To: J3 J3/23-113
From: Brad Richardson
Subject: SCAN and CO_SCAN
Date: 2023-February-06
#Reference:
Introduction
============
SCAN is a common operation closely related to REDUCE. Both SCAN and
REDUCE apply a binary operation to a sequence. SCAN returns the
sequence of results of the operations, where REDUCE returns only the
final result. Whether a SCAN is INCLUSIVE or EXCLUSIVE determines
whether an element in the resulting sequence includes the result of
the binary operation with the corresponding element in the input
sequence or not, respectively.
Typical applications that make us of scan operations include
design of binary adders, polynomial interpolation, simulation of
parallel algorithms that assume the ability for multiple processors
to access the same memory cell at the same time, on parallel machines
that forbid simultaneous access. Additional applications can be found
on the Wikipedia page for "Prefix sum":
https://en.wikipedia.org/wiki/Prefix_sum
Proposal
========
Provide a SCAN intrinsic function and CO_SCAN collective subroutine.
These function and subroutine shall implement scan operations
analogous to the REDUCE function and CO_REDUCE subroutine.
Descriptions
============
SCAN(ARRAY, OPERATION[, DIM, MASK, SEGMENT, EXCLUSIVE,
REVERSED, ORDERED]) OR
SCAN(ARRAY, OPERATION, IDENTITY[, DIM, MASK, SEGMENT, EXCLUSIVE,
REVERSED, ORDERED])
Description. Generalized scan of an array.
Class. Transformational function
Arguments.
ARRAY shall be an array of any type.
OPERATION shall be a pure function with exactly two arguments; each
argument shall be a scalar, nonallocatable, nonpointer,
nonpolymorphic, nonoptional dummy data object. The declared
type and type parameters of the first argument shall be the
same as IDENTITY if it is present, otherwise it shall be the
same as ARRAY. The declared type and type parameters of the
second argument shall be the same as ARRAY. If one argument
has the ASYNCHRONOUS, TARGET, or VALUE attribute, the other
shall have that attribute. Its result shall be a
nonpolymorphic scalar and have the same declared type and
type parameters as IDENTITY if it is present, otherwise it
shall have the same declared type and type parameters as
ARRAY. OPERATION should implement a mathematically
associative operation. It need not be commutative.
IDENTITY shall be a scalar data object.
DIM (optional) shall be an integer scalar with a value in the range
1 <= DIM <= n, where n is the rank of ARRAY.
MASK (optional) shall be of type logical and shall be conformable
with ARRAY.
SEGMENT (optional) shall be of type logical with the same shape as
ARRAY.
EXCLUSIVE (optional) shall be a logical scalar.
REVERSED (optional) shall be a logical scalar.
ORDERED (optional) shall be a logical scalar.
Result Characteristics. The result has the same rank and shape as
ARRAY. The result has the same declared type and type parameters as
IDENTITY if it is present, otherwise it has the same declared type
and type parameters as ARRAY.
Result Value. The value of each element of the result is determined by
the values of selected elements of ARRAY, and IDENTITY if present. The
optional arguments DIM, MASK, SEGMENT, EXCLUSIVE and REVERSED affect
the selection of elements of ARRAY for each element of the result. If
no elements of ARRAY are selected for a given element of the result
that result element is set to the value of IDENTITY if it is present,
otherwise error termination is initiated. The value of the result
element is determined by successively applying OPERATION to all
selected values which contribute to that element.
For any given element r of the result, let a be the corresponding
element of ARRAY. IDENTITY contributes to every element r if it is
present. Every element of ARRAY contributes to r unless disqualified
by one of the following rules.
1. If the argument REVERSED is present with the value true, no element
that precedes a in the array element ordering of ARRAY contributes to
r. Otherwise no element that follows a in the array element ordering
of ARRAY contributes to r.
2. If the DIM argument is provided, an element z of ARRAY does not
contribute to r unless all its indices, excepting only the index for
dimension DIM, are the same as the corresponding indices of a. If the
DIM argument is present, then a family of completely independent scan
operations are carried out along the selected dimension of ARRAY.
3. If the MASK argument is provided, an element z of ARRAY contributes
to r only in the element of MASK corresponding to z is true. It follows
that array elements corresponding to positions where the MASK is false
do not contribute anywhere to the result. However, the result is
nevertheless defined at all positions, even positions where the MASK
is false.
