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15
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 46 (19 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.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
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Cited by 24 (10 self)
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CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
Limit laws for partial match queries in quadtrees
 ANN. APPL. PROBAB
, 2001
"... It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for th ..."
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Cited by 10 (4 self)
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It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.
Digital Trees and Memoryless Sources: from Arithmetics to Analysis
 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10), Discrete Math. Theor. Comput. Sci. Proc
, 2010
"... Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstr ..."
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Cited by 7 (1 self)
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Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the memoryless source, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters’ expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous Dirichlet series. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific arithmetic properties, especially irrationality measures, of the sources under consideration.
Compression of Graphical Structures: Fundamental Limits, Algorithms, and Experiments
, 2009
"... Information theory traditionally deals with “conventional data,” be it textual data, image, or video data. However, databases of various sorts have come into existence in recent years for storing “unconventional data” including biological data, social data, web data, topographical maps, and medical ..."
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Cited by 6 (4 self)
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Information theory traditionally deals with “conventional data,” be it textual data, image, or video data. However, databases of various sorts have come into existence in recent years for storing “unconventional data” including biological data, social data, web data, topographical maps, and medical data. In compressing such data, one must consider two types of information: the information conveyed by the structure itself, and the information conveyed by the data labels implanted in the structure. In this paper, we attempt to address the former problem by studying information of graphical structures (i.e., unlabeled graphs). As the first step, we consider the ErdősRényi graphs G(n, p) over n vertices in which edges are added randomly with probability p. We prove that the structural entropy of G(n, p) is n
Precise asymptotic analysis of the Tunstall code
 Proc. 2006 International Symposium on Information Theory (Seattle
"... A variabletofixed length encoder partitions the source string over an mary alphabet A into a concatenation of variablelength phrases. Each phrase except the last one is constrained to belong to a given dictionary D of source strings; the last phrase is a nonnull prefix of a dictionary entry. On ..."
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Cited by 5 (2 self)
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A variabletofixed length encoder partitions the source string over an mary alphabet A into a concatenation of variablelength phrases. Each phrase except the last one is constrained to belong to a given dictionary D of source strings; the last phrase is a nonnull prefix of a dictionary entry. One common constraint on a dictionary is that it leads to a unique parsing of any string over A. We will assume that all dictionaries are uniquely parsable. It is convenient to represent a uniquely parsable dictionary by a complete parsing tree T, i.e., a tree in which every internal node has all m children nodes in the tree. The dictionary entries d ∈Dcorrespond to the leaves of parsing tree. The encoder represents each parsed string by the fixed length binary code word corresponding to its dictionary entry. If the dictionary D is has M entries, then the code word for each phrase has
Tunstall Code, Khodak Variations, and random Walks
, 2008
"... A variabletofixed length encoder partitions the source string into variablelength phrases that belong to a given and fixed dictionary. Tunstall, and independently Khodak, designed variabletofixed length codes for memoryless sources that are optimal under certain constraints. In this paper, we s ..."
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Cited by 4 (2 self)
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A variabletofixed length encoder partitions the source string into variablelength phrases that belong to a given and fixed dictionary. Tunstall, and independently Khodak, designed variabletofixed length codes for memoryless sources that are optimal under certain constraints. In this paper, we study the Tunstall and Khodak codes using analytic information theory, i.e., the machinery from the analysis of algorithms literature. After proposing an algebraic characterization of the Tunstall and Khodak codes, we present new results on the variance and a central limit theorem for dictionary phrase lengths. This analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates.
An analytic approach to the asymptotic variance of trie statistics and related structures
, 2013
"... ..."
A Master Theorem for Discrete Divide and Conquer Recurrences
"... Divideandconquer recurrences are one of the most studied equationsin computerscience. Yet, discrete versions of these recurrences, namely T(n) = an + m∑ bjT (⌊pjn+δj⌋) j=1 for some known sequence an and given bj, pj and δj, present some challenges. The discrete nature of this recurrence (represen ..."
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Cited by 2 (0 self)
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Divideandconquer recurrences are one of the most studied equationsin computerscience. Yet, discrete versions of these recurrences, namely T(n) = an + m∑ bjT (⌊pjn+δj⌋) j=1 for some known sequence an and given bj, pj and δj, present some challenges. The discrete nature of this recurrence (represented by the floor function) introduces certain oscillations not captured by the traditional Master Theorem, for example due to Akra and Bazzi who primary studied the continuous version of the recurrence. We apply powerful techniques such as Dirichlet series, MellinPerron formula, and (extended) Tauberian theorems of WienerIkehara to provide a complete and precise solution to this basic computer science recurrence. We illustrate applicability of our results on several examples including a popular and fast arithmetic coding algorithm due to Boncelet for which we estimate its average redundancy. To the best of our knowledge, discrete divide and conquer recurrences were not studied in this generality and such detail; in particular, this allows us to compare the redundancy of Boncelet’s algorithm to the (asymptotically) optimal Tunstall scheme.
AVERAGECASE ANALYSIS OF COUSINS IN m–ARY TRIES
 APPLIED PROBABILITY TRUST (13 MAY 2008)
, 2008
"... We investigate the average similarity of random strings as captured by the average number of “cousins” in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m– ary ..."
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Cited by 1 (0 self)
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We investigate the average similarity of random strings as captured by the average number of “cousins” in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m– ary, and the corresponding trees are m–ary trees. Certain analytic issues arise in the m–ary case that do not have an analog in the binary case.