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JOURNAL OF ALGORITHhfS Average 1, 187-208 (1980) Running Time of the Fast Fourier Transform PERSI DIACONIS Bell Laboratories, Murray Hill, New Jersey; Stanforcr! California and Stanford Uniwrsity, Received May 9, 1979; and in revised form October 29, 1979 We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations” of the original Cooley-Tukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n A(n) (n2/9)(x2/log x). The average is not a good indication of the number of operations. For example, it is shown that for about half of the integers n less than x, the number of “operations” is less than n i 61. A similar analysis is given for Good’s algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirp-z transform and the mixed-radix algorithm that computes the transform of a series of prime length p in O(p log p) operations. 1. INTRODUCTION The main results of this paper give approximations to the running time of several algorithms for computation of the discrete Fourier transform (DFT) of n numbers. In Section 2 we discuss the need for exact computation of the DFT versus “padding.” We also describe the available algorithms for computing the DFT. Direct computation of the DFT is shown to involve approximately 2n2 operations-multiplications and additions. If an algorithm is to be used for many different values of n, the average running time is of interest. For direct computation, the average is Several variants of the fast Fourier transform (FFT) involve approximately 2n ,4(n) operations. Here A(n) = Z,+J is the sum of the prime divisors of n counted with multiplicity (so A( 12) = 2 + 2 + 3 = 7). In Section 3 we show that the average number of operations satisfies 187 0196-6774/80/020187-22$02.00/O Copyright 0 1980 by Academic F’rcss, Inc. Ai: rigirij of ~sprduciion in any iorm reserved. 188 PERSI DIACONIS Thus, on the average, these versions of the FFT do not seem to speed things up very much. We will argue that the average is a bad indication of the size of n A(n). Theorem 3 shows that the proportion of integers n less than x such that n A(n) is smaller than n’+Y tends to a limit L(y): $i{n I x : n A(n) 5 n’+Y}I- L(y). The distribution function L(y) is supported on 0 I y I 1 and, for example, L(0.61) = 0.5. Thus, approximately half of the integers less than or equal to x have n A(n) I n t6’ . The results in Section 3 show that, up to lower-order terms, Good’s version of the FFT has the same average case behavior as n A(n). Section 4 analyzes two algorithms for computing the DFT in O(n log n) operations. These are the chirp-z algorithm and the mixed-radix algorithm which uses the chirp-z (or number theoretic) transform for series of prime length. Neither approach dominates. Both algorithms have average running time proportional to x log, x. For the chirp-z approach, the “constant” of proportionality is a bounded oscillating function of x which oscillates around the constant of proportionality of the mixed-radix approach. For individual n, the better algorithm can speed things up by a factor of lf to 2. Some Notation Throughout this paper, p is a prime, Z,,,, means a sum over the distinct prime divisors of n, Ep,+, means a sum over the prime divisors of n counted with multiplicity. The 0,o notation will be used with, for example, 0, meaning that the implied constant depends on k; f(x) - g(x) means f(x)/g(x) + 1. We write 1x1 for the largest integer less than or equal to x, 1x1 for the smallest integer greater than or equal to x, and {x} for the fractional part of x. The number of elements in a finite set S is denoted IS/. 