Algorithms for Programmers - Ideas, Source Code by J. Arndt

By J. Arndt

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The crucial modification of the use of the MFA is not to choose R and C as close as possible to n as is usually done. Instead one chooses R to be minimal so that the row length C corresponds to the biggest data set that fits into the available RAM. We now analyze how the number of seeks depends on the choice of R and C: In what follows it is assumed that the data lies in memory as row0 , row1 , . . , rowR−1 . In other words, the data of each rows lies contiguous in memory. Further let α ≥ 2 be the number of times the data set exceeds the available RAM size.

An−1 ] and b = [b0 , b1, . . 2) (mod n) The last equation may be rewritten as n−1 hτ := ax b(τ −x) x=0 That is, indices τ − x wrap around, it is a cyclic convolution. Pseudo code to compute the cyclic convolution of a[ ] with b[ ] using the definition, the result is returned in c[ ]: procedure convolution(a[],b[],c[],n) { for tau:=0 to n-1 { s := 0 for x:=0 to n-1 { tx := tau - x if tx<0 then tx := tx + n s := s + a[x] * b[tx] } c[tau] := s } } // modulo reduction For length-n sequences this procedure involves proportional n2 operations, therefore it is slow for large values of n.

These are (following the notation in [8]) denoted by h(1) and h(0) respectively. 8c) x≤τ h(1) = x>τ There is a simple way to separate h(0) and h(1) as the left and right half of a length-2 n sequence. This is just what the acyclic convolution (or linear convolution) does: Acyclic convolution of two (length-n) sequences a and b can be defined as that length-2 n sequence h which is the cyclic convolution of the zero padded sequences A and B: A := [a0 , a1 , a2 , . . , an−1 , 0, 0, . . 9) Same for B.

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