The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
Data structures for mobile data
Journal of Algorithms
Journal of Algorithms - Analysis of algorithms
The height of a random binary search tree
Journal of the ACM (JACM)
An analytic approach to the height of binary search trees II
Journal of the ACM (JACM)
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Smoothed analysis of binary search trees
Theoretical Computer Science
On Smoothed Analysis of Quicksort and Hoare's Find
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Smoothed analysis of left-to-right maxima with applications
ACM Transactions on Algorithms (TALG)
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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 nreal 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 nand dlies 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茂戮驴 1/nmay 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 $\Theta(\frac{n}{d+1} \sqrt{n/d} + n \log n)$.