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arXiv · 2207.02476

Improving Order with Queues

Andreas Karrenbauer·Kurt Mehlhorn·Pranabendu Misra·Paolo Luigi Rinaldi·Anna Twelsiek·Alireza Haqi·Siavash Rahimi Shateranloo
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Abstract

Given a sequence of $n$ numbers and $k$ parallel First-in-First-Out (FIFO) queues, how close can one bring the sequence to sorted order? It is known that $k$ queues suffice to sort the sequence if the Longest Decreasing Subsequence (LDS) of the input sequence is at most $k$. But, what if the number of queues is too small for sorting completely? - We give a simple algorithm, based on Patience Sort, that reduces the LDS by $k - 1$. We also show, that the algorithm is optimal, i.e., for any $L > 0$ there exists a sequence of LDS $L$ such that the LDS cannot be reduced below $L - k + 1$ with $k$ queues. - Merging two sorted queues is at the core of Merge Sort. In contrast, two sequences of LDS two cannot always be merged into a sequence of LDS two. We characterize when it is possible and give an algorithm to decide whether it is possible. Merging into a sequence of LDS three is always possible. - A down-step in a sequence is an item immediately followed by a smaller item. We give an optimal algorithm for reducing the number of down-steps. The algorithm is online. Our research was inspired by an application in car manufacturing.

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Andreas Karrenbauer, Kurt Mehlhorn, Pranabendu Misra, Paolo Luigi Rinaldi, Anna Twelsiek, Alireza Haqi, Siavash Rahimi Shateranloo. 2022-07-06. Improving Order with Queues. https://arxiv.org/abs/2207.02476

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