On compiling algorithms for arithmetic expressions

Abstract

This paper deals with algorithms concerning arithmetic expressions used in a FORTRAN IV compiler for a HITAC-5020 computer having n accumulators. The algorithms generate an object code which minimizes the frequency of storing and recovering the partial results of the arithmetic expressions in cases where there are several accumulators.

The reduction of binary trees by means of an input-restricted deque

“…17, n° 3,1983 254 R-KEMP (B) Convert the labelled tree into a program n which évaluâtes T. The following algorithm is of this type and represents an optimal strategy (with respect to the number of variables appearing in II) for the réduction of binary trees ( [9,10]). …”

“…The algorithm uses an auxiliary store H, an input-restricted deque ( [8]; p. 238) of length keN, a counter containing the current position of the input pointer and an auxiliary cell Z. The deque list symbols are triples (a, b 9 1) e (IU L) x (ƒ U L) x { 0,1} representing the subtree Tg. If the réduction of the substree T% has required an auxiliary cell in H, then t-l; otherwise t = 0.…”

“…Using the approximation (8 a), qi^iri) is exponentially small and therefore negligible, being O (n~m) for all m>0. A comparison of q(n) with (9) shows that q(n) = h o (n). Thus using theorem 5 (case fe^n 0 -5 " 6 , 0<6^1/2), we obtain:…”

“…We consider the sum given in (9). Choosing s >0 with e<8, we obtain immediately R(k + l)>n 05+e for sufficient large n and therefore by the approximation (Sb):…”

“…Next, if the algorithm is in state $, then we do not know the left brother of y. In this case, the input pointer goes back to the first position of PO (T) 9 the triple (x,y 9 1) is removed from the deque, (x 9 y) is stored in an empty auxiliary cell and the algorithm goes in state y.…”