2. THE FAST FOURIER TRANSFORM The discrete Fourier transform of n real numbers x0, x1, . . . , x,,-, is the sequence n-l +tk) = 2 xjdk? k = 0, 1, 2, . . . , n - 1; q, = e2=jln. (2.1) j-0 The usual assumption is that the numbers dk are stored (or available for FAST FOURIER TRANSFORM 189 free). Then, for each k, direct computation of c+(k) involves n multiplications and n additions to good approximation. Computing +(k) for k = and n2 addi0, 1, 2, . . . ) n - 1 involves approximately nz multiplications tions. We will say that approximately 2n2 operations are involved for direct computation. The FFT is a collection of algorithms for computing the DFT. The basic papers on the FFT are collected together in [ 141. A discussion from a modern algorithmic point of view with applications and references is in [ 1, Chap. 71. It is useful to divide the ideas behind FFT algorithms into two types. Type 1 concerns methods of “pasting together” transforms of shorter series. Type 2 concerns methods of transforming the sum in (2.1) into a convolution. First consider the Type 1 ideas. When n is composite, some of the products +qjk are calculated many times. Suppose n = pq. The CooleyTukey and Tukey-Sandy algorithms allow computation of the DFT via computation of p transforms of length q and q transforms of length p. If the shorter transforms are computed directly, this leads top2q2 + q2p2 = 2n(p + q) operations approximately. In general, when n = II:,,pF, the number of operations is 2nE,,+p = 2n A(n). (2.2) We will see later that it is possible to compute the shorter transforms in O(p logp) operations instead of O(p2) operations that direct computation entails. Direct computation is suggested by many writers and implemented in published algorithms such as that of Singleton [18]. Another way of linking together shorter transformations, suggested by Good [20], also falls under the Type 1 ideas. Good’s algorithm requires that the length of the shorter series be relatively prime. For n = II:,,piq, the algorithm computes the DFT of series of lengthpg. If the transforms of length pi%are computed directly using 2p,? operations, then the number of operations is approximately 2n i pi” = 2n G(n). (2.3) i=l The equality in (2.3) defines the function G(n). Expressions (2.2) and (2.3) may be regarded as the dominant term of the result of a more careful count of operations as presented by Rose [16]. When n is a power of 2, n = 2k, A(n) = 2k, and from (2.2) the number of operations is 4n log, n. This is the oft-quoted result “the FFT allows computation of the DFT in O(n log n) operations.” As we have seen, this statement holds only when n is a power of 2. 190 PERU DIACONIS When n is not a power of 2, the technique of padding a series by zeros to the next highest power of 2 can be used. If m is the smallest power of 2 larger than n, the DFT of the sequence of length m: This yields &k) = (x0, x,, x*, * *. 3 xn-,, 0, 0, . . . 9 0) is computed. Z;p-;xjsjmk, k = 0, 1,2, . . . , m - 1, instead of +(k) as defined by (2.1). In many applications, (i, can be used as effectively as +. The difference between C+and 6 is sometimes important. One example occurs in spectral analysis where one looks at +(k) in the hope of detecting periodic oscillations in the sequence xi. If the period divides n, the difference between + and 4 can matter. To give a simple example, let a be a positive integer and for 0 I j < n, define xi by xj = 1 if j is a multiple = 0 Then +(k) = ~~“/a’J-‘q~k that J-0 ’ * of a otherwise. If a divides n, then an easy computation +(k) = k if k = n/a = 0 if k # n/a. shows Thus, Q(k) clearly identifies a series of period a. This clear identification destroyed if n is not a multiple of a, for then is akt”/aJ +(k) = ’ ; ” qak n 2 and e(k) is never 0. In the language of spectral analysis, there is leakage at all frequencies. Such leakage can cause problems, and n is often chosen as a multiple of a period of interest instead of a convenient power of 2. Further discussion of the need for exact computation of the DFT can be found in [6, 