A direction-independent algorithm for determining the forward
and backward compute point for a term or subscript during
compilation
By Joseph T. Ryan'
This paper describes a second-scan procedure for determining the forward and backward compute
points for all of the elements of subscripted variables. Here, the forward and backward compute
points are the earliest time at which a subscript can be computed and the latest point at which it
can still be used. The procedure is independent of the direction of the second scan. (The first
scan is assumed to be a forward read of the source program.) The subscripted variables may
occur at any level within the nesting of DO-type loops. The optimality and collapsibility of loops
is unaffected by the algorithm.
Definitions
In order to clarify the terminology used in this paper,
the following definitions are made:
Until recently most compilers interpreted the sourcelanguage input file just to sequence and categorize the
source statements which were to be stored in memory
or to be put onto an intermediate file for later use in the
compilation. Each source statement was carefully dissected in accordance with the appropriate syntactical
and orthographical rules but little or no attention was
given to the flow analysis of the source program until
after the source file was entirely read. Some more recent
compilers have been advertized as "one-pass" compilers,
but these, for the most part, have been inefficient because
the program flow analysis and the compute point determination for subscripted variables have not been oriented
in a mutually sequential fashion. Several subpasses
were required to orient the two operations and this
increased rather than decreased the compilation time.
Further, the optimality and collapsibility of loops were
determined to be virtually impossible (see Flores,
1965).
It is possible, however, to at least provide (conveniently) both the syntax analysis and program flow
analysis during the first pass. This is because the
BEGIN DO's and END DO's (and their associated
nest levels) occur sequentially according to prescribed
rules. The output file from this first phase can be read
backward to perform the necessary subscript analysis
and compute point determination; alternatively the output file can be rewound (in the case of a tape file) and the
subscript analysis and compute point determination can
be combined with the code generation and preassembly
edit phase of the compilation. This paper presents an
algorithm that can be used for determining the compute
point for terms and subscripts which allows for either
contingency; i.e. the algorithm is independent of the
direction of the second pass. Further, the optimality
and collapsibility of loops is unaffected by the
algorithm.
DEFINITION 1. The forward and backward compute
points are the sequence numbers associated with the
earliest point at which a subscript can be computed
and the latest point at which it can still be used,
respectively.
2. A loop is called unsafe if there is a
transfer into the loop from a point in the source
program at level zero as well as a transfer from within
the loop out to level zero; otherwise, the loop is
called safe.
DEFINITION
3. An inner loop is optimal with respect
to the next outer loop if every execution of the outer
loop specifies the number of times that the inner loop
is executed. The following two segments of code are
samples of an optimal and a non-optimal loop-pair,
respectively.
DEFINITION
Temp l=3A,+
4Aj
1
DO pj = Aj,Bj,
TX)<xI = Af,B,,C,
r
Subscript Occurrence
(3/ + 4/)
a CONTINUE
'•— Temp 1 = Temp 1
j8 CONTINUE
Temp 1 = Temp 1
Optimal
• Looppair
+ 4 C , - X*3C, J
(Note: X — Number of times inner loop is executed per
outer loop execution.)
'Research Triangle Institute, Durham, North Carolina.
157
Compilation algorithm
Temp 1 = AAj
J
Temp 2 = 3Ar
+ Temp 1
DO a / = Aj, B,,
NonSubscript Occurrence optimal
(3/+4/)
Looppair
a CONTINUE
— Temp 2 = Temp 2
j8 CONTINUE
Temp 1 = Temp 1
DEFINITION 4. A loop-pair is collapsible if the following conditions hold:
(1) the loop-pair is optimal,
(2) the two loops coincide,
(e.g., DO a / = A,, Bi, C,
DO a / = Aj, Bj, Cj
a CONTINUE
(3) The inner loop is incremented to the exact point
where it is also correct for the outer increment,
i.e., where
Temp 1 + 4 Q - X * 3C, = 0
(1)
in the first example of DEFINITION 3 above.
