# Automated Recurrence Analysis for Almost-Linear Expected-Runtime Bounds

## TL;DR

This paper introduces a fast, sound algorithm for automatically deriving almost-linear expected-runtime bounds for recurrence relations in randomized algorithms, improving analysis efficiency over traditional methods.

## Contribution

The paper presents a simple linear-time algorithm that automatically infers optimal expected-runtime bounds for classical randomized algorithms, enhancing analysis efficiency and accuracy.

## Key findings

- Efficiently derives expected-runtime bounds for classical randomized algorithms.
- Successfully infers asymptotically optimal bounds for algorithms like QUICK-SORT and COUPON-COLLECTOR.
- Implemented approach demonstrates practical efficiency in experimental evaluations.

## Abstract

We consider the problem of developing automated techniques for solving recurrence relations to aid the expected-runtime analysis of programs. Several classical textbook algorithms have quite efficient expected-runtime complexity, whereas the corresponding worst-case bounds are either inefficient (e.g., QUICK-SORT), or completely ineffective (e.g., COUPON-COLLECTOR). Since the main focus of expected-runtime analysis is to obtain efficient bounds, we consider bounds that are either logarithmic, linear, or almost-linear ($\mathcal{O}(\log n)$, $\mathcal{O}(n)$, $\mathcal{O}(n\cdot\log n)$, respectively, where n represents the input size). Our main contribution is an efficient (simple linear-time algorithm) sound approach for deriving such expected-runtime bounds for the analysis of recurrence relations induced by randomized algorithms. Our approach can infer the asymptotically optimal expected-runtime bounds for recurrences of classical randomized algorithms, including RANDOMIZED-SEARCH, QUICK-SORT, QUICK-SELECT, COUPONCOLLECTOR, where the worst-case bounds are either inefficient (such as linear as compared to logarithmic of expected-runtime, or quadratic as compared to linear or almost-linear of expected-runtime), or ineffective. We have implemented our approach, and the experimental results show that we obtain the bounds efficiently for the recurrences of various classical algorithms.