Automated Tail Bound Analysis for Probabilistic Recurrence Relations

Abstract Probabilistic recurrence relations (PRRs) are a standard formalism for describing the runtime of randomized algorithm. Given PRR and time limit $$\kappa $$ κ , we consider tail probability $$\Pr [T \ge \kappa ]$$ Pr [ T ≥ ] i.e., that T exceeds . Our focus is formal analysis bounds aims at finding tight asymptotic upper bound $$u \Pr [T\ge u To address this problem, classical most well-known approach cookbook method by Karp (JACM 1994), while other approaches mostly limited to deriving specific PRRs via involved custom analysis. In work, propose novel common exponentially-decreasing whose preprocessing random passed sizes observe discrete or (piecewise) uniform distribution recursive call either single procedure divide-and-conquer. We first establish theoretical Markov’s inequality, then instantiate with template-based algorithmic refined treatment exponentiation. Experimental evaluation shows our capable (i) asymptotically tighter than Karp’s method, (ii) match best-known manually-derived QuickSelect, (iii) only slightly worse (with $$\log \log n$$ log n factor) manually-proven optimal QuickSort. Moreover, handles all examples (including realistic such as QuickSort, DiameterComputation, etc.) in less 0.1 s, showing efficient practice.