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<p><b>Abstract</b>—The classification problem transforms a set of <tmath>N</tmath> numbers in such a way that none of the first <tmath>\frac{N}{2}</tmath> numbers exceeds any of the last <tmath>\frac{N}{2}</tmath> numbers. A comparator network that solves the classification problem on a set of <tmath>r</tmath> numbers is commonly called an <tmath>r{\hbox{-}}classifier</tmath>. This paper shows how the well-known Leighton's Columnsort algorithm can be modified to solve the classification problem of <tmath>N=rs</tmath> numbers, with <tmath>1 \le s \le r</tmath>, using an <tmath>r{\hbox{-}}{\rm{classifier}}</tmath> instead of an <tmath>r{\hbox{-}}{\rm{sorting}}</tmath> network. Overall, the <tmath>r{\hbox{-}}{\rm{classifier}}</tmath> is used <tmath>O(s)</tmath> times, namely, the same number of times that Columnsort applies an <tmath>r{\hbox{-}}{\rm{sorter}}</tmath>. A hardware implementation is proposed that runs in optimal <tmath>O(s + \log r)</tmath> time and uses an <tmath>O(r\log r(s + \log r))</tmath> work. The implementation shows that, when <tmath>N= r\log r</tmath>, there is a classifier network solving the classification problem on <tmath>N</tmath> numbers in the same <tmath>O(\log r)</tmath> time and using the same <tmath>O(r\log r)</tmath> comparators as an <tmath>r{\hbox{-}}{\rm{classifier}}</tmath>, thus saving a <tmath>\log r</tmath> factor in the number of comparators over an <tmath>(r\log r){\hbox{-}}{\rm{classifier}}</tmath>.</p>
Comparator network, classifier, classification problem, hardware algorithm.
Alan A. Bertossi, M. Cristina Pinotti, Stephan Olariu, Si-Qing Zheng, "Classifying Matrices Separating Rows and Columns", IEEE Transactions on Parallel & Distributed Systems, vol. 15, no. , pp. 654-665, July 2004, doi:10.1109/TPDS.2004.16
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