D. Knuth, and A. Yao. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sct. USA, volume 72, No 1, pages 4720-472, Dec. 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [10] and finally in distribution by Hensley [20] who proved in 1994 that the Euclidean algorithm has Gaussian behaviour; see Knuth s and Shallit s vivid accounts [26, 39] The Centered algorithm (K) has been analysed by Rieger [35] The Subtractive algorithm (T ) was studied by Knuth and Yao [51], and Vardi [49] analysed the By Excess Algorithm (L) by comparing it to the Subtractive Algorithm. The methods employed so 1 far are rather disparate, and their applicability to new situations is somewhat unclear. Here, we design a unifying framework that additionally provides new results on ....

Yao, A.C., and Knuth, D.E. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975) pp 4720-4722. 31

.... bit complexity of the Subtractive Algorithm (T ) on the set of valid inputs of denominator less than N is asymptotically of log cubed order: e CN (T ) CN (T ) 2 log 2 2 2 log 3 2 N The first assertion has been proven (with direct combinatorial methods) by Yao and Knuth in 1975 [41]. The second and third assertions provide new results. 5.4. Two Pseudo Euclidean Algorithms related to random continued fractions. The general framework that we propose here can be easily adapted to other Euclidean Algorithms, the ones that we have called Pseudo Euclidean in previous papers [37, ....

A.C. Yao, D.E. Knuth, Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975), 4720-4722.

....that M(k; i) a 1 X l=0 k l k l 1 ; 1) where k 0 = k; k 1 = i and k 2 ; k 3 ; k a is the sequence of nonzero remainders in the Euclid s algorithm for the gcd of k 0 and k 1 . This claim will immediately imply the theorem, as Knuth mentions [23] exercise 4.5.3. No. 35 and [33]) that summing up the right hand sides of (1) for i = 1; 2; 3; k 1 yields 6 2 k(ln k) 2 O(k ln k(ln ln k) 2 ) Proof of the claim. We show that if k = ui r; 0 r i then M(k; i) k i M(i; r) 2) 6 where we assume by the de nition that M(x; 0) 0, for x 1: ....

Yao, A., Knuth, D.: Analysis of the subtractive algorithm for greatest common divisors. Proceedings of National Academy of Sciences, USA, 72:4720-4722 (1975)

....that M(k; i) a 1 X l=0 k l k l 1 ; 1) where k 0 = k; k 1 = i and k 2 ; k 3 ; k a is the sequence of nonzero remainders in the Euclid s algorithm for the gcd of k 0 and k 1 . This claim will immediately imply the theorem, as Knuth mentions [25] exercise 4.5.3. No. 35 and [36]) that summing up the right hand sides of (1) for i = 1; 2; 3; k 1 yields 6 2 k(ln k) 2 O(k ln k(ln ln k) 2 ) Proof of the claim. We show that if k = ui r; 0 r i then M(k; i) k i M(i; r) 2) where we assume by the de nition that M(x; 0) 0, for x 1: ....

Yao, A., Knuth, D.: Analysis of the subtractive algorithm for greatest common divisors. Proceedings of National Academy of Sciences, USA, 72:4720-4722 (1975) 16

.... 1. If the continuous case models the discrete one, then Kinchin s result implies that the sum of continued fraction coecients S(p=q) should satisfy S(p=q) 12 2 log q log log q ; for almost all p q N . This has not yet been proved. In fact, there are some subtleties as Knuth and Yao [51] showed that the average value of S(p=q) taken over all 0 p q is 1 q X 0 p q S(p=q) 6 2 (log q) 2 : What is going on is that a small number of large coecients are in ating the average. The fact that most values of S(p=q) are much smaller than the average was shown in [82] For 1 ....

D.E. Knuth and A. Yao, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. 72 (1975), 4720-4722.

....number of subtractive steps in the following algorithm for finding the greatest common divisor of m and n by mutual subtraction: Replace the larger number by the difference of the two numbers until both are equal; then the common value is the gcd. Yao and Knuth proved the following beautiful result[11]: X mn s(m; n) 6 2 n(log n) 2 O(n log n(log log n) 2 ) This implies that for any n, the average number of steps the above subtractive algorithm takes for any pair of integers m n is asymptotically 6 2 (log n) 2 . We want to show that the above estimate is also valid for ....

A. C. Yao and D. E. Knuth, "Analysis of the subtractive algorithm for greatest common divisors", Proc. Nat. Acad. Sci USA 72:12 (1975), 4720--4722. 14

.... average bit complexity of the Subtractive Algorithm (T) on the set of valid inputs of denominator less than N is asymptotically of log cubed order: e CN (T ) CN (T ) 2 log 2 2 2 log 3 2 N The first assertion has been proven (with direct combinatorial methods) by Yao and Knuth in 1975 [38]. The second and third assertions provide new results. 5.4. Three other Euclidean Algorithms related to random continued fractions. The general framework that we propose here can be easily adapted to other Euclidean Algorithms, the ones that we have called fast in a previous paper [35] for ....

Yao, A.C., and Knuth, D.E. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975) pp 4720-4722. 29

.... in Section 2, and are called Even (E) Odd (O) Ordinary (U) and Centered (C) The complexity of the first four algorithms is now known: The two classical algorithms (G) and (K) have been analysed by Heilbronn, Dixon and Rieger [16] The Subtractive algorithm (T) was studied by Knuth and Yao [25], and Vardi [24] analysed the By Excess Algorithm (L) by comparing it to the Subtractive Algorithm. The methods used are rather disparate, and their applicability to new situations is somewhat unclear. Here, we design a unifying framework that also provides new results on the distribution of ....

Yao, A.C., and Knuth, D.E. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975) pp 4720-4722.

