A new parallel algorithm for the solution of a general linear recurrence is described.  Its relation to the work of Kogge and Stone is discussed

Heller, D. E.

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AD/A-002 248 
ON THE EFFICIENT COMPUTATION OF RECURRENCE RELATIONS 
Don E. H e l ! e r  
C a r n e g i e - M e l l o n  U n i v e r s i t y  
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P. AuTMORfa) 
Don E. I l c  1 ler 
B. PfRFORutr rG ~ ~ ( G A N ~ Z A T I O ? ~  MAME AND AODRESS 
Carnegic-Xcllon University 
Department of Computer Science 
Pittsbutgh. PA 15213 
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Office of Naval Research 
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Approved for Public Release; Distribution unlimited. 
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J m e  13, 1974 
13. UUMOER OC PAGES 7 
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B. SUPPLEMENTARY NOTES 
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NATIONAL TECHNICAL 
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Spnngldd. VA. 21151 
D. & e s T r l i C T  (Con~Inu* on rot r r s *  alde If nrcoammry and td.nlrly bv blocb number) 
A new parallel a l w r i  thm for t h e  solution of  a ~ c n c r a l  linear recurrence i s  
dcscribcd. TCS rc1.1tion to the work of Kol:):c and Stone is discussed. 
afE EFFICIENT COWUTXTION 3F RE(XRRENCB RELATIONS 
Don E. Heller 
June 13, 1974 
Catnegie-Nellor. Universi tv 
Department of Computer Science 
Pittsburgh, Pa. 15213 
This paper was prepared i n  part while the author w2s i n  residence a t  
:CASE, KXSA Langley Research Center under NASA Grant NGR47-102-001 and 
in  part a t  Carnegie-Mellon University under Sational Science Foundation 
Grant GJ32111 and Office of Naval Research Contract Ny14-67-A-031b-0010, 
NR 044-422. 
OW THE E F F I C I E N T  COMPIJTAT ION OF RECURRENCE RELATIONS 
Recently much progress has been made i n  the formulation o f  pa ra l le l  
a l g o r f t h s  wkich clnpute the terms of a sequence (yi) defined by 
Yo 4(ven, 
(1 1 
Yf = fi(yo. Yl,.*..Yi-1). i = 1 ,.... N. 
The germinal po in t  o f  t h i s  work i s  the now well-known "log-sum" algorithm 
N 
which computes Ti-l - ai i n  f loq2~' ]  para l le l  addit ion steps. given F121 
processors, Here the underlying recurrence i s  
Yo = 0 
y = yi1 + a i = 1 ...., N; 
yN i s  the desired resul t .  
Tm, apparently d i s t i n c t  generalizations o f  the log-sum algorithm 
have appeared. Kogge and Stone [l] have considered the case 
where f i s  associative, g d is t r ibutes over f, and there i s  a function h 
such that  g(x,g(y,z)) = g(h(x,y),z). Seemingly res t r i c ted  t o  f i r s t  order 
recurrences, by a sui tab1 e mapping tnth order recurrences are a1 SG treated. 
Hel ler [2] has studied the case 
Thfs problem i s  equivalent t o  the solut ioa o f  a lower t r iangular  l i near  
system o f  equations. I n  t h i s  note we give an improved para l le l  algorithm 
f o r  (3) and show a re lat ionship between the two generalizations. 
Rewrite (3) as (I-L)y=b, where L i s  a s t r i c t l y  lower t r iangular  
matrix, and I i s  the ident i ty .  y and h are (N + 1)-vectors. Since 
LN+l 
= 0, 
N (I-L)" = (I+LCL~+* .+L ) 
= ( I + L ~  ( 1 + ~ @  ) . . . ( I+L) 
where 2 m 1 ~ < ~ @ 1 .  Thus we have the a l p r i t hm :  
