MergeSort

Motivation

Sorting is a problem that is well solved by divide and conquer algorithms. Merge sort is a elegant example which can be parallelized in a straight-forward manner.

Finally, parallelization of the combine step, while not trivial, is possible (and left as an optional fun exercise for those so inclined).

Background

In computer science, merge sort refers to a sorting algorithm which splits an array of data into continuously smaller halves until the arrays reach a size of 1, after which the elements are continuously compared and merged into a sorted array.

For more information on how this process works, visit the wikipedia page on merge sort.

Visualization

If you are unclear on how merge sort works, take a look at the visualgo explanation and visualization of merge sort.

Code to Use

Both the sequential and parallel merge sorts will be passed a Combiner. When you are done with the divide and conquer phases, invoke combiner.combineRange(data, min, mid, maxExclusive) to merge the two sorted sub-problem results.

Code to Implement

You will need to implement merge sort sequentially and in parallel, but you will need to do both implementations recursively. The kernel method should call itself using recursion, but each public mergesort method should only call its kernel once to do the work.

class:	MergeSort.java
methods:	sequentialMergeSort sequentialMergeSortKernel parallelMergeSort parallelMergeSortKernel
package:	sort.studio.merge
source folder:	student/src/main/java

Sequential Implementation

The only thing you will need to edit is the kernel method. As this is a sequential implementation, there is no need to call an async or finish. Using the description of merge sort provided to you in the background, try to consider what the base case would be. Then, further consider how to call the method recursively. When the array reaches the base case, you will need to complete the “merging” aspect of merge sort. As mentioned above, there is a method in the utils class that should help with this.

method: public static void sequentialMergeSort(int[] data, Combiner combiner) (sequential implementation only)

method: private static void sequentialMergeSortKernel(int[] data, int lowInclusive, int highExclusive, Combiner combiner) (sequential implementation only)

Parallel Implementation

This approach should behave much like the sequential implementation, but it should also include async/finish calls. Think carefully about where to place them to maximize concurrency.

When the length of the range has fallen below threshold convert to sorting the array sequentially.

method: public static void parallelMergeSort(int[] data, int threshold, Combiner combiner) (parallel implementation required)

method: private static void parallelMergeSortKernel(int[] data, int lowInclusive, int highExclusive, int threshold, Combiner combiner) (parallel implementation required)

(Optional) Challenge Parallel Combiner

class:	ParallelCombiner.java
methods:	parallelCombine
package:	sort.fun.merge
source folder:	student/src/main/java

You can divide and conquer the combine step in merge sort. The work should remain ${\mathcal {O}}(n\lg {}n)$ while the critical path length can be cleanly improved from ${\mathcal {O}}(n)$ down to ${\mathcal {O}}(\lg ^{3}{}n)$.

Sadly, you cannot simply divide both sides down the middle and merge the lower halves together and the upper halves together (as the data will likely not line up perfectly).

Tip:Being able to perform operations like merge sort combine in parallel often comes down to the question "How can I figure out where to put the current item?"

You can divide one side (choose the longer of the two) and then binary search for the value in the other side.

Warning: SPOILER: Arrays.binarySearch(a) returns a negative number if it does not find the exact number.

Testing Your Solution

Correctness

class:	MergeSortTestSuite.java
package:	sort.studio.merge
source folder:	testing/src/test/java

Optional Fun Parallel Combiner Correctness

class:	ParallelMergeSortParallelCombinerTestSuite.java
package:	sort.fun.merge
source folder:	testing/src/test/java

Performance

class:	SortTiming.java
package:	sort
source folder:	src/main/java

Note: do not be concerned if your implementations run slower than Java's highly optimized Dual Pivot Quicksort.