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Reduction Variables

MATLAB® supports an important exception, called reduction, to the rule that loop iterations must be independent. A reduction variable accumulates a value that depends on all the iterations together, but is independent of the iteration order. MATLAB allows reduction variables in parfor-loops.

Reduction variables appear on both sides of an assignment statement, such as any of the following, where expr is a MATLAB expression.

X = X + exprX = expr + X
X = X - expr See Associativity in Reduction Assignments in Requirements for Reduction Assignments
X = X .* exprX = expr .* X
X = X * exprX = expr * X
X = X & exprX = expr & X
X = X | exprX = expr | X
X = [X, expr]X = [expr, X]
X = [X; expr]X = [expr; X]
X = min(X, expr)X = min(expr, X)
X = max(X, expr)X = max(expr, X)
X = union(X, expr)X = union(expr, X)
X = intersect(X, expr)X = intersect(expr, X)

Each of the allowed statements listed in this table is referred to as a reduction assignment. By definition, a reduction variable can appear only in assignments of this type.

The general form of a reduction assignment is

X = f(X, expr)X = f(expr, X)

The following example shows a typical usage of a reduction variable X.

X = ...;            % Do some initialization of X
parfor i = 1:n
    X = X + d(i);
end

This loop is equivalent to the following, where you calculate each d(i) by a different iteration.

X = X + d(1) + ... + d(n)

In a regular for-loop, the variable X would get its value either before entering the loop or from the previous iteration of the loop. However, this concept does not apply to parfor-loops.

In a parfor-loop, the value of X is never transmitted from client to workers or from worker to worker. Rather, additions of d(i) are done in each worker, with i ranging over the subset of 1:n being performed on that worker. The results are then transmitted back to the client, which adds the partial sums of the workers into X. Thus, workers do some of the additions, and the client does the rest.

Notes About Required and Recommended Guidelines

If your parfor code does not adhere to the guidelines and restrictions labeled as Required, you get an error. MATLAB catches some of these errors at the time it reads the code, and others when it executes the code. These errors are labeled as Required (static) or Required (dynamic) respectively. Guidelines that do not cause errors are labeled as Recommended. You can use MATLAB Code Analyzer to help parfor-loops comply with the guidelines.

Basic Rules for Reduction Variables

The following requirements further define the reduction assignments associated with a given variable.

Required (static): For any reduction variable, the same reduction function or operation must be used in all reduction assignments for that variable.

The parfor-loop on the left is not valid because the reduction assignment uses + in one instance, and [,] in another. The parfor-loop on the right is valid.

InvalidValid
parfor i = 1:n
   if testLevel(k)
      A = A + i;
   else
      A = [A, 4+i];
   end
   % loop body continued
end
parfor i = 1:n
   if testLevel(k)
      A = A + i;
   else
      A = A + i + 5*k;
   end
   % loop body continued
end
Required (static): If the reduction assignment uses *, [,], or [;], then X must be consistently specified as the first or second argument in every reduction assignment.

The parfor-loop on the left is not valid because the order of items in the concatenation is not consistent throughout the loop. The parfor-loop on the right is valid.

InvalidValid
parfor i = 1:n
   if testLevel(k)
      A = [A, 4+i];
   else
      A = [r(i), A];
   end
   % loop body continued
end
parfor i = 1:n
   if testLevel(k)
      A = [A, 4+i];
   else
      A = [A, r(i)];
   end
   % loop body continued
end

Requirements for Reduction Assignments

Reduction Assignments. In addition to the specific forms of reduction assignment listed in the table in Reduction Variables, the only other (and more general) form of a reduction assignment is

X = f(X, expr)X = f(expr, X)
Required (static): f can be a function or a variable. If f is a variable, then you cannot change f in the parfor body (in other words, it is a broadcast variable).

If f is a variable, then for all practical purposes its value at run time is a function handle. However, as long as the right side can be evaluated, the resulting value is stored in X.

The parfor-loop on the left does not execute correctly because the statement f = @times causes f to be classified as a temporary variable. Therefore f is cleared at the beginning of each iteration. The parfor-loop on the right is correct, because it does not assign f inside the loop.

InvalidValid
f = @(x,k)x * k;
parfor i = 1:n
   a = f(a,i);
   % loop body continued
   f = @times;  % Affects f
end
f = @(x,k)x * k;
parfor i = 1:n
   a = f(a,i);
   % loop body continued
end

The operators && and || are not listed in the table in Reduction Variables. Except for && and ||, all the matrix operations of MATLAB have a corresponding function f, such that u op v is equivalent to f(u,v). For && and ||, such a function cannot be written because u&&v and u||v might or might not evaluate v. However, f(u,v) always evaluates v before calling f. Therefore && and || are excluded from the table of allowed reduction assignments for a parfor-loop.

