Parallel and Distributed Computing Introduction Exercise Guide (3)

Exercise guidance

Equivalence

The equivalence relationship reflects the relationship between increasing the processor and increasing the problem size in order to maintain the parallel efficiency.
Under a certain parallel implementation of the problem, write $T(n,p)$ is the time required to use k processors to calculate a problem of scale n,$T_0(n,p)$ is the time spent on communication and redundant calculations for a problem with a scale of n using k processors. The equivalent relationship of the problem is as follows:
$$T(n,1)\ge C{T}_0(n,p)$$
where$C=E/(1-E)$，$E$ is parallel efficiency. Since the subsequent calculations are mainly concerned with the magnitude of the function, there is no need to substituteEE in theinequality$In\; the$
actual calculation of$E$ , the time complexity of the original serial algorithm (that is, the baseline originally used) is$g\left(n\right)$ , the communication time complexity is$h(n,p)$ , can be simplified as
$$g(n)\ge Ch(n,p)$$
Among them, g and h can be directly replaced by the complexity metric level, and the degree of the highest complexity term can be directly incorporated into C
to simplify the inequality to obtain
$$n\ge f\left(p\right)$$
This is the equal speedup relationship of the parallel system

Scalability function

For a problem of size n, mark $M\left(n\right)$ is the memory required by the problem, then the scalability function of the parallel implementation of the problem is
$$M\left(f\right(p\left)\right)/p$$
This function indicates how the memory capacity required by each processor increases as a function of p in order to maintain efficiency

Scalability of parallel implementation

Parallel implementation of functions with a higher level of scalability has a faster increase in memory and will reach the actual memory of the machine earlier, resulting in poor scalability; the opposite is better