DEFINITION
(1)
(2)
(3)
(4)
(5)
5. A term is any one of the following:
relative constant subscript,
initial value,
increment value,
test value,
subscripted variable.
DEFINITION 6. The level associated with a given point
in the source program is equal to the number of
BEGIN DO's less the number of END DO's preceding the given point in the source program.
7. The plateau associated with a given
point in the source program is equal to the number
of BEGIN DO's and END DO's preceding the given
point in the source program.
DEFINITION
8. A unique sequence number is transmitted to the interscan file for each item put on the
file during the first scan through the source program.
In the context of this paper, sequence numbers will
increase in the forward direction through the source
program.
DEFINITION
158
The algorithm
This section describes the algorithm in detail. Flow
charts of the algorithm together with a legend defining
the symbols used in the flow charts are presented. For
simplicity, FORTRAN terminology is used throughout the
discussion. The technique, however, is applicable to
any FORTRAN-type algebraic compiler.
The algorithm described is invoked at each occurrence
of a term during the scan. If all of the elements comprising the term (we shall call these elements variables)
are at compute level 0, then the forward compute point
for the term is set to zero and the backward compute
point for the term is set to infinity; otherwise, the
algorithm will be used to determine these compute points.
Since the flow diagrams of Figs. 1 and 2 embody the
scheme which is described here, references to boxes in
the flow charts will be made in this section when necessary. The algorithm begins by initializing the forward
and backward compute points to zero and to infinity,
respectively (Box 1, Fig. 1). Next, all of the variables
comprising the term are tabulated and listed. The
individual variables are designated V; (i = 1 , 2 , . . . , KT),
where KT is the number of such variables (Boxes 2 and 3,
Fig. 1). Box 4 of Fig. 1 contains the only decision
junction based on the direction of the scan. The previous
sequence number of an occurrence of each of the
variables that comprise the term is used to provide a
first estimate for the forward and backward compute
points. If these points are modified on the basis of
previous variable occurrences, the modifications are
made in Boxes 7 and 13 (Fig. 1).
Occurrences of variables in EQUIVALENCE, COMMON variables, and CALL'S next influence the compute
points. If the forward compute point is improved on
the basis of the sequential order of such occurrences,
the changes are made in Boxes 13 through 17 (Fig. 1).
If the backward compute point is improved on the basis
of such occurrences, the changes are made in Boxes 7
through 12 (Fig. 1).
The two basic rules used in this stage of the algorithm
are as follows:
(1) If the source program contains no variables in
both COMMON and EQUIVALENCE, then a
variable in EQUIVALENCE will redefine all
variables in EQUIVALENCE and each COMMON variable is redefined at all CALL'S.
(2) If the source program contains at least one
variable in both COMMON and EQUIVALENCE,
a COMMON variable will redefine all of EQUIVALENCE, and CALL'S will redefine all of
COMMON. (The algorithm could be made
simpler—but less efficient—if COMMON variables
also redefine all of COMMON.)
The forward compute point then is improved further
and the compute level is determined on the basis of the
flow analysis which was performed during the first scan.
Theflowdiagram for this is shown in Fig. 2. In Box 18,
level and plateau counters are initialized. Then, based
Compilation algorithm
FCPT - 0
BCP- - -
List the terminal
variables comprising
Term T and Denote then
v
i <1-1-2
V
: SEQ(P')?
HZ)
19
Is the DO loop f o r \
plateau P1 safe?
I
As the scaA
I forward? J
No
SEQ(P')
I Yes
• No
•
13
(
PSV^KCPT
Is P' a s s o c i a t e d \
with a BEGIN DO?
J
11
SEQ(P') < FCPT?
BED - ^ B C P T
1
L pn
No
-1 BCP
Is the DO loop for
plateau P " safe?
\
J
f BED?I
22
23
C L
T=L'
F' —>FCP_
FCP = point just
prior to SEQ(P')
Clj. - V - 1
Yes I
Is the DO loop for
plateau P' safe?
