Area Complexity of Multilective Merging

Abstract

Lower bounds on the area A(n,m,k,r) required for merging of two sorted sequences of k-bit numbers with length n and m respectively, when the inputs can be replicated up to r times (r ≤ n), are given:

$$A(n,m,k,r) = \left\{ {\begin{array}{*{20}{c}} {\Omega \left( {\frac{n}{r}} \right) for {2^k} \geqslant \frac{n}{r} and n \geqslant m \geqslant \frac{n}{r}} \\ {\Omega (m((\log \frac{{{2^k}}}{m}) + 1)) for {2^{\frac{3}{8}k}} \geqslant \frac{n}{r} and \frac{n}{r} \geqslant m} \\ {\Omega (m((\log \frac{{{2^k}}}{m}) + 1))for\frac{n}{r} \geqslant m and \frac{n}{r} \geqslant {2^{\frac{3}{8}k}} and {2^{(\frac{{3.({8^K}) - 1}}{{{3^{K + 1}} - 1}})}} \geqslant m} \end{array}} \right.$$