UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
UIUCDCS-R -7^-675 
 
 /yuLZ>4 
 
 K 
 
 August 1974 
 
 ON THE EVALUATION OF POWERS 
 
 by 
 
 Andrew Chi -Chin Yao 
 
UIUCDCS-R -7U-675 
 
 ON THE EVALUATION OF POWERS 
 
 by 
 
 Andrew Chi -Chin Yao 
 
 August 197^ 
 
 Department of Computer Science 
 
 University of Illinois at Urbana -Champaign 
 
 Urbana, Illinois 6l801 
 
 Research supported by NSF GJ-41538. 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/onevaluationofpo675yaoa 
 
1. Introduction 
 
 An addition chain (of length r) is a sequence of r+1 
 integers a , a ,a , ...,a such that (i) a = 1 and (ii) for each i, 
 a. = a. + a, for some j ^ k < i. It is well known [k] that, for any 
 r and any set of integers (n ,n , . ..,n }, there exists an addition 
 
 chain of length r which contains the values n^n , ...,n if and only if 
 
 ni n.^ n-p 
 
 there exists a procedure which, for any input x, computes [x , x , ...,x } 
 
 in r operations using only multiplications. A theorem by Brauer [1] [k-] 
 
 states that, for any n, there exists an addition chain of length 
 
 Sgn + O(0gn/ ggitn) which contains the value n. This then implies the 
 
 existence of a corresponding procedure to compute x in 2gn + 0(£gn/£g^n) 
 
 multiplications. Furthermore, it was shown by Erdbs [2] [k] that the 
 
 above result is nearly the best possible. In an open problem posed in 
 
 Knuth [3], it is asked if there are fast procedures to compute 
 
 n l n 2 n p 
 (x ,x ,...,x *} for p ^ 2. This problem cannot be solved by a direct 
 
 extension of the technique used by Brauer in the proof of his theorem. 
 
 In this paper we show that, for any positive integers 
 
 n, ,n p , ...,n there exists a procedure using only multiplications which, 
 
 n l n 2 n v 
 for any input x, computes (x ,x ,...,x ^} in ^N + constant x 
 
 P 
 
 Z [8gn./ 8g£$(n. +2) ] multiplications where N = max{n.}. This gives a 
 i=1 1 1 ± ' 1 
 
 solution to Knuth' s problem and leads to a corresponding theorem on 
 addition chains which generalizes Brauer' s theorem mentioned earlier. 
 
 2. Definition 
 
 Let e., UiSJ, and f ., 1 , 3* q, be positive integers. 
 
 e l e -n ^1 -^a 
 
 We shall say that (x , ...,x fj is computable from (x ,...,x H ) in 
 
 r multiplications (r ^ 0) if there exists a set of r positive integers, 
 
 &g. is logarithm to the base 2. 
 
-2- 
 
 ( f n ..... f } , such that 
 1 q+1' ' q+r J ' 
 
 i ) for all i = q+1, . . . , q+r, 
 
 f4 f4 fv 
 
 x = x d • x " for some j ^ k < i. 
 ii) {x \ . ..,x p } c {x , ...,x q+r }. 
 
 Since the exponents are additive when two powers of x are 
 multiplied, the above definition corresponds to a natural generalization 
 
 of the definition of addition chains (cf. Section l). The exponents 
 
 f-, f 
 appearing in {x , ...,x Q) correspond to a set of numbers initially 
 
 available in the chain, as opposed to a single number, 1, in the earlier 
 
 definition. 
 
 n-, n„ 
 3. The Computation of (x , ...,x ^} 
 
 The following lemma is well-known [k]. 
 
 Lemma 1 For any integer i > 0, {y } is computable from {y} in less than 
 
 2[^iJ multiplications. 
 
 Proof Let the binary representation of i be 
 
 v 
 i = Z b. ' 2 J (1) 
 
 where v = [^ij. Then, 
 
 y 1 = n y 2d . (2) 
 
 b.=l 
 
 3 
 
 2^8 2 V 
 Thus, we first compute y ,y ,y ,...,y sequentially in v multiplications. 
 
 Then we can compute y by equation (2) in no more than v-1 multiplications, 
 
 The total number of multiplications is no greater than 2v-l. □ 
 
 Theorem 2 For any integers m,n where < m ^ n, {x } is computable from 
 
 2^8 oW n \ i / n 
 
 (x,x ,x ,x ,...,xr } in less than c^n/^#?(n+2) multiplications for 
 
 some constant c. 
 
-3- 
 Proof Assume n^ k. Define the following quantities: 
 
 k = f(^n)/2j (3) 
 
 D = 2 k 00 
 
 t = [% D nj (5) 
 
 Let the D-ary representation for m be 
 
 v3 
 
 m = E a.D J where g a. g D-l for j = 0,1, . . .,t . (6) 
 
 We partition the set of integers (0, 1, ...,t} into D disjoint subsets 
 S(0),S(1),...,S(D-1) by letting 
 
 S(i) = U|a = i} for i = 0,1, ...,D-1. 
 
 it follows from (6) that 
 
 D-l D-l 
 
 m = S i • [ S Tr] = Z i • m. (7) 
 
 1=1 teS(i) 1=1 x 
 
 where 
 
 m. = E D^ (8) 
 
 1 teS(i) 
 
 From (7) and (8), we obtain the following two equations: 
 
 x = n x for i = 1,2, ...,D-1 (9) 
 
 teS(i) 
 
 D" 1 m, i 
 x m = n (x^) (10) 
 
 1=1 
 
 rj-t 2 4 
 
 Since all the sr in (9) are available in the set {x, x , x , 
 
 8 2^ n J, . . * * L m 
 
 x ,...,x 1 ^ we can construct a procedure to compute x as follows: 
 
 Step 1: For i = 1,2,..., D-l do the following: 
 
 (a) Compute x" from (9) in less than |S(i)| multiplications. 
 
