Book ; Online: Density functions for QuickQuant and QuickVal
2021
Abstract: We prove that, for every $0 \leq t \leq 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the $t$th quantile in a randomly ordered list has a Lipschitz continuous density ... ...
| Abstract | We prove that, for every $0 \leq t \leq 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the $t$th quantile in a randomly ordered list has a Lipschitz continuous density function $f_t$ that is bounded above by $10$. Furthermore, this density $f_t(x)$ is positive for every $x > \min\{t, 1 - t\}$ and, uniformly in $t$, enjoys superexponential decay in the right tail. We also prove that the survival function $1 - F_t(x) = \int_x^{\infty}\!f_t(y)\,\mathrm{d}y$ and the density function $f_t(x)$ both have the right tail asymptotics $\exp [-x \ln x - x \ln \ln x + O(x)]$. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant. Comment: 72 pages; submitted for publication in September, 2021 |
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| Keywords | Mathematics - Probability ; Computer Science - Data Structures and Algorithms ; 68P10 (Primary) 60E05 ; 60C05 (Secondary) |
| Subject code | 519 |
| Publishing date | 2021-09-29 |
| Publishing country | us |
| Document type | Book ; Online |
| Database | BASE - Bielefeld Academic Search Engine (life sciences selection) |
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