Non-asymptotic distributional bounds for the Dickman approximation of the running time of the Quickselect algorithm

Abstract

Given a non-negative random variable $W$ and $\theta >0$, let the generalized Dickman transformation map the distribution of $W$ to that of \[ W^*=_d U^{1/\theta }(W+1), \] where $U \sim{\cal U} [0,1]$, a uniformly distributed variable on the unit interval, independent of $W$, and where $=_d$ denotes equality in distribution. It is well known that $W^*$ and $W$ are equal in distribution if and only if $W$ has the generalized Dickman distribution ${\cal D}_\theta $. We demonstrate that the Wasserstein distance $d_1$ between $W$, a non-negative random variable with finite mean, and $D_\theta $ having distribution ${\cal D}_\theta $ obeys the inequality \[ d_1(W,D_\theta ) \le (1+\theta )d_1(W,W^*). \] The specialization of this bound to the case $\theta =1$ and coupling constructions yield \[ d_1(W_{n,1},D_1) \le \frac{8\log (n/2)+10} {n} \quad \mbox{for all $n \ge 1$, where for $m \ge 1$} \quad W_{n,m}=\frac{1} {n}C_{n,m}-1, \] and $C_{n,m}$ is the number of comparisons made by the Quickselect algorithm to find the $m^{th}$ smallest element of a list of $n$ distinct numbers. A similar bound holds for $W_{n,m}$ for $m \ge 2$, and together recover and quantify the results of [12] that show distributional convergence of $W_{n,m}$ to the standard Dickman distribution ${\cal D}_1$ in the asymptotic regime $m=o(n)$. By comparison to an exact expression for the expected running time $E[C_{n,m}]$, lower bounds are provided that show the rate is not improvable for $m \not = 2$.