is never more e#cient than the (1+1) EA. Moreover, all lower bounds grow with . On a more complicated function, however, a small increase of provably decreases the expected runtime drastically.
Degree distribution of the FKP network model
 In International Colloquium on Automata, Languages and Programming
, 2003
"... Abstract. Recently, Fabrikant, Koutsoupias and Papadimitriou [7] introduced a natural and beautifully simple model of network growth involving a tradeoff between geometric and network objectives, with relative strength characterized by a single parameter which scales as a power of the number of nod ..."
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Cited by 22 (2 self)
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Abstract. Recently, Fabrikant, Koutsoupias and Papadimitriou [7] introduced a natural and beautifully simple model of network growth involving a tradeoff between geometric and network objectives, with relative strength characterized by a single parameter which scales as a power of the number of nodes. In addition to giving experimental results, they proved a powerlaw lower bound on part of the degree sequence, for a wide range of scalings of the parameter. Here we prove that, despite the FKP results, the overall degree distribution is very far from satisfying a power law. First, we establish that for almost all scalings of the parameter, either all but a vanishingly small fraction of the nodes have degree 1, or there is exponential decay of node degrees. In the former case, a power law can hold for only a vanishingly small fraction of the nodes. Furthermore, we show that in this case there is a large number of nodes with almost maximum degree. So a power law fails to hold even approximately at either end of the degree sequence range. Thus the power laws found in [7] are very different from those given by other internet models or found experimentally [8]. 1
Profiles of random trees: planeoriented recursive trees
, 2005
"... We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of ..."
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Cited by 17 (5 self)
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We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between the profile of planeoriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).
Poisson approximation for functionals of random trees
 and Alg
, 1996
"... We use Poisson approximation techniques for sums of indicator random variables to derive explicit error bounds and central limit theorems for several functionals of random trees. In particular, we consider (i) the number of comparisons for successful and unsuccessful search in a binary search tree a ..."
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Cited by 14 (2 self)
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We use Poisson approximation techniques for sums of indicator random variables to derive explicit error bounds and central limit theorems for several functionals of random trees. In particular, we consider (i) the number of comparisons for successful and unsuccessful search in a binary search tree and (ii) internode distances in increasing trees. The Poisson approximation setting is shown to be a natural and fairly simple framework for deriving asymptotic results.
Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model
, 1995
"... Multiway trees, also known as mary search trees, are data structures generalizing binary search trees. A common probability model for anlayzing the behavior of these structures is the random permutation model. The probability mass function Q on the set of mary search trees under the random permut ..."
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Cited by 8 (6 self)
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Multiway trees, also known as mary search trees, are data structures generalizing binary search trees. A common probability model for anlayzing the behavior of these structures is the random permutation model. The probability mass function Q on the set of mary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty mary search tree. We study some basic properties of the functional Q, which serves as a measure of the "shape" of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values. 1 Research was carried out while the first author was a postdoctoral research associate at the National Institute of Standards and Technology, Statistical Engineering Division. The first author's institution will change its name to Truman State University in July, 1996. Research ...
On the Expected Depth of Random Circuits
, 1998
"... In this paper we analyze the expected depth of random circuits of fixed fanin f: Such circuits are built a gate at a time, with the f inputs of each new gate being chosen randomly from among the previously added gates. The depth of the new gate is defined to be one more than the maximal depth of it ..."
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Cited by 8 (0 self)
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In this paper we analyze the expected depth of random circuits of fixed fanin f: Such circuits are built a gate at a time, with the f inputs of each new gate being chosen randomly from among the previously added gates. The depth of the new gate is defined to be one more than the maximal depth of its input gates. We show that the expected depth of a random circuit with n gates is bounded from above by eflnn and from below by 2:04: : : flnn:
Long and short paths in uniform random recursive dags
, 2009
"... Abstract. In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in prob ..."
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Cited by 7 (3 self)
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Abstract. In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in probability as n → ∞. We also show that max1≤i≤n Si / logn → σ in probability. Keywords and phrases. Uniform random recursive dag. Randomly generated circuit. Random web model. Longest paths. Probabilistic analysis of algorithms. Branching process.
One, Two And Three Times log n/n For Paths In A Complete Graph With Random Weights
 Combin. Probab. Comput
, 1998
"... . Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being e.g. uniformly distributed. It is shown that, asymptotically, this is log n=n for two given points, that the maximum if one point is fixed and the other varies is 2 ..."
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Cited by 6 (0 self)
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. Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being e.g. uniformly distributed. It is shown that, asymptotically, this is log n=n for two given points, that the maximum if one point is fixed and the other varies is 2 log n=n, and that the maximum over all pairs of points is 3 log n=n. Some further related results are given too, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths. 1. Introduction Let a random weight T ij be assigned to every edge ij of the complete graph K n . (Thus T ji = T ij . We do not define T ij for i = j.) We assume that the \Gamma n 2 \Delta weights T ij , 1 i ! j n, are independent and identically distributed; moreover we assume that they are nonnegative and that their distribution function P(T ij t) = t + o(t) as t & 0; the main examples being the uniform U(0; 1) and the exponential Exp(1) distributions. L...