181. We next turn to Type 2 ideas. These involve transforming the summation index in the sum (2.1) for cp(k) to convert the sum into a convolution. Convolutions for a series of any length n can be computed exactly by using the FFT on an appropriately extended series. For example, the chirp-z approach discussed by Rabiner et al. [13] and Aho et al. [2] makes the change of variables jk = (j2 + k2 - (j - k)2)/2. Then n-l c$(k) = c$‘~ This is a convolution of the sequence (xjd’i2) (2.4) with the sequence (q,_‘2/2), 191 FAST FOCJRIER TRANSFORM premultiplied by q,k2/2. The idea n is prime, Rader [ 151 proposed transformation. By definition, n),k= 1,2 )...) n-2,gn-‘= g”,k = gb (mod n). Then does not depend using a primitive g is an integer 1 (mod n). Make on n being prime. When root g (mod n) to do the such that gk f 1 (mod the transformation j = n-l cp(k) = c#(g”) = xg + 2 Xgaq#fa+b. a=1 (2.5) The sum is a convolution of the sequence (x,+) with the sequence (q,f’). It is worth pointing out that an integer n can be factored in less than O(n’/‘) operations and that primitive roots can be found in less than O(n’/‘) time. Lehmer [12] gives the number theoretic details. After transforming to a convolution, the FFT can be used to compute the convolution on an extended series. The extended series must be of length approximately the smallest power of 2 larger than 2n. For definiteness, we will use the algorithm given by Rader [ 151. This requires the use of the FFT on a series of length the smallest power 2 larger than or equal to 2n - 4. Three FFTs are required to perform convolutions. To get a simple approximation to the number of operations of these algorithms we will neglect lower-order terms. (2.6) Define T(n) to be the smallest power of 2 larger than 2n - 4. Let C(n) = 3 T(n) log, z-(n) = n log, n if n # 2k if n = 2k. (2.7) ASSUMPTION. The number of operations of the chirp-z transform (or the approach using (2.5)) is well approximated by @Z(n) for some fixed j3 > 0. A reasonable further assumption is to take p = 2 (for additions and multiplications). Since C(n) is O(n log n), when Assumption (2.7) is valid, the DFT of n numbers can be computed in O(n log n) operations, even when n is prime. The availability of methods for computing the DFT in O(n log n) operations immediately suggests a question. Which is more efficient: direct use of the chirp-z idea on a series of length n or splitting a series of length n into pieces of prime length, computation of the shorter transforms by a fast algorithm, and pasting the pieces back together using a mixed-radix algorithm? A detailed analysis of this problem is given in Section 4. To compare the approaches we make a simple approximation to the number of operations required by the mixed-radix approach. The approximation is based on (2.7) and (2.2). 192 PERSI DIACONIS (2.8) ASSUMPTION. The mixed-radix approach which uses a fast algorithm to compute series of prime length uses /3F(n) = /3nZ,.,,(C(p)/p) operations, with j3 as in (2.7). Some numerical examples of C(n) and F(n) are given in Section 4. Note that in making comparison of the relative size of PC(n) and PF(n), fi cancels out. We also note that none of the asymptotic comparisons depend on the use of 2n - 4 in (2.6). Using 2n - c for any fixed c leads to essentially the same results. 