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In parallel computing, I know the speed up equation is

$$ \frac{1}{ s + \frac{1-s}{p} } $$

But what is meant by superlinear speed up? Is it something theoritical? Could you explain it with equations?

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  • $\begingroup$ In what context have you come across the term "superlinear speedup"? $\endgroup$ – David Richerby Apr 4 '16 at 15:27
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    $\begingroup$ @DavidRicherby It's a pretty common term in the high-performance computing community. $\endgroup$ – Anton Trunov Apr 4 '16 at 17:20
  • $\begingroup$ @AntonTrunov OK, great. But many of us aren't in the high-performance computing community and none of those words appears in the question. In other contexts, "linear speedup" means other things. $\endgroup$ – David Richerby Apr 4 '16 at 17:22
  • $\begingroup$ @DavidRicherby The OP mentions "parallel computing". The HPC people prefer the term "HPC" nowadays , not "parallel computing", since parallel hardware and software are just means to achieve high performance. $\endgroup$ – Anton Trunov Apr 4 '16 at 17:27
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With equation: not really.

Superlinear speedup comes from exceeding naively calculated speedup even after taking into account the communication process (which is fading, but still this is the bottleneck).

For example, you have serial algorithm that takes $1t$ to execute. You have $1024$ cores, so naive speedup is $1024x$, or it takes $t/1024$, but it should be calculated like in your equation taking into account memory transfer, slight modifications to the algorithm, parallelisation time.

So speedup should be lower than 1024x, but sometimes it happens that speedup is bigger, then we call it $superlinear$.

Where it comes from?
From several places: cache usage (what fits into registers, main memory or mass storage, where very often more processing units gives overall more registers per subtask), memory hit patterns, simply better (or a slightly different) algorithm, flaws in the serial code.
For example, a random process that searches space for a result is now divided into $1024$ searchers covering more space at once so finding the solution faster is more probable. There are byproducts (if you swap elements like in bubble sort and switch into GPU it swaps all pairs at once, while serial only up to point).

On the distributed system communication is even more costly, so programs are changed to make memory usage local (which also changes memory access, divides problem differently than in sequential application).

And the most important, the sequential program is not ideally the same as the parallel version - different technology, environment, algorithm, etc. so it is hard to compare them.

Excerpt from "Introduction to Parallel Computing" Second edition by Ananth Grama, 2003

Theoretically speedup can never exceed the number of processing elements $p$.
If the best sequential algorithm takes $T_s$ units of time to solve a given problem on a single processing element, then a speedup of $p$ can be obtained on $p$ processing elements if none of them spends more than time $T_s/p$.
A speedup greater than $p$ is possible only if each processing element spends less than time $T_s/p$ solving the problem.
In this case, a single processing element could emulate the $p$ processing elements and solve the problem in fewer than $T_s$ units of time.
This is a contradiction because speedup, by definition is computed with respect to the best sequential algorithm.

So the name "superlinear" in this context comes from the definition of speedup.

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My layman explanation can help with the imagination of how it works.

Certain types of algorithms lead to superlinear speedup if HPC interconnect architecture makes it possible to have references between internal states of cores. The reason is information (and thermodynamic) theoretical and can be simply explained with the following description.

Take two systems containing 3 bits of possible states in each:

011 and 101

The maximum number of possible states of this system is 2 * 2^3 = 16.

Now let's combine these two ensembles into a composed one, simply having 6 bits of possible state space:

011101

The maximum number of possible states of this system is 2^6 = 64. Combined system has available entropy increase of 64 / 16 = 4 times.

Composition of ensembles having bigger variety of internal states leads to exponentially bigger "speedups", because cores can hypothetically have more mutual references.

In chemistry, energy is released when systems combine (think nuclear fusion). That energy comes exactly from the possibility to address more states for combined systems.

The sources I've read were all describing parallel algorithms on isolated cores, where there were no ability to have mutual references between state bits of different cores. From the point of view of entropy increase of combined states, I can imagine how superlinear speedup is possible.

For information-theoretical details, see "The mixing paradox" in the Wikipedia article https://en.wikipedia.org/wiki/Gibbs_paradox on the exact thermodynamic nature of this behavior.

This has some philosophical implications for human-human communication and the importance of live speech conversations versus typing text and reading books.

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