Testing of sequences by simulation

TL;DR

This paper introduces several algorithms to test whether a random integer sequence contains all distinct elements, analyzing their efficiency through simulation for small and large sequence lengths.

Contribution

It presents new algorithms for sequence goodness testing and evaluates their performance via simulation across different input sizes.

Findings

Algorithms effectively determine sequence distinctness.

Performance varies with input size and algorithm.

Simulation provides insights into comparison counts and runtime.

Abstract

Let $\xi$ be a random integer vector, having uniform distribution \[\mathbf{P} \{\xi = (i_1,i_2,...,i_n) = 1/n^n \} \ \hbox{for} \ 1 \leq i_1,i_2,...,i_n\leq n.\] A realization $(i_1,i_2,...,i_n)$ of $\xi$ is called \textit{good}, if its elements are different. We present algorithms \textsc{Linear}, \textsc{Backward}, \textsc{Forward}, \textsc{Tree}, \textsc{Garbage}, \textsc{Bucket} which decide whether a given realization is good. We analyse the number of comparisons and running time of these algorithms using simulation gathering data on all possible inputs for small values of $n$ and generating random inputs for large values of $n$.