3. AVERAGE RUNNING TIME OF MIXED-RADIX THAT COMPUTE THE FFT OF PRIMES IN O(p’) ALGORITHMS OPERATIONS In this section, approximations for the mean, variance, and distribution of the number of operations of some mixed-radix algorithms are derived. As explained in Section 2, 2n A(n) and 2n G(n) are reasonable approximations to the number of operations used by the Cooley-Tukey and Good algorithms, respectively. Theorems 1 and 2 provide approximations to the first and second moments of n A(n) and n G(n). Here, if H(n) is any function, the first and second moments of H(n) are and The variance of H(n) can be computed from the first and second moments via 4X4=-IJnixW(n)- CL&))* =W(X)- (pH(x))*. In Theorems 1 and 2 we write S(s) for Riemann’s zeta function. THEOREM 1. Let n A(n) be defined by (2.2), n G(n) be defined by (2.3). As x tends to infinity, there is a c > 0 such that the first moment of n A(n) or n G(n) equals / I/*1S(s+ 1)g&s+*( x2 exp( - c(log x)““)). (3.1) For any fixed k 2 1, + ‘k( (log:;*+l)’ (3*2) FAST FOURIER 193 TRANSFORM with m a1=--5-’ fi THEOREM 2. As x tends to [email protected], moment of n A(n) or n G(n) equals there is a c > 0 such that the second I l/2l S(s+ 2) g+Y-+o( x4 exp( - c(log x)““)). (3.3) For any fixed k 2 1, s+3 ’ s(s+2)~ds=x4i / l/2 j= b,=ip, bj = (- b,_ I (log + Ok( (log;)k+l)9 1) j( sI”;;‘)“‘~,-,. The first moment and variance are not good indicators of and n G(n), for the variance is close to the square moment. This suggests that these functions have fluctuations mean which are of the same size as their mean. The next theorem gives the limiting distribution of n A(n) The results show that the proportion of numbers such that n G(n)) is smaller than n’+’ tends to a limit. n A(n) THEOREM (3.4) X) ’ the sizes of of the first about their and n G(n). n A(n) (or 3. As x + oo, for any fired z, 0 I z I 1, ${n 5 x : n A(n) I ~J’+~}I ry L(z), (3.5) -${n 5 x : n G(n) I n’+‘}l (3.6) w L(z), where L(z) is the distribution function of an absolutely continuous measureon [0, I] with density L’(z). The density satisfies L’(z) = (l/z)p(l/z - 1), where p(y) = 1 for 0 < y 5 1 and p(y) satisfies the differential dsfference equation yp’(y) = - p(y - 1). Remarks. A few values of L(z) are z / 0.33 0.47 0.61 0.78 0.95 L(z) 1 0.05 0.25 0.50 0.75 0.95 The density L’(y) is drawn in Fig. 1. PERSI DIACONIS 194 0 .2 .4 .6 .8 1 FIG. 1.Graphof L’(y). The function p(y) was introduced by Dickman [lo] in connection with the largest prime divisor of n. It is thoroughly discussed by de Bruijn [8, 91, Billingsley [7], and Knuth and Trabb-Pardo [ 111. Bellman and Kotkin [5] and Van de Lune and Wattel [19] give tables of p(y) which were used to compute L(z) and L’(z) as given above. Theorems 1, 2, and 3 are closely related to two other number theoretic functions which we now define: Let n = II:, ,p,” be the prime decomposition of n. Suppose pr < p2 < . . . -c P,. (3.7) Define A*(n) = i pi. We also write A*(n) = xp. i=l (3.8) Define P,(n) Pb to be the kth largest prime divisor of n. Thus, PI(n) = P,, P*(n) = p,(n/p,(n)), f’,(n) = p,(n/[p,(n) . P,(n)]) . . . with the convention that p,(n) = 1 if n has fewer than i prime divisors. The functions A(n) and A l (n) have been discussed by Alladi and Erdos [3]. They prove the following theorem, which will be needed in the proof of Theorem 3. THEOREM (Alladi and Erdbs). For all m 2 1, as x -+ 00, where k, is a rational multiple of {( 1 + (1 /m)). 195 FAST FOURIER TRANSFORM When m = 1, this gives X,,,/(n) - k,(x2/log show that k, = r2/12. They also show that ns*A(n) - A*(n) x). Alladi and Erdos = x log log x + O(x). (3.9) Saffari [17, p. 2131 has given an asymptotic expansion for the mean of A*(n). The techniques of this paper yield somewhat more precise results: THEOREM 4. As x ten& to [email protected], there is a c > 0 such that the first moment of A(n), A*(n), G(n), or PI(n) equals JI/*’ S(s+ 1)&is+o( x exp( - c(log x)““)) asx+ao. (3.10) For any fixed k 2 1, + ‘k( ~,og~~k+,)’ (3*11) with c,=Yj--’ W) cj = (_ 1)’ Hs ( + s + 1) 1 O’) 1 /I_,- THEOREM 5. As x ten& to infinity, there is a c > 0 such that the second moment of A(n), A*(n), G(n), or P,(n) equals s+l / ’ [(s + 2)% ds + 0(x2 exp( - c(log x)““)). I/* (3.12) For any fixed k 2 1, JI/2’ {(s + 2)Sds = x2x (log4x)j + ‘k( ~~ogx:)k+‘). (3’13) with Remarks. Theorems modifications 1, 2, 4, and 5 are all proved by using