< FED?]
No
17
OJ
FED -s. FCPT
N
Fig. 1.—Flow chart of algorithm to determine preliminary
compute points for terms
°
f
SEQ(P') <FCP T ?
J
Fig. 2.—Flow chart of algorithms for improving the forward
compute point and determining the compute level on the basis
of first scan flow analysis
159
Compilation algorithm
on the current nest level and the sequence numbers
associated with the next outer BEGIN DO, a new
forward compute point is determined. This determination as well as the compute level also depends on
whether or not each of the loops nesting the current
variable occurrence are safe loops (Boxes 19, 20, and 21,
Fig. 2). The forward compute point and compute level
are finally prescribed in either Box 22 or 23 (Fig. 2).
At the occurrence of labels (in the FORTRAN sense) all
level-dependent subscripts and terms whose computed
level equals the current level will be put out with a
compute sequence equal to that of the label.
BCP r = Backward compute point for term T (sequence
number)
BED = Backward EQUIVALENCE definition (sequence
number)
BPL = The plateau of the BEGIN DO associated with
level L
I = > V is in COMMON
0 = > V is not in COMMON
1 = > at least 1 variable is in COMMON
or EQUIVALENCE
7. CEFLAG = 0 = > no variable is in COMMON or
EQUIVALENCE
CL r = Compute level for term T
CSj/ = Sequence number of current occurrence of
variable V
1 = > V is a variable in EQUIVALENCE
fcv = > y j s n o t a v a r i a b l e in EQUIVALENCE
11.
F' = Temporary storage slot
12. FCD = Forward COMMON definition (sequence
number)
13. FCO = Forward CALL occurrence (sequence number)
14. FCP r = Forward compute point for term T (sequence
number)
15. FED = Forward EQUIVALENCE definition (sequence
number)
K r = The number of terminal variables comprising
16.
termT
17.
L = Current nest level
L' = Temporary storage slot
18.
Plateau of END DO if P is
plateau of BEGIN DO of
same loop
19.
Lp = Linked plateau = Plateau of BEGIN DO if
P is plateau of END DO
of same loop
Np = Plateau of the BEGIN DO of the next outer
20.
loop
21.
P = Plateau (associated with a BEGIN or END DO)
22.
P' = Temporary storage slot
23.
P" = Temporary storage slot
24. PS,/ = Sequence number of previous occurrence of
variable V
25. SEQ (P) = Sequence number of plateau P
Summary and conclusions
The algorithm described above was implemented in
an experimental FORTRAN-type compiler, FTNX-I,
written for the UNIVAC 1107 computer. The algorithm
comprises the second pass of the compiler. In the few
tests which have been made to date (object program
decks of up to 150 cards with as many as twenty subscripted variables), the compiled programs were the
same whether the second pass read the output file from
the first pass backwards or whether the output tape
from the first pass was rewound and read in a forward
manner. Confirmation of the efficiency of the algorithm
is presently being considered. A mathematical proof of
the optimality of the program, with respect to minimizing the number of times that subscripts must be
computed, is impossible since there are an infinite
number of possible loop configurations within a program. The standard criteria which prescribe the "best"
placement of subscript compute points are used by the
algorithm (see Hill, Langmaack, Schwarz, and Seegmuller, 1962).
Symbols used in Figs. 1 and 2
1. BCD = Backward COMMON definition (sequence
number)
2. BCO = Backward CALL occurrence (sequence number)
References
FLORES, I. (1965). Computer Software: Programming Systems for Digital Computers, New York: Prentice-Hall.
HILL, U., LANGMAACK, H., SCHWARZ, H. R., and SEEGMULLER, G. (1962). "Efficient Handling of Subscripted
Variables in
ALGOL 60 Compilers," Proc. 1962 Rome Symposium on Symbolic Languages in Data Processing, New York: Gordon and
Breach Publishing Co.
160