Smoothed Analysis of Binary Search Trees and Quicksort under Additive Noise
Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divide-and-conquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0,1], each number is...
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| Published in | Lecture notes in computer science Vol. 5162; pp. 467 - 478 |
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| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Germany
Springer Berlin / Heidelberg
2008
Springer Berlin Heidelberg |
| Series | Lecture Notes in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3540852379 9783540852377 |
| ISSN | 0302-9743 1611-3349 1611-3349 |
| DOI | 10.1007/978-3-540-85238-4_38 |
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| Summary: | Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divide-and-conquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0,1], each number is individually perturbed by adding a value drawn uniformly at random from an interval of size d, and the resulting numbers are inserted into a search tree. An analysis of the smoothed tree height subject to n and d lies at the heart of our paper: We prove that the smoothed height of binary search trees is $\Theta (\sqrt{n/d} + \log n)$ , where d ≥ 1/n may depend on n. Our analysis starts with the simpler problem of determining the smoothed number of left-to-right maxima in a sequence. We establish matching bounds, namely once more $\Theta (\sqrt{n/d} + \log n)$ . We also apply our findings to the performance of the quicksort algorithm and prove that the smoothed number of comparisons made by quicksort is $\Theta(\frac{n}{d+1} \sqrt{n/d} + n \log n)$ . |
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| Bibliography: | Original Abstract: Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divide-and-conquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0,1], each number is individually perturbed by adding a value drawn uniformly at random from an interval of size d, and the resulting numbers are inserted into a search tree. An analysis of the smoothed tree height subject to n and d lies at the heart of our paper: We prove that the smoothed height of binary search trees is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta (\sqrt{n/d} + \log n)$\end{document}, where d ≥ 1/n may depend on n. Our analysis starts with the simpler problem of determining the smoothed number of left-to-right maxima in a sequence. We establish matching bounds, namely once more \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta (\sqrt{n/d} + \log n)$\end{document}. We also apply our findings to the performance of the quicksort algorithm and prove that the smoothed number of comparisons made by quicksort is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta(\frac{n}{d+1} \sqrt{n/d} + n \log n)$\end{document}. |
| ISBN: | 3540852379 9783540852377 |
| ISSN: | 0302-9743 1611-3349 1611-3349 |
| DOI: | 10.1007/978-3-540-85238-4_38 |