slight of the proof of the prime number theorem. By using the 196 PERSI DIACONIS usual modifications of this proof, it is possible to improve the error terms. Using the Riemann hypothesis, the error terms can be further improved. For example, I believe that, on the Riemann hypothesis, the error term in (3.10) becomes O(x”2(log x)‘). The results are given with the error involving (log x) I/” to allow the proof to rely on the proof of the prime number theorem given by Ayoub [4] without modification. As with n A(n) and n G(n), the mean and variance are not good indicators of the sizes of A(n), A*(n), G(n), and P,(n). Asymptotically, these functions all have the same distribution. The next result implies Theorem 3. THEOREM 6. Let H(n) be any one of the functions A(n), A*(n), G(n), or P,(n). As x tendr to infinity, for any fixed y, 0 5 y I 1, -${n Ix : H(n) I ny}I- L(y), where L(y) was defined in Theorem 3. Proof of Theorems 1,2,4, and 5. The approach used here is the classical technique using Dirichlet series. The following identities are needed: LEMMA 7. Let A(n), G(n), A*(n), and P,(n) be as defined in (2.2), (2.3), (3.7), and (3.8), respectively. m A*(n) x-;;;tZ=l re s > 2, g (3.14) (A*(n))’ ns n-1 re s > 3, (3.15) re s > 2, (3.16) 2IX Ay’ - &)EP n-1 ps+2 + SW 7 fq)z. ( (p” - 1)2 res > 3, (3.17) 5 n-1 G(n) -=,,&(l n* --$)(l p PS -+-‘=&,(S), p”- 1 res > 2, (3.18) FAST FOURIER 197 TRANSFORM 2O”F =S(s)?(-+(I --$)(l -pLJ-’ tI=l -&(l -+i)i(l- j$-‘} P + W)( y(s))*, 4s) (3.19) > 3. (3.20) (3.21) Lemma 7 can be proved by a number of arguments. approach seems to be useful somewhat generally. The following Proof of Lemma 7. For fixed real s > 1, define a probability P, on the space D = { 1, 2, 3, . . . } by P,(j) = (1 /{(s))(l/j’), measure where S(s) = ZJ’Z.,1 /j”. F or each prime number p, let X, : Q + 52 u { 0} be defined by X,,(n) = p if pin, X, = 0 otherwise. Thus P,(X, = p) = I/p’, P,(X, = 0) = 1 - l/p’. The random variables X,, are easily seen to be independent var(X,) = (l/p’-*)(l - l/p’). Let A* = x,X,. with E(X,) = l/ps-‘, For s > 1, the Borel-Cantelli lemma implies that A* is almost surely finite. A* has finite mean if and only ifs > 2. A* has finite variance if and only if s > 3. For s > 2, 1-g I(s) n-l A*(n) ns = qA*) = 2 E(X,) = 2 P 1 P ps-’ This implies (3.14). To prove (3.15) note that for s > 3, Lx S(s) (A::)‘2 - (E(A*))* = var(A*) = 2 var(X,) P = - 1 =4 P ps * 1-L PS 1 * To prove (3.16) and (3.17) consider the random variables YP : SI + S2u {0}, where, if n = IIp 4(“), Y,(n) = a,(n)p. Thus for a = 0, 1, 2, . . . ; P,( Yp = ap) = (1 - 1/p”)( l/p”“). It is straightforward to check that E( 5) = p/(p” - l), var( Y,) = ps+*/(ps - l)*. To prove (3.18) and (3.19), consider the random variables 2,:&I + Q u (0) where if n = IIp4’“), Z,(n) = ~4~“). Thus, P,(Z, = 0) = 1 - l/p” and for a = 1, 2,, . . , P, (Z, = p”) 198 PERSI DIACONIS = (1 - 1/p”)( 1/p “). Again it is easy to check that E(Z,) = (1 -$)--$(l Jqzp’) = (1 - -+o ---$)-I and - --+y. The argument used to prove (3.14) and (3.15) now leads to a proof of (3.18) and (3.19). Finally, to prove (3.20) and (3.21), use the fact that for any prime P, the product being over all primes q larger than p. It follows that Since also (3.20) and (3.21) follow. The arguments prove the identities for all large real s and thus, by analytic continuation, for all s such that the right sides are analytic. The validity for the half planes given follows easily from the known behavior of the function I&, 1/p” (see, for example, [4, Chap. 2, Sect. 4, (16)]). q Proof of Theorem 1. The argument used here will follow Landau’s proof of the prime number theorem as presented by Ayoub [4, Chap. 21. Since we make constant use of Ayoub’s arguments, the reader is advised to follow the present proof with a copy of Ayoub’s book in hand. First consider the function n