-k- 
 
 m_. 
 
 (b) Compute (x - 1 ) in less than 2[^iJ multiplications 
 (by Lemma l) . 
 Step 2: Compute x m from (10) in D-2 multiplications. 
 
 Let M be the total number of multiplications in the above procedure. 
 
 Then, 
 
 D-l 
 M < Z (|s(i)| + 2[^iJ) + D-2 
 i=l 
 
 D-l 
 =£ 2 |S(i)| + 2(D-l)%(D-l) +D-2 (ll) 
 
 i=l 
 
 Noting that the S(i)'s form a partition of the set {0, 1, ...,t}, we 
 
 obtain from (ll) that, 
 
 M < t + 1 + 2(D-l)^. (D-l) + D -2 (12) 
 
 (12), together with equations (3), (k-) and (5), implies 
 
 M < 2(%n/^^.n) + 1 + hi^n) 1 ' 2 Sg. Qo,n + 2(%n) 1 / 2 . (13) 
 
 It follows from (13) that there exists a constant c such that 
 
 M < c^n/%.^(n+2) (1*0 
 
 Thus the theorem is true if ni k . Obviously we can choose c so that 
 the theorem is also true for n = 1,2,3. □ 
 
 Theorem 3 For any set of positive integers (n ,n , ...,n }, 
 
 n l ^2 n_ 
 (x , x , ...,x^>} is computable from input {x} in less than 
 
 P 
 £g.~N + c £ [2gn./0g. 2g. (n. +2) ] multiplications for some constant c, 
 i=l 
 
 where N = maxfn.l . 
 i 1 
 
-5- 
 
 n l n 2 "-p 
 Corollary {x , x ,...,x } is computable from {x} in less than 
 
 gg.~N + cp^N/^^(N+2) multiplications. 
 
 2 4 8 
 Proof of Theorem 3 and Corollary First we compute {x, x , x , x , •••, 
 
 x } from input x in [ ^Nj multiplications. For each i, according 
 
 n n ,2k '?[^9- n n I 
 
 to Theorem 2, x x is computable from [x,x ,x , ...,x c - 1 - } in 
 
 ... r 2 k p|%N| 
 
 c 0gn./0g. 2g. (n. +2 J multiplications, and also from {x, x , x , ...,x J ] in 
 
 c^N/^^(N+2) multiplications for some constant c. The theorem and 
 
 corollary then follow immediately. □ 
 
 In terms of addition chains, Theorem 3 and its 
 
 corollary give the following generalization of Bauer's theorem [1] [k]. 
 
 Theorem k- For any positive integers n,,n p , ...,n , there exists an 
 
 P 
 addition chain of length less than ^N + c £ Sgn./Pg. 2/g. (n. +2) containing 
 
 1=1 X X 
 
 the values n,,n„,...,n for some constant c, where N = maxfn.}. 
 1' 2' ' p ' j_ i 
 
 Corollary For positive integers n..,n , ...,n , there exists an addition 
 chain of length less than ^N + cp^N/^^(N+2) containing n,,n p , ...,n . 
 
 k. Conclusion 
 
 n l n 2 n p 
 
 We have shown that {x , x ,...,x } can be computed in 
 
 ^N + cp^N/^^(N+2) multiplications for input x where N = max{n.} 
 
 i 1 
 
 and c is a constant. On the other hand, it is well-known that to evaluate 
 
 nn no rw, 
 {x , x , ...,x V) by arithmetic operations, at least Sg.~N operations are 
 
 n l n ? n D 
 necessary. Thus, our procedures for evaluating {x ,x ,...,x *} are 
 
 nearly the best possible when p « ^^(N+2). It remains an interesting 
 
 n, no ru 
 
 open problem to determine the complexity of computing (x ± , x , ...,x P] 
 
 for general p. 
 
-6- 
 
 References 
 
 [1] Brauer, A., Bull. AMS k-5 (1939), pp. 736-739. 
 
 [2] Erdbs, P., Acta Arithmetica 6 (i960), pp. 77-81. 
 
 [3] Knuth, D. E., The Art of Computer Programming, Vol. 2, 
 Add! son -Wesley, 1969, Section k.6.3 Ex. 32. 
 
 [4] Knuth, D. E., The Art of Computer Programming, Vol. 2, 
 Addis on -Wesley, I969, pp. 398-I+I8. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-7^-675 
 
 3. Recipient's Accession No. 
 
 4. Title and Subtitle 
 
 ON THE EVALUATION OF POWERS 
 
 5. Report Date 
 
 August 197^ 
 
 7. Author(s) 
 
 Andrew Chi -Chin Yao 
 
 8. Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urbana -Champaign 
 
 Urb'ana, Illinois 618OI 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract/Grant No. 
 
 NSF GJ-^1538 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D . C . 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 It is shown that for any set of positive integers {n_,n , ...,n ), there exists 
 
 n-, n P n^ L 2 P 
 
 a procedure which computes (x ,x , ...,x *} for any input x in less than 
 
 P 
 
 Sg~N + c s [0gn./(!g 2g (n.+2) ] multiplications for some constant c, where N = max{n.}, 
 i=l - 1 x i x 
 
 This gives a solution to an open problem in Knuth [3] and generalizes Brauer's 
 theorem on addition chains. 
 
 17. Key Words and Document Analysis. 17o. Descriptors 
 
 addition chains, Brauer's theorem 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 FORM NTIS-35 (10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 
 Page 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7 1 
 
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