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25
Phase Change of Limit Laws in the Quicksort Recurrence Under Varying Toll Functions
, 2001
"... We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "to ..."
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Cited by 54 (18 self)
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We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n## log E(Tn )/ log n 1/2), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an insitu permutation algorithm to tree traversal algorithms, etc.
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
, 2005
"... We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only con ..."
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Cited by 28 (11 self)
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We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only convergence of finite moments when ˛ 2.1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for ˛ D 0 and a “quicksort type ” limit law for ˛ D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.
Total Path Length for Random Recursive Trees
, 1998
"... Total path length, or search cost, for a rooted tree is defined as the sum of all roottonode distances. Let T n be the total path length for a random recursive tree of order n. Mahmoud (1991) showed that W n := (T n \Gamma E[T n ])=n converges almost surely and in L 2 to a nondegenerate limiting ..."
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Cited by 25 (0 self)
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Total path length, or search cost, for a rooted tree is defined as the sum of all roottonode distances. Let T n be the total path length for a random recursive tree of order n. Mahmoud (1991) showed that W n := (T n \Gamma E[T n ])=n converges almost surely and in L 2 to a nondegenerate limiting random variable W . Here we give recurrence relations for the moments of W n and of W and show that W n converges to W in L p for each 0 ! p ! 1. We confirm the conjecture that the distribution of W is not normal. We also show that the distribution of W is characterized among all distributions having zero mean and finite variance by the distributional identity W d = U(1 +W ) + (1 \Gamma U)W \Gamma E(U); where E(x) := \Gammax ln x \Gamma (1 \Gamma x) ln(1 \Gamma x) is the binary entropy function, U is a uniform(0; 1) random variable, W and W have the same distribution, and U; W , and W are mutually independent. Finally, we derive an approximation for the distribution of W usi...
The Wiener index of random trees
, 2001
"... The Wiener index is analyzed for random recursive trees and random binary search trees in the uniform probabilistic models. We obtain the expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures ..."
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Cited by 24 (3 self)
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The Wiener index is analyzed for random recursive trees and random binary search trees in the uniform probabilistic models. We obtain the expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixedpoint equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.
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.
Density Approximation and Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
 Adv. Appl. Probab
, 2002
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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Cited by 11 (6 self)
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.
On the Distribution of Distances in Recursive Trees
 J. Appl. Probab
, 1996
"... Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree T n with n labeled nodes is a recursive tree if n = 1; or n ? 1 and T n can be constructed by joining node n to a node of some recursive tree T n\Gamma1 . ..."
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Cited by 9 (1 self)
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Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree T n with n labeled nodes is a recursive tree if n = 1; or n ? 1 and T n can be constructed by joining node n to a node of some recursive tree T n\Gamma1 . For arbitrary nodes i ! n in a random recursive tree we give the exact distribution of X i;n , the distance between nodes i and n. We characterize this distribution as the convolution of the law of X i;i+1 and n \Gamma i \Gamma 1 Bernoulli distributions. We further characterize the law of X i;i+1 as a mixture of sums of Bernoullis. For i = i n growing as a function of n, we show that X i n;n is asymptotically normal in several settings. 1 AMS 1991 subject classifications. Primary 05C05; secondary 60C05 2 Keywords and phrases. recursive trees, Stirling numbers of the first kind. 1 1 Introduction and summary A tree on n nodes (vertices) labeled 1; 2; : : : ; n is a recursive tree if t...
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 8 (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.
On the Depth of Randomly Generated Circuits
 In Proceedings of Fourth European Symposium on Algorithms, Lecture
, 1996
"... This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean circuits with descriptions length d that consist of gates with a fixed fanin f and a constant number of inputs. Assuming uniform distribution of descriptions, we show ..."
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Cited by 4 (0 self)
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This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean circuits with descriptions length d that consist of gates with a fixed fanin f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically ef ln n with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discretetime processes. We are able to establish the result by embedding the processes in suitable continuoustime branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average NC. Key Words: random circuits, depth, recursive trees, domination by coupling, continuous Poisson process. 1 The Problem and Motivation A circuit is a ...