1 Department of Mathematics, Statistics and Computer Science [Chicago]
2 Department of Computer Science [Purdue]
We consider the standard Quicksort algorithm that sorts n distinct keys with all possible n! orderings of keys being equally likely. Equivalently, we analyze the total path length L(n) in a randomly built \emphbinary search tree. Obtaining the limiting distribution of L(n) is still an outstanding open problem. In this paper, we establish an integral equation for the probability density of the number of comparisons L(n). Then, we investigate the large deviations of L(n). We shall show that the left tail of the limiting distribution is much ''thinner'' (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics.
Towards Analytic Information Theory: Data Compression, Prediction and Universal Coding Through Analytic Methods; Funder: National Science Foundation; Code: 9804760
Data Compression from the String Matching Perspective: Second Order Properties; Funder: National Science Foundation; Code: 9415491
James Allen Fill, 2013, Distributional convergence for the number of symbol comparisons used by QuickSort, The Annals of Applied Probability, 23, 3, 10.1214/12-aap866, https://doi.org/10.1214/12-aap866.
james Allen fill, 2010, Distributional Convergence for the Number of Symbol Comparisons Used by QuickSort (Extended Abstract), Discrete Mathematics & Theoretical Computer Science, DMTCS Proceedings vol. AM,..., Proceedings, 10.46298/dmtcs.2800, https://doi.org/10.46298/dmtcs.2800.
James Allen Fill;Svante Janson, 2009, Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees, arXiv (Cornell University), 12, 4, pp. 403-416, 10.1007/s00026-009-0006-0, https://arxiv.org/abs/math/0701259.