Improving Order with Queues

TL;DR

This paper explores how to improve sequence sorting using multiple FIFO queues, providing optimal algorithms for LDS reduction, merging sequences, and reducing down-steps, with applications in manufacturing.

Contribution

It introduces an optimal patience sort-based algorithm for LDS reduction with k queues, characterizes mergeability of sequences based on LDS, and offers an online algorithm for reducing down-steps.

Findings

The LDS can be reduced by up to k-1 using k queues.

Merging two sequences with LDS 2 is not always possible, but always possible with LDS 3.

An optimal online algorithm reduces down-steps in sequences.

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.