Quicksort Is Optimal For Many Equal Keys

I prove that the average number of comparisons for median-of-$k$ Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant $\alpha_k$ times worse than the lower bound for sorting random multisets with $\Omega(n^\varepsilon)$ duplicates of each value (for any $\varepsilon>0$). The constant is $\alpha_k = \ln(2) / \bigl(H_{k+1}-H_{(k+1)/2} \bigr)$, which converges to 1 as $k\to\infty$, so Quicksort is asymptotically optimal for inputs with many duplicates. This resolves a conjecture by Sedgewick and Bentley (1999, 2002) and constitutes the first progress on the analysis of Quicksort with equal elements since Sedgewick's 1977 article

Combinatorial problems related to sequences with repeated entries

Student Number : 9708525G - PhD thesis - School of Mathematics - Faculty of ScienceSequences of numbers have important applications in the field of Computer Science. As a result they have become increasingly regarded in Mathematics, since analysis can be instrumental in investigating algorithms. Three concepts are discussed in this thesis, all of which are concerned with ‘words’ or ‘sequences’ of natural numbers where repeated letters are allowed: • The number of distinct values in a sequence with geometric distri- bution In Part I, a sample which is geometrically distributed is considered, with the objective of counting how many different letters occur at least once in the sample. It is concluded that the number of distinct letters grows like log n as n → ∞. This is then generalised to the question of how many letters occur at least b times in a word. • The position of the maximum (and/or minimum) in a sequence with geometric distribution Part II involves many variations on the central theme which addresses the question: “What is the probability that the maximum in a geometrically distributed sample occurs in the first d letters of a word of length n?” (assuming d ≤ n). Initially, d is considered fixed, but in later chapters d is allowed to grow with n. It is found that for 1 ≤ d = o(n), the results are the same as when d is fixed. • The average depth of a key in a binary search tree formed from a sequence with repeated entries Lastly, in Part III, random sequences are examined where repeated letters are allowed. First, the average left-going depth of the first one is found, and later the right-going path to the first r if the alphabet is {1, . . . , r} is examined. The final chapter uses a merge (or ‘shuffle’) operator to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths

Analytic combinatorics : functional equations, rational and algebraic functions

This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families. All semantic annotations used in the book are freely available

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families. All semantic annotations used in the book are freely available

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families. All semantic annotations used in the book are freely available

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families

The semantic transparency of English compound nouns

What is semantic transparency, why is it important, and which factors play a role in its assessment? This work approaches these questions by investigating English compound nouns. The first part of the book gives an overview of semantic transparency in the analysis of compound nouns, discussing its role in models of morphological processing and differentiating it from related notions. After a chapter on the semantic analysis of complex nominals, it closes with a chapter on previous attempts to model semantic transparency. The second part introduces new empirical work on semantic transparency, introducing two different sets of statistical models for compound transparency. In particular, two semantic factors were explored: the semantic relations holding between compound constituents and the role of different readings of the constituents and the whole compound, operationalized in terms of meaning shifts and in terms of the distribution of specifc readings across constituent families. All semantic annotations used in the book are freely available

El Rocío: A Case Study of Music and Ritual in Andalucía

Music is central to the processional pilgrimage of El Rocío, which attracts hundreds of thousands of pilgrims to Andalusia, Spain, late each spring. The pilgrimage affords a unique view, in microcosm, of the relationships between music and ritual from both ritual-studies and ethnomusicological perspectives. Based on extensive fieldwork and other research, this dissertation explores the nexus of the Catholic ritual system in Andalusia, flamenco, and the specific music of El Rocío: the Sevillanas Rocieras. That nexus becomes clear through exploration of three particular features of the pilgrimage: (1) the devotional processions that generate a single, focused, collective emotion; (2) the Andalusian musical form called the palo; and (3) the informal musical gatherings called juergas, which take place nightly along the route. Analysis of structural and morphological relationships between ritual, music, and emotion yields surprising realizations about how these three elements come together as embodied aesthetics within a communitas to generate popular culture. Another important finding of this work is the necessity of placing, at the center of the inquiry, the religious experience—including the curious Andalusian phenomenon of the “chaotic” emotional procession and its role within the overall pilgrimage and ritual system. The dissertation concludes with two theoretical positions. The first addresses the process of “emotional structuring” and its role within the musical rituals of El Rocío and, by extension, Andalusia. The second advances a theory of ritual relations with potential application to ritual systems beyond Andalusia. The author presents both positions within an evolutionary framework based on the tenets of biomusicology, neurophenomenology, and Peircean semiotics