A(n). The identity (3.16) together with Theorem 3.1 of Ayoub [4] for expressing the sum of the coefficients of a Dirichlet series yields for nonintegral x and any (Y > 3, (3.22) with f(s) = sts - 1) T p’_p- 1* FAST FOURIER 199 TRANSFORM Changing the variable of integration nonintegral x, and any 4 > 1; from s to s + 2 in (3.22) gives, for (3.23) with g(s) = S(s + 1)X p p ps+’ - 1 - In what follows the path re s = a in (3.23) will be deformed so that part of it lies slightly to the left of the line re s = 1. We now show that Ayoub’s bounds for log l(s) apply to g(s). Observe that for re s > i, the function {(s + 1) is uniformly bounded, in absolute value, by {(+). Further 2p p’+’ p- 1 = 7 $ + F p’(p’j* _ 1)= logs(s)+ h(s), (3.24) where h(s)= 2 P 1 ps(ps+’ - 1) Thus h(s) is analytic in the half plane re s > f and uniformly bounded in any half plane re s > b, with b > f . Suppose the path of integration is now deformed exactly as in [4, Chap. 2, Sect. 51. Ayoub’s arguments yield bounds for all parts of the path except along the cut running from b + k to b - k, where b is 1 - c loge9 T as in [4, Eq. (I), p. 651. Along the cut, make the substitution g(s) = Z(s + 1) log SW + w + lM( s1 with h(s) defined by (3.24). Since {(s + l)h(s) is analytic and single valued along the cut, the integral along the upper side of the cut cancels the integral along the lower side. From here, the argument in [4, p. 691 yields hi 1 s CUt = Il/2 ’ {(s + l)h x+2 G!S+ 0(x3 exp( - c(log x)““)). (3.25) The last equality follows from the choice of b given in [4, p. 701. This completes the proof of (3.1). Equation (3.2) follows by routine integration by parts. 200 PER.9 DIACONIS The argument for n G(n) is virtually the same, and is omitted. The arguments for Theorems 2, 4, and 5 are also virtually identical to the proof of Theorem 1. In each case, the identity for the Dirichlet series is used together with the inversion theorem (3.1) of Ayoub [4] as in (3.22). Then a change of variables is made to move the path of integration to re s = a with a > 1. Again the integrand differs from log S(s) by a bounded analytic function. Thus, Ayoub’s argument can be used to bound the integrals away from s = 1. Along the cut the argument given for Theorem 1 holds essentially word for word. Further details are omitted. 0 Proof of Theorems 3 and 6. It is useful to have another way to express the limiting relations to be proved. LEMMA 8. Let H(n) denote one of the functions A*(n), A(n), PI(n), or G(n). Then, the following two conditions are equivalent. As x + CO, J-I{n<x:H(n)<nY}l+L(y) forO<yIl. $I{n for0 <y lx:H(n) Ixy}l+L(y) (3.26) I 1. (3.27) Prooj Heuristically, Lemma 8 is true because most integers less than x are “large.” We argue that (3.27) implies (3.26): Clearly {n 5 x:H(n) I ny} c {n I x : H(n) I xy}. But, {n 2 x:H(n) 2 xy} = {n <x/log x : H(n) I x’} u {x/log x 5 n 2 x : H(n) I xy, H(n) > n’} u l- X log x I n I x : H(n) 2 ny I = s, u s, u s,. The set S, is negligible and S,c n<x:i XY (1% XY I H(n) I xy . I If (3.27) holds, this last set, and so S,, has density 0. Finally, S, differs from {n I x : H(n) I ny } by a set of density 0. This completes the proof that (3.27) implies (3.26). The proof of the reverse implication is similar and is omitted. 0 The results for A(n) and G(n) given in Theorem 6 imply Theorem 3; thus, we need only prove Theorem 6. Theorem 6, for the function P&n), was proved by de Bruijn 191. Nice discussion and simplified proofs are given by Billingsley [7] and Knuth and Trabb-Prado [Ill. The idea of the FAST FOURIER 201 TRANSFORM proof of Theorem 6 is to use the known results for P,(n) by showing that A(n), A*(n), and G(n) differ from P,(n) by a “small” amount. We now prove Theorem 6 for A(n). Recall that we write P,.(n) for the ith largest prime divisor of n. Let y E (0, 1) be fixed, and choosk‘an integer m so large that l/m <~~.