Density functions for QuickQuant and QuickVal

Abstract

We prove that, for every 0≤t≤1, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the tth quantile in a randomly ordered list has a Lipschitz continuous density function ft that is bounded above by 10. Furthermore, this density ft(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−Ft(x)= ∫x∞ft(y)dy and the density function ft(x) both have the right tail asymptotics exp[−xlnx−xlnlnx+O(x)]. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.