+or i - 0  step 1 u n t i l  m - l  do 
-C __C -
The algorithm i s  sequential i n  i, a d  wi th in the braces operations are 
performed concurrently. When completed, we have the desired y, and 
(I-L)-' i s  stored i n  L i .  L I  may now be used t o  compute y '  given h'.  
3 I t  4s easi ly  shown that, w i th  o(N ) processors, the calculat ion may 
2 be done i n  m + 3m + 1 para l le l  steps o f  addit ion and mu1 t i p l i ca t i on .  
(We use the fac t  that  matr ix products may be computed i n  logarithmic 
time wi th  su f f i c ien t l y  many processors. ) The previous resu l t  required 
4 O(N ) prccersars and m2 + 4n + 2 operation steps. 
We now turn t o  the Kogge - Stone resul ts.  Rcwrite (2 )  as 
Here g i s  replaced by the binary operation 61, and f by 8. Assume tha t  
@ i s  associative, Q d is t r ibutes over 8, and there i s  a Q' such that  
a Q (b O c) = (a 0' b) Q c, 1-et abe a symbol d i s t i n c t  from a l l  others, 
and def ineae x = x @ a=  x, aQ x = x O a  a f o r  a l l  x, Definean 
operator L on (N+l)-vectors by 
Then y = Ly d b by (2 '  ). It i s  observed t h a t  L i s  an add i t i ve  operator 
N+1 
since 0 d is t r i bu tes  over 8 and by the d e f i n i t i o n  o f  a. Yoreover, L =a, 
i i 
since f o r  any z, and i = 1 ,..., N+1, (L z ) ~  = a and for  l < j < i ,  (L z ) ~  = 
i -1 i - 1  (L(L z )J j  = aj O (L z ) ~ - , =  a. Qa=a. Therefore, J 
2 2 Since L~ = (L )L = L(L ), 0' behaves as an associat ive operation, and SO 
= (a. P' aj-, P'"'0' aj-2i+l) D yj-2?,2i,j,N. 
Similarly, 3 - - 
2i+1 
and the "coeff ic ients" o f  L may be computed from the "coefficients" 
21 of L i n  one 0' operation step. Thus an a lgo r i  tb s im i la r  t o  the previous 
one nay be used t o  canpute y. If the operator ( L ~  .@I) i s  not fomed. 
the computation time i s  0(log2N) w i th  O(N) processors. In fac t ,  i f  
N y' Ly' 6 b', i t  i s  less e f f i c i c ? t  t o  d i r e c t ? y  apply (L e"'8 1) than 
t o  use the above method. 
3 
The general recurrence (1) m y  be w r i t t e n  as y = Lly, where L1 i s  
a s t r i c t l y  lower t r i angu la r  operatcr i n  the sense that,  f o r  any z, 
(Llz)l i s  independent of zi , zi+l ,. . . ,z By an induct ion argument N' 
L,~" i s  a constant operator, and sc the so lu t i on  may be found by 
y = L:~+'z f o r  any r .  The special  cases (2)  2nd (3) a l low the simple 
computation of the powers o f  L1 when L,z = Lz @ b, and L i s  l inear .  
Kung [31 has shown tha t  f o r  non-l inear recurrences, i t i s  no t  possible, 
i n  general, t o  decrease the cowputstion t ime by more than a constant 
f ac to r  by the use o f  para1 l e l  ism. 
REFERENCES 
1, Kogge, P, M. and Stone, H, S., "A Pa ra l l e l  Algorithm f o r  the E f f i c i e n t  
Solut ion o f  a General Class o f  Recurrence Equations." IEEE  Trans. on 
Computers, C-22 (i 373). pp. 786-793. 
2, Hel ler,  !I,, "A Determinant Theorem w i t h  Appl icat ions t o  Pa ra l l e l  
, 
Algorithms,' To appear, S IAM Journal Num, Anal,, 1974. 
3. Kung, H. T., "New Algorithms and Lower Rounds fo r  the Para l le l  
Evaluation o f  Certain Rational Expressions. " Proceedings, ACM S-mpos ium 
on Theory o f  Computing, 1974. 


On the efficient computation of recurrence relations

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