# Reduction

Problem

Perform an associative reduction operation across a data set.

Context

Many serial algorithms sweep over a set of items to collect summary information.

Forces

The summary can be expressed as an associative operation over the data set, or at least is close enough to associative that reassociation does not matter.

Solution

Two solutions exist in oneAPI Threading Building Blocks (oneTBB). The choice on which to use depends upon several considerations:

- Is the operation commutative as well as associative?

- Are instances of the reduction type expensive to construct and destroy. For example, a floating point number is inexpensive to construct. A sparse floating-point matrix might be very expensive to construct.

Use `oneapi::tbb::parallel_reduce` when the objects are inexpensive to construct. It works even if the reduction operation is not commutative.

Use `oneapi::tbb::parallel_for` and `oneapi::tbb::combinable` if the reduction operation is commutative and instances of the type are expensive.

If the operation is not precisely associative but a precisely deterministic result is required, use recursive reduction and parallelize it using `oneapi::tbb::parallel_invoke`.

Examples

The examples presented here illustrate the various solutions and some tradeoffs.

The first example uses `oneapi::tbb::parallel_reduce` to do a + reduction over sequence of type `T`. The sequence is defined by a half-open interval [first,last).

    T AssociativeReduce( const T* first, const T* last, T identity ) {
       return oneapi::tbb::parallel_reduce(
           // Index range for reduction
           oneapi::tbb::blocked_range<const T*>(first,last),
           // Identity element
           identity,
           // Reduce a subrange and partial sum
           [&]( oneapi::tbb::blocked_range<const T*> r, T partial_sum )->float {
               return std::accumulate( r.begin(), r.end(), partial_sum );
           },
           // Reduce two partial sums
           std::plus<T>()
       );
    }

The third and fourth arguments to this form of `parallel_reduce` are a built in form of the agglomeration pattern. If there is an elementwise action to be performed before the reduction, incorporating it into the third argument (reduction of a subrange) may improve performance because of better locality of reference. Note that the block size for agglomeration is not explicitly specified; `parallel_reduce` defines blocks automatically with the help of implicitly used `oneapi::tbb::auto_partitioner`.

The second example assumes the + is commutative on `T`. It is a good solution when `T` objects are expensive to construct.

    T CombineReduce( const T* first, const T* last, T identity ) {
       oneapi::tbb::combinable<T> sum(identity);
       oneapi::tbb::parallel_for(
           oneapi::tbb::blocked_range<const T*>(first,last),
           [&]( oneapi::tbb::blocked_range<const T*> r ) {
               sum.local() += std::accumulate(r.begin(), r.end(), identity);
           }
       );
       return sum.combine( []( const T& x, const T& y ) {return x+y;} );
    }

Sometimes it is desirable to destructively use the partial results to generate the final result. For example, if the partial results are lists, they can be spliced together to form the final result. In that case use class `oneapi::tbb::enumerable_thread_specific` instead of `combinable`. The `ParallelFindCollisions` example in Divide and Conquer demonstrates the technique.

Floating-point addition and multiplication are almost associative. Reassociation can cause changes because of rounding effects. The techniques shown so far reassociate terms non-deterministically. Fully deterministic parallel reduction for a not quite associative operation requires using deterministic reassociation. The code below demonstrates this in the form of a template that does a + reduction over a sequence of values of type `T`.

    template<typename T>
    T RepeatableReduce( const T* first, const T* last, T identity ) {
       if( last-first<=1000 ) {
           // Use serial reduction
           return std::accumulate( first, last, identity );
       } else {
           // Do parallel divide-and-conquer reduction
           const T* mid = first+(last-first)/2;
           T left, right;
           oneapi::tbb::parallel_invoke(
               [&]{left=RepeatableReduce(first,mid,identity);},
               [&]{right=RepeatableReduce(mid,last,identity);}
           );
           return left+right;
       }
    }

The outer if-else is an instance of the agglomeration pattern for recursive computations. The reduction graph, though not a strict binary tree, is fully deterministic. Thus the result will always be the same for a given input sequence, assuming all threads do identical floating-point rounding.

`oneapi::tbb::parallel_deterministic_reduce` is a simpler and more efficient way to get reproducible non-associative reduction. It is very similar to `oneapi::tbb::parallel_reduce` but, unlike the latter, builds a deterministic reduction graph. With it, the `RepeatableReduce` sample can be almost identical to `AssociativeReduce`:

    template<typename T>
    T RepeatableReduce( const T* first, const T* last, T identity ) {
       return oneapi::tbb::parallel_deterministic_reduce(
           // Index range for reduction
           oneapi::tbb::blocked_range<const T*>(first,last,1000),
           // Identity element
           identity,
           // Reduce a subrange and partial sum
           [&]( oneapi::tbb::blocked_range<const T*> r, T partial_sum )->float {
               return std::accumulate( r.begin(), r.end(), partial_sum );
           },
           // Reduce two partial sums
           std::plus<T>()
       );
    }

Besides the function name change, note the grain size of 1000 specified for `oneapi::tbb::blocked_range`. It defines the desired block size for agglomeration; automatic block size selection is not used due to non-determinism.

The final example shows how a problem that typically is not viewed as a reduction can be parallelized by viewing it as a reduction. The problem is retrieving floating-point exception flags for a computation across a data set. The serial code might look something like:

    feclearexcept(FE_ALL_EXCEPT);
    for( int i=0; i<N; ++i )
        C[i]=A[i]*B[i];
    int flags = fetestexcept(FE_ALL_EXCEPT);
    if (flags & FE_DIVBYZERO) ...;
    if (flags & FE_OVERFLOW) ...;
    ...

The code can be parallelized by computing chunks of the loop separately, and merging floating-point flags from each chunk. To do this with `tbb:parallel_reduce`, first define a “body” type, as shown below.

    struct ComputeChunk {
       int flags;          // Holds floating-point exceptions seen so far.
       void reset_fpe() {
           flags=0;
           feclearexcept(FE_ALL_EXCEPT);
       }
       ComputeChunk () {
           reset_fpe();
       }
       // "Splitting constructor"called by parallel_reduce when splitting a range into subranges.
       ComputeChunk ( const ComputeChunk&, oneapi::tbb::split ) {
           reset_fpe();
       }
       // Operates on a chunk and collects floating-point exception state into flags member.
       void operator()( oneapi::tbb::blocked_range<int> r ) {
           int end=r.end();
           for( int i=r.begin(); i!=end; ++i )
               C[i] = A[i]/B[i];
           // It is critical to do |= here, not =, because otherwise we
           // might lose earlier exceptions from the same thread.
           flags |= fetestexcept(FE_ALL_EXCEPT);
       }
       // Called by parallel_reduce when joining results from two subranges.
       void join( Body& other ) {
           flags |= other.flags;
       }
    };

Then invoke it as follows:

    // Construction of cc implicitly resets FP exception state.
       ComputeChunk cc;
       oneapi::tbb::parallel_reduce( oneapi::tbb::blocked_range<int>(0,N), cc );
       if (cc.flags & FE_DIVBYZERO) ...;
       if (cc.flags & FE_OVERFLOW) ...;
       ...