Average-Case Analysis of Cousins in <i>m</i>-ary Tries

Mahmoud, Hosam M.; Ward, Mark Daniel

doi:10.1239/jap/1222441835

Journal of Applied Probability

2008

DOI: 10.1239/jap/1222441835

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Average-Case Analysis of Cousins in m-ary Tries

Hosam M. Mahmoud

Mark Daniel Ward²

Abstract: 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.

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“…They have numerous applications in storing computer files (see Knuth (1998, pp. 116-119)), detecting similarity of DNA strands (see Mahmoud and Ward (2008)), etc. Tries also provide models for the design and analysis of several important algorithms, such as extendible hashing (see Fagin et al (1979)).…”

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Analysis of swaps in radix selection

Elmasry

Mahmoud

2011

Adv. Appl. Probab.

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Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p = 1 2 ) and the unbiased case (p = 1 2 ). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.Keywords: Random structure; algorithm; order statistic; recurrence; perpetuity; phase change; digital data; digital sorting; selection; Mellin transform; poissonization; depoissonization 2010 Mathematics Subject Classification: Primary 60C05 Secondary 60F05; 68P10 Radix methodsRadix Sort is a sorting technique based on analyzing the digital composition of keys. Digits (possibly bits at the lowest machine level) are extracted from keys and used to classify the keys. Radix Sort dates back to the nineteenth century and can be found in the work of Hollerith (1894) on tabulating machines. It provides a good alternative for comparison-based sorting algorithms like Quick Sort and Merge Sort, where keys are compared according to an ordering relation; see Knuth (1998, pp. 116-119), Mahmoud (2000, pp. 148-151), and Mehlhorn (1984, pp. 42-68) for a broad discussion of these sorting algorithms.Two different variants of Radix Sort are known, the least-significant-digit (LSD) Radix Sort and the most-significant-digit (MSD) Radix Sort. The LSD Radix Sort starts with the leastsignificant digit and orders the keys accordingly. It repeatedly orders the keys moving towards the most-significant digit and using one digit at a time; see, for example, Cormen et al. (2001, p. 178) for details. To guarantee the correctness of the LSD variant, every ordering round

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Analysis of swaps in radix selection

Elmasry

Mahmoud

2011

Adv. Appl. Probab.

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Analysis of swaps in radix selection

Elmasry

Mahmoud

2011

Advances in Applied Probability

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On unary nodes in tries

Wagner¹

2010

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience The difference between ordinary tries and Patricia tries lies in the fact that all unary nodes are removed in the latter. Their average number is thus easily determined from earlier results on the size of tries/Patricia tries. In a well-known contention resolution algorithm, whose probabilistic model is essentially equivalent to tries, unary nodes correspond to repetitions, i.e., steps in the algorithm that do not resolve anything at all. In this paper, we take an individual's view on such repetitions: we consider the distribution of the number of repetitions a certain contender encounters in the course of the algorithm―-which is equivalent to the number of unary nodes on the path from the root to a random string in a trie. We encounter an example of a sequence of distributions that does not actually converge to a limit distribution, but rather oscillates around a (discrete) limit distribution.

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Average-Case Analysis of Cousins in m-ary Tries

Cited by 3 publications

References 14 publications

Analysis of swaps in radix selection

Analysis of swaps in radix selection

Analysis of swaps in radix selection

On unary nodes in tries

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