Presburger Arithmetic and Its Use in Verification

2.2. MULTICORE PARALLELISM ON .NET FRAMEWORK
•Thesetwomergedarraysareconcatenatedtogetthefinalresult.
In parallelism’s point of view, two calls of the merge function here are independent
and easy to parallelize. One rough version of parallel MergeSort is sketched as
follows:
let rec pmsort ls =
if Array.length ls pmsort ls1)
let ls2’ =pmsort ls2
pmerge (ls1’.Result, ls2’)
where pmerge is a parallel variant of merge with similar way of parallelization.
The idiom used here is the Future pattern, and ls1’ is a future value which will be
executed if its Result property is invoked later on. In our example, ls1’.Result
is always executed; therefore, two tasks are always created and executed in parallel.
To explicitly wait for all tasks to complete, we use Task.WaitAll from
System.Threading.Tasks module. Here Task.Factory.StartNew creates a task
for running computation by wrapping around a function closure. For example, we
have a similar code fragment as the Future pattern:
let ls1’ =Task.Factory.StartNew(fun () −> pmsort ls1)
let ls2’ =Task.Factory.StartNew(fun () −> pmsort ls2)
Task.WaitAll(ls1’, ls2’)
pmerge (ls1’.Result, ls2’.Result)
Our rough parallel version shows that the CPU usage is better but overall execution
time is not improved compared to the sequential variant. The reason is that
tasks creation is always done regardless of the size of input arrays, which leads to
spawning too many tasks for just doing trivial jobs. Thus, we introduce one way to
control degree of parallelism:
let rec pmsortUtil (ls, depth) =
if Array.length ls pmsortUtil (ls1, depth−1))
let ls2’ =pmsortUtil (ls2, depth−1)
pmergeUtil (ls1’.Result, ls2’, depth)
let DEPTH =4
let rec pmsort1 ls =
pmsortUtil(ls, DEPTH)
Parallelism is only done until a certain depth. When the size of the array
is rather small, we fall back to the sequential version. Benchmarking done with
different combinations of depth and array size gives us results described in Figure
2.3. Detailed source code can be found in Appendix A.2.
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