Question1: a). Suppose you want to perform two sums: one is a sum of 10 scalar v

Question

 Question1: a). Suppose you want to perform two sums: one is a sum of 10 scalar variables, and one is a matrix sum of a pair of two-dimensional arrays, with dimensions 10 by 10. What speed-up do you get with 10 versus 100 processors? Next, calculate the speedups assuming the matrices grow to 100 by 100. Which are strong scaling and which are weak scaling in this exercise?  b).To achieve the speed-up on the exercise a) with 100 processors, we assumed the load was perfectly balanced. That is, each of the 100 processors had 1% of the work to do. Instead, show the impact on speed-up if one processor’s load is higher than all the rest. Calculate at 2% and 5%. 

Explanation / Answer

Suppose if we assume performance is a function of time for an addition, t, then there are 10 additions that do not benefit from 100 parallel processors.

Single processor is 110t, the execution time for 10 processors is:
Speed up from 10 to 100 processors
For Single processor: Time = (10 + 100) × tadd

a)10 processors
Time = 10 × tadd + 100/10 × tadd = 20 × tadd
Speedup = 110/20 = 5.5 (55% of potential)

100 processors
Time = 10 × tadd + 100/100 × tadd = 11 × tadd
Speedup = 110/11 = 10 (10% of potential)

b) What if matrix size is 100 × 100?
Single processor: Time = (10 + 10000) × tadd

10 processors
Time = 10 × tadd + 10000/10 × tadd = 1010 × tadd
Speedup = 10010/1010 = 9.9 (99% of potential)

100 processors
Time = 10 × tadd + 10000/100 × tadd = 110 × tadd
Speedup = 10010/110 = 91 (91% of potential).

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