4. If the SEGMENT argument is provided, an element z of ARRAY does
not contribute if there is some intermediate element w of ARRAY,
possibly z itself, with all of the following properties:
a. If the argument REVERSED is present with the value true, w does
not follow z but does follow a in the array element ordering;
otherwise w does not precede a in the array element ordering;
b. If the DIM argument is present, all the indices of w, excepting
only the index for dimension DIM, are the same as the corresponding
indices of a;
c. The element of SEGMENT corresponding to w does not have the same
value as the element of SEGMENT corresponding to a. In other words, z
can contribute only if there is an unbroken string of SEGMENT values,
all alike, extending from z through a.
5. If the EXCLUSIVE argument is provided and is true, then a itself
does not contribute to r.
6. If the argument ORDERED is provided and is true, then each element
of the result is calculated strictly in array element order, or in
reverse array element order if REVERSED is also provided and is true.
These general rules lead to the following important cases:
Case (i): If ARRAY has rank one, element i of the result of
SCAN(ARRAY, OPERATION) is determined by the first i
elements of ARRAY; element SIZE(ARRAY)-i+1 of the result
of SCAN(ARRAY, OPERATION, REVERSED = .TRUE.) is determined
by the last i elements of ARRAY.
Case (ii): If ARRAY has rank greater than one, then each element of
result of SCAN(ARRAY, OPERATION) has a value determined by
the corresponding element a of the ARRAY and all elements
of ARRAY that precede a in array element order. For
SCAN(ARRAY, OPERATION, REVERSED = .TRUE.), a is determined
by the elements of ARRAY that correspond to or follow a in
array element order.
Case (iii): Each element of the result of
SCAN(ARRAY, OPERATION, MASK=MASK) is determined by selected
elements of ARRAY, namely the corresponding element a of
the ARRAY and all elements of ARRAY that precede a in array
element order, but an element of ARRAY may contribute to
the result only if the corresponding element of MASK is
true. If this restriction results in selecting no array
elements to contribute to some element of the result, then
error termination is initiated. For
SCAN(ARRAY, OPERATION, IDENTITY, MASK=MASK), if no elements
of ARRAY contribute to some element of the result, then
that element of the result is set to IDENTITY. Note that
in the general case
SCAN(ARRAY, OPERATION, IDENTITY, MASK=MASK, ...) ==
SCAN(MERGE(ARRAY, IDENTITY, MASK), OPERATION, ...)
Case (iv): Each element of the result of
SCAN(ARRAY, OPERATION, DIM=DIM) is determined by selected
elements of ARRAY, namely the corresponding element a of
the ARRAY and all elements of ARRAY that precede a along
dimension DIM; SCAN(A(1:N,1:N), ADD, DIM=2), result element
(i1, i2) could be computed as SUM(A(i1,1:i2)). More
generally, the values of the section
(s1, s2, ..., sDIM-1, :, sDIM+1, ..., sn) of
SCAN(ARRAY, OPERATION, DIM=DIM) are equal to
SCAN(ARRAY(s1, s2, ..., sDIM-1, :, sDIM+1, ..., sn),
OPERATION).
Case (v): If ARRAY has rank one, then element i of the result of
SCAN(ARRAY, OPERATION, EXCLUSIVE=.TRUE.) is determined by
the first i-1 elements of ARRAY.
Case (vi): It is possible for SCAN(ARRAY, OPERATION) to have a result
that differs from SCAN(ARRAY, OPERATION, ORDERED = .TRUE.)
Case (vii): If EXCLUSIVE is provided and is true, then IDENTITY shall be
provided, since at least one element of the result will have
no elements of ARRAY selected to contribute to it.
Case (viii): The options may be used in any combination.
Examples. The following examples use the function ADD, which returns
the sum of its two integer arguments, or the function COND_INC which
returns the second argument plus one if the first argument is .TRUE.,
and the second argument otherwise.