Every reduction assignment has an associated function f. The properties of f that ensure deterministic behavior of a parfor statement are discussed in the following sections.

Associativity in Reduction Assignments. The following practice is recommended for the function f, as used in the definition of a reduction variable. However, this rule does not generate an error if not adhered to. Therefore, it is up to you to ensure that your code meets this recommendation.

Recommended: To get deterministic behavior of parfor-loops, the reduction function f must be associative.

To be associative, the function f must satisfy the following for all a, b, and c.

f(a,f(b,c)) = f(f(a,b),c)

The classification rules for variables, including reduction variables, are purely syntactic. They cannot determine whether the f you have supplied is truly associative or not. Associativity is assumed, but if you violate this rule, each execution of the loop might result in different answers.

Note

The addition of mathematical real numbers is associative. However, the addition of floating-point numbers is only approximately associative. Different executions of this parfor statement might produce values of X with different round-off errors. You cannot avoid this cost of parallelism.

For example, the statement on the left yields 1, while the statement on the right returns 1 + eps:

(1 + eps/2) + eps/2           1 + (eps/2 + eps/2)

Except for the minus operator (-), all special cases listed in the table in Reduction Variables have a corresponding (approximately) associative function. MATLAB calculates the assignment X = X - expr by using X = X + (-expr). (So, technically, the function for calculating this reduction assignment is plus, not minus.) However, the assignment X = expr - X cannot be written using an associative function, which explains its exclusion from the table.

Commutativity in Reduction Assignments. Some associative functions, including +, .*, min, and max, intersect, and union, are also commutative. That is, they satisfy the following for all a and b.

f(a,b) = f(b,a)

Noncommutative functions include * (because matrix multiplication is not commutative for matrices in which both dimensions have size greater than one), [,], and [;]. Noncommutativity is the reason that consistency in the order of arguments to these functions is required. As a practical matter, a more efficient algorithm is possible when a function is commutative as well as associative, and parfor is optimized to exploit commutativity.

Recommended: Except in the cases of *, [,], and [;], the function f of a reduction assignment must be commutative. If f is not commutative, different executions of the loop might result in different answers.

Violating the restriction on commutativity in a function used for reduction could result in unexpected behavior, even if it does not generate an error.

Unless f is a known noncommutative built-in function, it is assumed to be commutative. There is currently no way to specify a user-defined, noncommutative function in parfor.

Overloading in Reduction Assignments. Most associative functions f have an identity element e, so that for any a, the following holds true.

f(e,a) = a = f(a,e)

Examples of identity elements for some functions are listed here.

FunctionIdentity Element
+0
* and .*1
[,] and [;][]
&true
|false

MATLAB uses the identity elements of reduction functions when it knows them. So, in addition to associativity and commutativity, also keep identity elements in mind when overloading these functions.

Recommended: An overload of +, *, .*, [,], or [;] must be associative if it is used in a reduction assignment in a parfor-loop. The overload must treat the respective identity element in the table (all with class double) as an identity element.
Recommended: An overload of +, .*, union, or intersect must be commutative.

There is no way to specify the identity element for a function. In these cases, the behavior of parfor is less efficient than for functions with a known identity element, but the results are correct.

Similarly, because of the special treatment of X = X - expr, the following is recommended.

Recommended: An overload of the minus operator (-) must obey the mathematical law that X - (y + z) is equivalent to (X - y) - z.

Using a Custom Reduction Function

Suppose that each iteration of a loop performs some calculation, and you are interested in finding which iteration of a loop produces the maximum value. This reduction exercise makes an accumulation across multiple iterations of a loop. Your reduction function must compare iteration results, until the maximum value can be determined after all iterations are compared.

First consider the reduction function itself. To compare one iteration result against another, the function requires as input the current result and the known maximum from other iterations so far. Each of the two inputs is a vector containing iteration results and iteration number.

function mc = comparemax(A, B)
% Custom reduction function for 2-element vector input

if A(1) >= B(1) % Compare the two input data values
    mc = A;     % Return the vector with the larger result
else
    mc = B;
end

Inside the loop, each iteration calls the reduction function (comparemax), passing in a pair of two-element vectors:

  • The accumulated maximum and its iteration index, which is the reduction variable cummax

  • The iteration value and index

If the data value of the current iteration is greater than the maximum in cummmax, the function returns a vector of the new value and its iteration number. Otherwise, the function returns the existing maximum and its iteration number.

Each iteration calls the reduction function comparemax to compare its own data [dat i] to data already accumulated in cummax. Try the following code for this loop.

% First element of cummax is maximum data value
% Second element of cummax is where (iteration) maximum occurs
cummax = [0 0];  % Initialize reduction variable
parfor ii = 1:100
    dat = rand(); % Simulate some actual computation
    cummax = comparemax(cummax, [dat ii]);
end
disp(cummax);

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