Run serially, portion A of program P consumes 33 s, while portion B

consumes 925 s. On a parallel computer, moderately serial portion A speeds up four times, while perfectly parallel portion B speeds up by the number of processors. i) What is the speedup of program P on 1,024 processors? ii) How many processors are required to achieve at least half the theoretical maximum possible speedup on P?

For i), I did:

Serial run-time of A: A_serial = 33s
Serial run-time of B: B_serial = 925s
Parallel run-time of A = A_parallel = A_serial/A_speedup = 33s/4 = 8.25s
Parallel run-time of B = B_parallel = B_serial/# processors = 925s/1024 = 0.90s

Serial run-time = T_serial = A_serial + B_serial = 33s + 925s = 958s
Parallel run-time = T_parallel = A_parallel + B_parallel = 8.25s + 0.90s = 9.15s
Total speedup = T_old/T_new = T_serial/T_parallel = 958s/0.90s = 104.66

Therefore, program P speeds up 104.66 times? Is this correct? Also, I wanted to know, since it seems from many sources that the theoretical maximum speedup is 20, can the calculated speedup ever be greater than 20? Because that is the result I got for this problem (104.66, which is >20). And what exactly does "moderately serial" mean?

For ii):

max speedup = 20 -> half max speedup = (50%)(20) = 10
10 = T_serial/T_parallel = 958s/T_parallel -> T_parallel = 958s/10 = 95.8s
95.8s = A_parallel + B_parallel = 8.25s + 925s/# processors -> # processors = 925s/(95.8s - 8.25s) = 10.57 processors

For i), your approach and calculations are correct. The speedup of program P on 1,024 processors would indeed be 104.66 times.

Regarding the theoretical maximum speedup, it is a measure of the maximum potential improvement that can be achieved when running a program in parallel. In theory, the maximum speedup is equal to the number of processors used. However, in practice, achieving a speedup equal to the number of processors is unlikely due to factors such as overhead, communication costs, and limitations of parallel algorithms. So, while the theoretical maximum speedup is 1,024 (equal to the number of processors), it is not always achievable.

In this case, the calculated speedup of 104.66 is greater than the commonly cited theoretical maximum of 20. There can be a few reasons for this. One possibility is that the theoretical maximum speedup of 20 is a rough estimate or average, and in certain cases, higher speedups can be achieved. Another possibility is that the given program P has certain characteristics that make it more amenable to parallelization and performance improvements.

Regarding the term "moderately serial," it refers to a portion of the program that cannot be fully parallelized, but still allows for some improvement through parallel execution. In this case, portion A is moderately serial and speeds up by a factor of four when run on a parallel computer. It means that it has some dependencies or sequential aspects that limit its parallelizability, but it can still benefit from parallel execution to some extent.

For ii), your calculation for the number of processors required to achieve at least half the theoretical maximum speedup is correct. Based on your formula and calculations, approximately 10.57 processors would be needed to achieve a speedup of at least 10 (half of the theoretical maximum speedup of 20). However, since the number of processors cannot be fractional, you would need to round up to the nearest whole number. Therefore, you would need at least 11 processors to achieve the desired speedup.