Case (i): SCAN([.TRUE., .FALSE., .TRUE., .TRUE.], COND_INC, 0)
is [1, 1, 2, 3].
| 1 2 3 |
Case (ii): If B is the array | 4 5 6 | then
| 7 8 9 |
| 45 33 18 |
SCAN(B, ADD, REVERSED = .TRUE.) is the array | 44 31 15 |.
| 40 26 9 |
Case (iii): If A is the array [3, 5, -2, -1, 7, 4, 8] then
SCAN(A, ADD, MASK = A < 6) is [3, 8, 6, 5, 5, 9, 9].
| 1 2 3 |
Case (iv): If B is the array | 4 5 6 | then
| 7 8 9 |
| 1 2 3 |
SCAN(B, ADD, DIM=1) is the array | 5 7 9 | and
| 12 15 18 |
| 1 3 6 |
SCAN(B, ADD, DIM=2) is the array | 4 9 15 |.
| 7 15 24 |
Case (v): SCAN([1, 3, 5, 7], ADD, IDENTITY = 0, EXCLUSIVE = .TRUE.) is
[0, 1, 4, 9].
| 1 2 3 4 5 |
Case (vi): If B is the array | 6 7 8 9 10 |, M is the array
| 11 12 13 14 15 |
| T T T T T | | T T F F F |
| F F T T T | and S is the array | F T T F F |, then
| T F T F F | | T T T T T |
SCAN(B, ADD, 0, DIM=2, MASK=M, SEGMENT=S, EXCLUSIVE=.TRUE.) is
| 0 1 0 3 7 |
| 0 0 0 0 9 |,
| 0 11 11 24 24 |
SCAN(B, ADD, 0, DIM=2, MASK=M, SEGMENT=S, EXCLUSIVE=.FALSE.) is
| 1 3 3 7 12 |
| 0 0 8 9 19 |,
| 11 11 24 24 24 |
SCAN(B, ADD, 0, DIM=2, MASK=M, EXCLUSIVE=.TRUE.) is
| 0 1 3 6 10 |
| 0 0 0 8 17 |,
| 0 11 11 24 24 |
SCAN(B, ADD, 0, DIM=2, MASK=M, EXCLUSIVE=.FALSE.) is
| 1 3 6 10 15 |
| 0 0 8 17 27 |,
| 11 11 24 24 24 |
SCAN(B, ADD, 0, DIM=2, SEGMENT=S, EXCLUSIVE=.TRUE.) is
| 0 1 0 3 7 |
| 0 0 7 0 9 |,
| 0 11 23 36 50 |
SCAN(B, ADD, DIM=2, SEGMENT=S, EXCLUSIVE=.FALSE.) is
| 1 3 3 7 12 |
| 6 7 15 9 19 |,
| 11 23 36 50 65 |
SCAN(B, ADD, 0, DIM=2, EXCLUSIVE=.TRUE.) is
| 0 1 3 6 10 |
| 0 6 13 21 30 |,
| 0 11 23 36 50 |
SCAN(B, ADD, DIM=2, EXCLUSIVE=.FALSE.) is
| 1 3 6 10 15 |
| 6 13 21 30 40 |,
| 11 23 36 50 65 |
SCAN(B, ADD, 0, MASK=M, SEGMENT=S, EXCLUSIVE=.TRUE.) is
| 0 11 0 0 0 |
| 0 13 0 4 5 |,
| 0 13 8 0 0 |
SCAN(B, ADD, 0, MASK=M, SEGMENT=S, EXCLUSIVE=.FALSE.) is
| 1 13 3 4 5 |
| 0 13 8 13 15 |,
| 11 13 21 0 0 |
SCAN(B, ADD, 0, MASK=M, EXCLUSIVE=.TRUE.) is
| 0 12 14 38 51 |
| 1 14 17 42 56 |,
| 1 14 25 51 66 |
SCAN(B, ADD, MASK=M, EXCLUSIVE=.FALSE.) is
| 1 14 17 42 56 |
| 1 14 25 51 66 |,
| 12 14 38 51 66 |
SCAN(B, ADD, 0, SEGMENT=S, EXCLUSIVE=.TRUE.) is
| 0 11 0 0 0 |
| 0 13 0 4 5 |,
| 0 20 8 0 0 |
SCAN(B, ADD, SEGMENT=S, EXCLUSIVE=.FALSE.) is
| 1 13 3 4 5 |
| 6 20 8 13 15 |,
| 11 32 21 14 15 |
SCAN(B, ADD, 0, EXCLUSIVE=.TRUE.) is
| 0 18 39 63 90 |
| 1 20 42 67 95 |, and
| 7 27 50 76 105 |
SCAN(B, ADD, EXCLUSIVE=.FALSE.) is
| 1 20 42 67 95 |
| 7 27 50 76 105 |.
| 18 39 63 90 120 |
NOTE X
If OPERATION is not computationally associative, SCAN without
ORDERED=.TRUE. with the same argument values might not always produce
the same result, as the processor can apply the associative law to the
evaluation.