~Observe that (3.28) Write Q,(S) for the proportion as the smallest set in (3.28) Q, 5 Pi(n) < 5 and of integers n I x such that n E S. Take S A(n) - i=l g P,(n) i=l 2 1 I I $ Qx( j, Pi(n) 25) - Qx( A(n) - s,Pi(n) >I). (3.29) Now, Markov’s inequality for positive random variables (for X > 0, P(X > c) I E(X)/c), together with the theorem of Alladi and Erdos quoted above, implies ,4(n) - 2 P,(n) > $ i=l Next we observe that the limiting distribution of CT:., P,(n) is the same as the limiting distribution of P,(n). This follows from the inclusions {n I x : P,(n) I xy} 1 ( n I x : $l 4(n) II {n I x : “P,(n) 5 xy) 5 xy} along with de Bruijn’s result which implies I{ n < x : mP,(n) XL(~) fory and m fixed as x + cc. Using the last observation, (3.29) in (3.28), completes the proof for A(n). The proof of Theorem 6 for A*(n) follows from the result just A(n) via Markov’s inequality together with the estimate (3.9) of Erdos for the mean of the difference A(n) - A*(n). 5 xy }I w (3.30) and proved for Alladi and 202 PERSI DIACONIS The last stage in the proof of Theorem 6 is to show that G(n) has the same limiting distribution as A*(n). The idea of the proof is to show that for almost all integers n, G(n) and A*(n) differ by at most a bounded amount. Toward this end, define u,(n) as the largest number such that p4”“)n. For integers m and 2 > 0, let S4 Z ={n:O<aP(n)<mfor21p~ZandOIaP(n)I It is straightforward lforp>Z}. to show that S,,. has density Since the product II,( 1 - l/p*) converges, a,,,,z can be made arbitrarily close to 1 by choosing Z and then m suitably large. Since G(n) 2 A*(n), Q, { G(n) 5 ny } I Q, (A *(n) I nY }. For the opposite inequality, let y be fixed and note that on S,,,,z 0 I G(n) - A*(n) I 2 pm G c(m, Z). PSZ Then, Qx{G(d 5 n’> 2 Qx{{S,n,z) n {G(n) 2 Q,{{S,,,.} n {A*(n) I ny - chz)}) 2 5 n’>) Q,{A*(n) I ny - c(m, Z>} - Q,{S,i.z>- I ny - c(m, Z)} - L(y) as x + cc. By choosing m and Z suitably large Q,{ Si, z} can be made arbitrarily small. This completes the proof of Theorem 6 for G(n). I-J Now, Q,(A*(n) 4.To SPLIT OR NOT TO SPLIT? The FFT of n numbers can be computed by using the chirp-z transform directly or by splitting n into prime factors, computing the FFT for each factor, and then putting the pieces back together. These two approaches were described in Section 2. Both approaches work in O(n log n) time. A more careful comparison will now be presented. As explained in Assumptions (2.7) and (2.8) of Section 2, a reasonable approximation for the number of operations used by the two algorithms is /K(n) for the chirp-z transform and /W(n) for the full factorization transform. Here /I > 0 is a constant which may be taken as 2 and, if T(n) 203 FAST FOURIER TRANSFORM is the smallest power of 2 larger than 2n - 4, C(n) and F(n) are defined by c (n ) = 37(n) log, r(n) = i n log, n ifn # 2k if n = 2k, F(n) = n x %. P”b p As a numerical (4.1) (4.2) example, when n = 100, C(100) = 3 . 256 * log, 256 = 6144, while F(100) = 100[2(C(2)/2) = 100[ 2(2/2) + 2(C(5)/5)] + 2(3 1 8 +3/5) 3 = 3080. Table I gives F(n) for 1000 I n I 1025. For each n in Table I (except 1024), C(n) = 67,584. The results in Table I suggest that the better algorithm speeds things up by a factor of approximately 5. Neither approach dominates. It is not clear how the approximations used to form Table I compare with actual running times. Some information about the two approaches can be gleaned from a comparison of averages. Recall that we write [x] for the smallest integer larger than x, lx] for the largest integer smaller than x, and {x} for the fractional part of x. THEOREM 7. Let C(n) and F(n) be defined by (4.1) and (4.2). For x > 0 define w(x) =; if {log,([x] 2-w3~2(L~J-*)) - 2)) = 0 otherwise. As x tends to infinity, - ’ 2 C(n) = 12w(x)( 1 - F)x X n5.x ; ~xF(n)= & log, x + O(X), x log, x + 0(x log log x). (4.3) (4.4) The factor w(x) oscillates boundedly between 1 and i so that 12w(x)(l - $w(x)) oscillates between 4 and 4.5, while 3/lag 2 = 4.33. 