.... (T) and Binary (B) The complexity of these algorithms (in terms of the number of arithmetical operations to be performed) is now well known: The two classical algorithms (G) and (K) have been analysed by Heilbronn, Dixon and Rieger [18] The Subtractive algorithm (T) was studied by Knuth and Yao [27], and Vardi [26] analysed the By Excess Algorithm (L) by comparing it to the Subtractive Algorithm. Brent [3] and Vall ee [24] have analysed the Binary algorithm. Methods. Our approach is a refinement of some methods that have been already used for instance in [4, 9, 24, 25] it consists in ....

Yao, A.C., and Knuth, D.E. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975) pp 4720-4722. 14

....In any case, n m = i j i = 1 j i . If i j, write j i = a 0 ; a k ] then n m = a 0 1; a k ] so s(n; m) s(j; i) 1. If i j, write i j = a 0 ; a l ] then n m = 1 1= i=j) 1; a 0 ; a l ] so s(n; m) s(i; j) 1. Yao and Knuth [YK75] showed that, for any positive integers m n, n X m=1 s(n; m) 6n(ln n) 2 2 O(n log n(log log n) 2 ) Theta(n(log n) 2 ) Let jX j = l, and a(X) b(X) form the major row of X . Therefore, P X;b(X) n s(b(X) a(X) Theta(n(log n) 2 ) Then P X;b(X) n jI(X)j = Theta(n(log ....

A. Yao and D. Knuth. Analysis of the subtractive algorithm for greatest common divisors. In Proceedings of National Academy of Sciences, USA, 72:4720--4722, 1975. 61

....l 1; a 0 ] so s(n; m) s(j; i) 1. If j i and i j = a l ; a 0 ] then n m = 1 1 i j = 1; a l ; a 0 ] so s(n; m) s(i; j) 1. 2 Lemma 6.12 jI(X)j is AP on PM. Proof Let s(n; m) be as in Lemma 6.11. We use the following strong result of Yao and Knuth [YK]: P m=n m=1 s(n; m) 6n= 2 ) ln n) 2 O(n(log n) log log n) 2 ) Theta(n(log n) 2 ) Let X be a matrix of size l 0, and let a(X) b(X) form the major row of the matrix X. Then P X : b(X) n s(b(X) a(X) Theta(n(log n) 2 ) By Lemma 6.11 P b(X) n jI(X)j = Theta(n(log n) ....

Andrew C. Yao and Donald E. Knuth, "Analysis of the subtractive algorithm for greatest common divisors", Proc. Nat. Acad. Sci USA 72:12 (1975), 4720-- 4722.

....j Gamma1 mod q and 1 d q. Then, s(a; q) 8 : 1 12 i a d q d 1 Gamma d 2 Sigma Delta Delta Delta Gamma ( Gamma1) d j Gamma 1 4 if is odd 1 12 i a Gammad q d 1 Gamma d 2 Sigma Delta Delta Delta Gamma ( Gamma1) d j if is even : Lemma 5. Knuth Yao[7]) Let N(a; q) be the sum of the partial quotients of the continued fraction expansion of a=q. Then X 1aq (a;q) 1 N(a; q) q log 2 q: Knuth and Yao actually give an asymptotic formula for this sum, but for our purposes the upper bound suffices. 4 J. B. CONREY ERIC FRANSEN ROBERT KLEIN ....

D. E. Knuth and A. C. C. Yao, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 4720--4722.

.... into account the steps with exchanges, so that the double generating function S(s; w) associated to the parameter number of exchanges can be expressed now as S(w; s) wG s (I Gamma wG s ) Gamma1 [1] 0) We then obtain an easy alternative proof of classical results of Yao and Knuth [48], Heilbronn [18] and Dixon [9] Theorem 5. The average number TN or e TN of subtractions performed by the subtractive Algorithm on the set the set f(u; v) 0 u v Ng; or on the set f(u; v) 0 u v N; gcd(u; v) 1g satisfies TN e TN 6 2 log 2 N: The average number GN or e GN ....

Yao, A.C., and Knuth, D.E. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sc. USA 72 (1975) pp 4720-4722.

....For instance, the ordinary Euclidean integer gcd algorithm can be understood in terms of a basis reduction algorithm on 2 Theta 2 integer matrices, where the reducing operations are elements of the modular group in the form (i) above. This connection allows us to apply a result of Yao and Knuth [17] concerning the integer gcd algorithm in our analysis. Some algorithms of Schonhage [14, 15] can be best understood in light of the modular group. A recent paper by Yap [18] is concerned with the modular group and its connection with lattice basis reduction algorithms. The basis reduction ....

....(ii) In x5 we show that the process of converting an input instance from representation (i) to representation (iii) and then executing the algorithm of x4 on the resulting data gives an average case polynomial time algorithm. This part of the argument relies on an estimate of Yao and Knuth [17]. The same techniques also handle other related groups such as SL 2 (Z) or the congruence subgroups of Gamma. We do not treat these cases in this paper. 2 Representations of Gamma To understand this work, one must first understand the relationships among the different representations (i) v) ....

[Article contains additional citation context not shown here]

A. C. Yao and D. E. Knuth, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. USA, 72 (1975), pp. 4720--4722.

No context found.

D. Knuth, and A. Yao. Analysis of the subtractive algorithm for greatest common divisors. Proc. Nat. Acad. Sct. USA, volume 72, No 1, pages 4720-472, Dec. 1975.

No context found.

Andrew C. Yao and Donald E. Knuth, "Analysis of the subtractive algorithm for greatest common divisors", Proc. Nat. Acad. Sci USA 72:12 (1975), 4720-- 4722.

No context found.

Yao, A., Knuth, D.: Analysis of the subtractive algorithm for greatest common divisors. Proceedings of National Academy of Sciences, USA, 72:4720-4722 (1975) 16

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC