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<p><b>Abstract</b>—Several new number representations based on a Residue Number System are presented which use the smallest prime numbers as moduli and are suited for parallel computations on a reconfigurable mesh architecture. The bit model of linear reconfigurable mesh with exclusive write and unit-time delay for broadcasting on a subbus is assumed. It is shown how to convert in <tmath>$O(1)$</tmath> time any integer, ranging between <tmath>$0$</tmath> and <tmath>$n-1$</tmath>, from any commonly used representation to any new representation proposed in this paper (and vice versa) using an <tmath>$n\times O\left({\log^2 n\over\log\log n}\right)$</tmath> reconfigurable mesh. In particular, some of the previously known conversion techniques are improved. Moreover, as a byproduct, it is shown how to compute in <tmath>$O(1)$</tmath> time the Prefix Sums of <tmath>$n$</tmath> bits by a reconfigurable mesh having the above mentioned size, thus improving previously known results. Applications to the Prefix Sums of <tmath>$N$</tmath><it>h</it>-bit integers and to Approximate String Matching with <tmath>$\alpha$</tmath> mismatches are also considered. The Summation and the Prefix Sums can be computed in <tmath>$O(1)$</tmath> time using <tmath>$O\left(h\log N+ {\log^2 N\over\log\log N}\right)\times Nh$</tmath> and <tmath>$O\left({h^2+\log^2 N\over\log(h+\log N)}\right)\times O(N(h+\log N))$</tmath> reconfigurable meshes, respectively. Moreover, it is shown for the first time how to find in <tmath>$O(1)$</tmath> time all the occurrences of a pattern of length <tmath>$m$</tmath> in a text of length <tmath>$n$</tmath>, allowing less than <tmath>$\alpha$</tmath> mismatches, using a reconfigurable mesh of size <tmath>$O(m\log|\Sigma|)\times O\left(n\left(\log|\Sigma|+{\log^2\alpha\over\log\log\alpha}\right)\right)$</tmath>, where the pattern and the text are strings over a finite alphabet <tmath>$\Sigma$</tmath> and <tmath>$\alpha<m\le n$</tmath>.</p>
Number representation, prefix sums, reconfigurable mesh, residue number system, string matching with $k$ mismatches, VLSI.
Alan A. Bertossi, Alessandro Mei, "A Residue Number System on Reconfigurable Mesh with Applications to Prefix Sums and Approximate String Matching", IEEE Transactions on Parallel & Distributed Systems, vol. 11, no. , pp. 1186-1199, November 2000, doi:10.1109/71.888638
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