204 PER.91 DIACONIS TABLE I Number of Operations Using Full Factorization of n With j3 = 1” n loo0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I5 16 17 I8 19 20 21 22 23 24 25 ODirect F(n) Factorization 2’5’ 7. 11. 13 2 . 3 . 167 17 . 59 22. 251 3 .5 .67 2 . 503 19 . 53 24. 32. 7 1009 2 . 5 . 101 3 . 337 22. 11 .23 1013 2.3. l32 5 .7 I 29 2’. 127 32. 113 2 .509 1019 22.3.5.17 1021 2 .7 ’ 73 3. 11 .31 2’0 52.41 use of the chirp-z 46,200 108,096 85,950 74,016 57,304 108,642 62,446 112,128 35.712 122,880 76,994 94,182 96,872 122,880 40,482 82,776 52,200 59,364 62,458 122,880 47,568 122,880 115,070 84,702 10,240 96,720 idea uses 67,584 operations. Proof of Theorem 7. To prove (4.3) it is easiest to approximate the distribution of C(n) and then calculate the mean from the distribution. The largest value C(n) can take as n ranges in 1 i n I x is determined by J = J(X) = [log2(2LxJ - 4)1; C(n) takes values 3J 2J, 3(5 - l)r-‘, ... . When n = 2k, C(n) = 2kk. Since this happens only once for each value of k, these values will not affect the computation. That is, ( I/x)X,,~,,,~~ x,k2k = @log x) and the error in (4.3) is O(x). Excepting powers of2, C(n) = 3k2k for 2k-2 + 2 < n I 2k-’ + 2. Write f(x) = f(x) - 2 - log, x = - 1 if x =2k+2 = - {1og*(p1 - 2)) + 0(1/x) otherwise. FAST FOURIER 205 TRANSFORM Then and similarly, for 1 I k I J, -$(n 5. x : C(n) The error term is uniform 3J2J ( 1 - w(x) +3(5 = 3(5 - k)2-‘-k)l in k. The mean of C(n) is + 0 + +3(5( )I - 2)Y-‘(+) = 3J2.‘( 1 - ;w(x)) = 3 * 4w(x)(l +oi ( x). = 9 1)2-‘-l + O(i)) +. ( w(x) 2+ 0 + ( )) . . +O(logx) + O(x) - :w(x))x log, x + O(x). This proves (4.3). To prove (4.4) write We now argue from (4.5) to the approximation nTx F(n) =$ c +-) P<X p + 0(x*). (4.6) To derive (4.6), we need the prime number theorem in the crude form z Pa+. log p = O(x) (see [4, Theorem 6.21, for example). We also use C(p) = O(p logp). First, Lx/p”J’ = (x/p”)* + 0(x/p”), so 2 C(P) Prr -T[x/pa]’ POIX P =; C(P) 2 :+0xX p’5.x P ( pa0 C(P) . P 1 (4.7) 206 PERSI DIACONIS But z y = o( .2X logp) = O(x). (4.8) P”lX Using (4.8) in (4.7) and again for the term involving side of (4.5) gives [x/p”J on the right (4.9) The proof of (4.6) is completed Next we need the following by the bound: bound: For w < z, as w -+ co, w,z<z+= (T&i - &) +o(wlig2w). t4-lo) To prove (4.10) write a(t) for the number of primes I t. Then, Using the prime number theorem in the form 7r(t) =- t ( t 1 t2 log i -n(t) +o z c----+0 1 w z log z also, 2 2-= r(r) dt sw t3 2 z s H’ This completes the proof of (4.10). ~ (log t)2 ’ I w log w 1 . ( w log2 w I3 207 FAST FOURIER TRANSFORM Finally, we complete our argument = [log, x]. We have +3(L from (4.6) to (4.4). Write L = L(x) 2 + 2)2L+= 2L+2<pjx The approximation 1 + O(1). (4.11) P2 (4.10) gives x 2L+2<p ‘=o& ( P2 IX and -= P2 t 2k-=+2<p12k-‘+2 2k-2 log 2k-2 - ’ - 2) log2 = 2k-‘(k ’ 2k-l ]+o(&) 2&-' log [ 1 + o(i)]. Using these bounds in (4.11) leads to Ix psx -C(P) = & y(1 + o(i)) + O(1) p2 6 log, x = ___ + O(log log x). log2 Using this in (4.6) completes the proof of Theorem 7. •J ACKNOWLEDGMENTS David Freedman, Andrew Odlyzko, Lawrence Rabiner, Don Rose, Larry Sbepp, and Charles Stein all helped at crucial times. I also thank a patient, careful referee. REFERENCES 1. A. AHO, J. HOPCRAFT, AND J. ULLMAN,“ The Design and Analysis of Computer Algorithms,” Addison-Wesley, Reading, Mass., 1974. 2. A. AHO, K. STEIOLI'IZ, AND J. ULLMAN, Evaluating polynomials at fixed sets of points, SIAM J. Computers 3. K. ALLADI 275-294. 4 (1975), 533-539. AND P. Et&s., On an additive arithmetic function, Pacific J. Math. 71 (1977), 208 PERSI DIACONIS 4. R. 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