# Divide and Conquer

Problem

Parallelize a divide and conquer algorithm.

Context

Divide and conquer is widely used in serial algorithms. Common examples are quicksort and mergesort.

Forces

- Problem can be transformed into subproblems that can be solved independently.

- Splitting problem or merging solutions is relatively cheap compared to cost of solving the subproblems.

Solution

There are several ways to implement divide and conquer in oneAPI Threading Building Blocks (oneTBB). The best choice depends upon circumstances.

- If division always yields the same number of subproblems, use recursion and `oneapi::tbb::parallel_invoke`.

- If the number of subproblems varies, use recursion and `oneapi::tbb::task_group`.

Example

Quicksort is a classic divide-and-conquer algorithm. It divides a sorting problem into two subsorts. A simple serial version looks like [1].

    void SerialQuicksort( T* begin, T* end ) {
       if( end-begin>1  ) {
           using namespace std;
           T* mid = partition( begin+1, end, bind2nd(less<T>(),*begin) );
           swap( *begin, mid[-1] );
           SerialQuicksort( begin, mid-1 );
           SerialQuicksort( mid, end );
       }
    }

The number of subsorts is fixed at two, so `oneapi::tbb::parallel_invoke` provides a simple way to parallelize it. The parallel code is shown below:

    void ParallelQuicksort( T* begin, T* end ) {
       if( end-begin>1 ) {
           using namespace std;
           T* mid = partition( begin+1, end, bind2nd(less<T>(),*begin) );
           swap( *begin, mid[-1] );
           oneapi::tbb::parallel_invoke( [=]{ParallelQuicksort( begin, mid-1 );},
                                 [=]{ParallelQuicksort( mid, end );} );
       }
    }

Eventually the subsorts become small enough that serial execution is more efficient. The following variation, does sorts of less than 500 elements using the earlier serial code.

    void ParallelQuicksort( T* begin, T* end ) {
       if( end-begin>=500 ) {
           using namespace std;
           T* mid = partition( begin+1, end, bind2nd(less<T>(),*begin) );
           swap( *begin, mid[-1] );
           oneapi::tbb::parallel_invoke( [=]{ParallelQuicksort( begin, mid-1 );},
                                 [=]{ParallelQuicksort( mid, end );} );
       } else {
           SerialQuicksort( begin, end );
       }
    }

The change is an instance of the Agglomeration pattern.

The next example considers a problem where there are a variable number of subproblems. The problem involves a tree-like description of a mechanical assembly. There are two kinds of nodes:

- Leaf nodes represent individual parts.

- Internal nodes represent groups of parts.

The problem is to find all nodes that collide with a target node. The following code shows a serial solution that walks the tree. It records in `Hits` any nodes that collide with `Target`.

    std::list<Node*> Hits;
    Node* Target;

    void SerialFindCollisions( Node& x ) {
       if( x.is_leaf() ) {
           if( x.collides_with( *Target ) )
               Hits.push_back(&x);
       } else {
           for( Node::const_iterator y=x.begin();y!=x.end(); ++y )
               SerialFindCollisions(*y);
       }
    }

A parallel version is shown below.

    typedef oneapi::tbb::enumerable_thread_specific<std::list<Node*> > LocalList;
    LocalList LocalHits;
    Node* Target;    // Target node

    void ParallelWalk( Node& x ) {
       if( x.is_leaf() ) {
           if( x.collides_with( *Target ) )
               LocalHits.local().push_back(&x);
       } else {
           // Recurse on each child y of x in parallel
           oneapi::tbb::task_group g;
           for( Node::const_iterator y=x.begin(); y!=x.end(); ++y )
               g.run( [=]{ParallelWalk(*y);} );
           // Wait for recursive calls to complete
           g.wait();
       }
    }

    void ParallelFindCollisions( Node& x ) {
       ParallelWalk(x);
       for(LocalList::iterator i=LocalHits.begin();i!=LocalHits.end(); ++i)
           Hits.splice( Hits.end(), *i );
    }

The recursive walk is parallelized using class `task_group` to do recursive calls in parallel.

There is another significant change because of the parallelism that is introduced. Because it would be unsafe to update `Hits` concurrently, the parallel walk uses variable `LocalHits` to accumulate results. Because it is of type `enumerable_thread_specific`, each thread accumulates its own private result. The results are spliced together into Hits after the walk completes.

The results will not be in the same order as the original serial code.

If parallel overhead is high, use the agglomeration pattern. For example, use the serial walk for subtrees under a certain threshold.

[1]

Production quality quicksort implementations typically use more sophisticated pivot selection, explicit stacks instead of recursion, and some other sorting algorithm for small subsorts. The simple algorithm is used here to focus on exposition of the parallel pattern.