CO_SCAN(A, OPERATION[, EXCLUSIVE, REVERSED, STAT, ERRMSG]) OR
CO_SCAN(A, OPERATION, IDENTITY[, EXCLUSIVE, REVERSED, STAT, ERRMSG])
Description. Generalized scan across images.
Class. Collective subroutine.
Arguments.
A shall not be polymorphic. It shall have the same shape,
type, and type parameter values in corresponding references.
It shall not be a coindexed object. It is an INTENT(INOUT)
argument. If A is scalar, the computed value is the result
of the scan operation of applying OPERATION to the
values of A in all corresponding references. If A is an
array, each element of the computed value is equal to the
result of the scan operation of applying OPERATION to
corresponding elements of A in all corresponding references.
The value assigned to an element of A is the element of the
computed result corresponding to the image number of the
executing image.
OPERATION shall be a pure function with exactly two arguments; the
result and each argument shall be a scalar, nonallocatable,
nonpointer, nonpolymorphic data object with the same type
and type parameters as A. The arguments shall not be
optional. If one argument has the ASYNCHRONOUS, TARGET, or
VALUE attribute, the other shall have that attribute.
OPERATION shall implement a mathematically associative
operation. OPERATION shall be the same function on all
images in the corresponding references.
The value of each element of the result is determined by the
values of selected images of A, and IDENTITY if present. The
optional arguments EXCLUSIVE and REVERSED affect the
selection of images for each element of the result. If no
images are selected for a given element of the result that
result element is set to the value of IDENTITY if it is
present, otherwise error termination is initiated. The value
of the result element is determined by successively applying
OPERATION to all selected values which contribute to that
element.
IDENTITY contributes to every element of the result if it is
present. Every image contributes to the result unless
disqualified by one of the following rules.
1. If the argument REVERSED is present with the value true,
no image that precedes the corresponding image in the result
in numerical order contributes to the result. Otherwise no
image that follows the corresponding image in the result
in numerical order contributes to the result.
2. If the EXCLUSIVE argument is provided and is true, then
the corresponding image itself does not contribute to the
result.
IDENTITY shall be a scalar with the same declared type and type
parameters as ARRAY. It shall not be a coindexed object. Its
value shall be the same in all corresponding references.
EXCLUSIVE (optional) shall be a logical scalar. Its presence, and value
if present, shall be the same in all corresponding
references.
REVERSED (optional) shall be a logical scalar.Its presence, and value
if present, shall be the same in all corresponding
references.
STAT (optional) shall be a noncoindexed integer scalar with a decimal
exponent range of at least four. It is an INTENT(OUT)
argument.
ERRMSG (optional) shall be a noncoindexed default scalar. It is an
INTENT(INOUT) argument.
The semantics of STAT and ERRMSG are described in 16.6.
Example. If the function MY_MULT returns the product of its two
integer arguments, the number of images in the current team is three,
and A is the array [1, 3, 5] on image 1, [2, 4, 6] on image 2,
and [7, 8, 9] on image 3, the value of A after executing the statement
CALL CO_SCAN(A, MY_MULT, 1, EXCLUSIVE=.TRUE.)
is [1, 1, 1] on image 1, [1, 3, 5] on image 2,
and [2, 12, 30] on image 3.
NOTE X
If the OPERATION function is not mathematically commutative, the result
of calling CO_SCAN can depend on the order of evaluations.
NOTE X
The consequences of the selection of contributing values to the result
are such that if EXCLUSIVE if present with the value .TRUE., then
IDENTITY shall be provided or error termination will be initiated.
Straw Vote
==========
How should SCAN be spelled?
A. SCAN and CO_SCAN
B. ACCUMULATE and CO_ACCUMULATE
C. Other