1. COMP60611 Fundamentals of Parallel and Distributed Systems
Lecture 6
Scalability Analysis
John Gurd, Graham Riley
Centre for Novel Computing
School of Computer Science
University of Manchester

2. Scalability What do we mean by scalability?
Scalability applies to an algorithm executing on a parallel computer, not simply to an algorithm!
How does an algorithm behave for a fixed problem size as the number of processors used increases?
This is known as strong scaling.
How does an algorithm behave as the problem size changes, in addition to changing the number of processors?
A key insight is to look at how efficiency changes.
22 February, 2019

3. Efficiency and Strong Scaling
Typically, for a fixed problem size, N, the efficiency of an algorithm decreases as P increases. (Why?)
Overheads typically do not get smaller as P increases. They remain ‘fixed’ or, worse, they may grow with P (e.g. the number of communications may grow – in an all-to-all communication pattern).
Recall that:
22 February, 2019

4. Efficiency and Strong Scaling
POP is the total overhead in the system.
Tref represents the true useful work in the algorithm.
Because it tends to decrease with fixed N, at some point (absolute) efficiency Eabs (i.e. how well each processor is being utilised) will drop below some acceptable threshold – say, 50%(?)
22 February, 2019

5. Scalability
No ‘real’ algorithm scales for all possible numbers of processors solving a fixed problem size on a ‘real’ computer.
Even ‘embarrassingly’ parallel algorithms will have a limit on the number of processors they can use.
For example, at the point where, with a fixed N, eventually there is only one ‘element’ of some large data structure to be operated on by each processor.
So we seek another approach to scalability which applies as both problem size N and the number of processors P change.
22 February, 2019

6. Isoscaling and Isoefficiency
A system is said to isoscale if, for a given algorithm and parallel computer, a specific level of efficiency can be maintained by changing the problem size, N, appropriately as P increases.
Not all systems isoscale!
e.g. a binary tree-based vector reduction where N = P (see later).
This approach is called scaled problem analysis.
The function (of P) describing how the problem size N must change as P increases in order to maintain a specified efficiency is known as the isoefficiency function.
Isoscaling does not apply to all problems.
e.g. weather modelling, where increasing problem size (resolution) is eventually not an option
or image processing with a fixed number of pixels
22 February, 2019

7. Weak Scaling
An alternative approach is to keep the problem size per processor fixed as P increases (total problem size N thus increases linearly with P) and see how the efficiency is affected
This is known as weak scaling.
Summary: strong scaling, weak scaling and isoscaling are three different approaches to understanding the scalability of parallel systems (algorithm + machine).
We will look at an example shortly, but first we need a means of comparing the behaviour of functions, e.g. performance functions and efficiency functions, over their entire domains.
These concepts will be explored further in lab exercise 2.
22 February, 2019

8. Comparison Functions: Asymptotic Analysis
Performance models are generally functions of problem size (N) and the number of processors (P).
We need relatively easy ways to compare models (functions) as N and P vary:
Model A is ‘at most’ as fast or as big as model B;
Model A is ‘at least’ as fast or as big as model B;
Model A is ‘equal’ in performance/size to model B.
We will see a similar need when comparing efficiencies and in considering scalability. (And you may also have encountered it if you studied complexity theory as an undergraduate.)
These are all examples of comparison functions.
We are often interested in asymptotic behaviour, i.e. the behaviour as some key parameter (e.g. N or P) increases towards infinity.
22 February, 2019

9. Comparison Functions – Example
From ‘Introduction to Parallel Computing’, Grama.
Consider the three functions below:
Think of these functions as modelling the distance travelled by three cars from time t=0. One car (A) has fixed speed and the others are accelerating – car C makes a standing start (zero initial speed).
22 February, 2019

10. Graphically
22 February, 2019

11. We can see that: For t > 45, B(t) is always greater than A(t).
For t > 20, C(t) is always greater than B(t).
For t > 0, C(t) is always less than 1.25*B(t).
22 February, 2019

12. Introducing ‘big-O’ Notation
It is often useful to express a bound on the growth of a particular function in terms of a simpler function.
For example, for t > 45, B(t) is always greater than A(t), we can express the relation between A(t) and B(t) using the Ο (Omicron or ‘big-O’) notation:
This means that A(t) is “at most” B(t) beyond some value of t.
Formally, given functions f(x), g(x),
f(x) = O(g(x))
if there exist positive constants c and x0 such that f(x) ≤ cg(x) for all x ≥ x0 [Definition from JaJa not Grama! – more transparent].
22 February, 2019

13. From this definition, we can see that:
A(t) = O(t) (“at most” or “of the order t”),
B(t) = O(t2) (“at most” or “of the order t2”),
Finally, C(t) = O(t2), too.
Informally, big-O can be used to identify the simplest function that bounds (from above) a more complex function, as the parameter gets (asymptotically) bigger.
22 February, 2019

14. Theta and Omega There are two other useful symbols:
Omega (Ω) meaning “at least”:
Theta (Θ) meaning “equals” or “goes as”:
For formal definitions, see, for example, ‘An Introduction to Parallel Algorithms’ by JaJa or ‘Highly Parallel Computing’ by Almasi and Gottlieb.
Note that the definitions in Grama et al. are a little misleading!
22 February, 2019

15. Performance Modelling – Example
The following slides develop performance models for the example of a vector sum reduction.
The models are then used to support basic scalability analysis of the resulting parallel systems.
Consider two parallel systems:
First, a binary tree-based vector sum when the number of elements (N) is equal to the number of processors (P), N=P.
Second, a version for which N >> P.
Develop performance models.
Compare the models.
Consider the resulting system scalability.
22 February, 2019

16. Vector Sum Reduction (N = P)
Assume that:
N = P, and
N is a power of 2.
Propagate intermediate values through a binary tree of ‘adder’ nodes (processors):
Takes log2N steps with N processors (at least one of the processors is busy at every step, waiting for a message then doing an addition; the other processors have some idle time).
Each step thus requires time for communication of a single word (cost ts+tw) and a single addition (cost tc):
22 February, 2019

17. Vector Sum Speedup (N = P)
Speedup is poor, but monotonically increasing.
If N=128, Sabs is ~18 (Eabs = Sabs/P = ~0.14, i.e. 14%),
If N=1024, Sabs is ~100 (Eabs = ~0.1, i.e. 10%),
If N=1M, Sabs is ~ 52,000 (Eabs = ~0.05, i.e. 5%),
If N=1G, Sabs is ~ 35M (Eabs = ~ 0.035, i.e. 3.5%).
22 February, 2019

18. Vector Sum Scalability (N = P)
Efficiency:
But, N = P in this case, so:
Strong scaling not ‘good’, as we have seen (Eabs << 0.5).
Efficiency is monotonically decreasing
Reaches 50% point, Eabs = 0.5, when log2 P = 2, i.e. when P = 4.
This does not isoscale, either!
Eabs gets smaller as P (hence N) increases and P and N must change together.
22 February, 2019

19. When N >> P
When N >> P, each processor can be allocated N/P elements (for simplicity, assume N is exactly divisible by P).
Each processor sums its local elements in a first phase.
A binary tree sum of size P is then performed to sum the P partial results.
The new performance model is thus:
22 February, 2019

20. Strong Scalability (N >> P)
Speedup:
Strong scaling??
For a given problem size N (>> P), the (log2P/N) term is always ‘small’ so speedup will fall off ‘slowly’.
P is, of course, limited by the value of N, but we are considering the case where N >> P.
22 February, 2019

21. Isoscalability (N >> P)
Efficiency:
Now, we can always achieve a required efficiency on P processors by a suitable choice of N.
22 February, 2019

22. Isoscalability (N >> P)
For example, for 50% Eabs, isoefficiency function is:
Or, for Eabs > 50%, isoefficiency function is:
As N gets larger for a given P, Eabs gets closer to 1!
The ‘good’ parallel phase (N/P work) thus dominates the log2P phase as N gets larger, leading to relatively good (iso)scalability.
22 February, 2019

23. Summary of Performance Modelling
Performance modelling provides insight into the behaviour of parallel systems (parallel algorithms on parallel machines).
Performance modelling allows the comparison of algorithms and gives insight into their potential scalability.
Two main forms of scalability:
Strong scaling (fixed problem size N as P varies)
There is always a limit to strong scaling for real parallel systems (i.e. a value of P at which efficiency falls below an acceptable limit).
Isoscaling (the ability to maintain a specified level of efficiency by changing N as P varies).
Not all parallel systems isoscale.
Asymptotic (‘big-O’) analysis makes comparison easier, but BEWARE the constants!
Weak scaling is related to isoscaling – aim to maintain a fixed problem size per processor as P changes and look at the effect on efficiency.
22 February, 2019