Bernstein Conditions for Detection of Parallelism

For implementation of instructions or block of instructions in parallel, it should be guaranteed that the instructions are independent of each other. These instructions can be / control dependent / data dependent resource dependent on each other.  We are considering only data dependency between the statements for taking decisions of parallel implementation. A.J. Bernstein has implicated the work of data dependency and resultant some conditions based on which we can settle on the parallelism of processes or instructions.

Bernstein conditions are rely on the subsequent two sets of variables:

i)  The input set or Read set RI that consists of memory locations examine by the statement of instruction I₁.

ii)  The output set or Write set WI  that consists of memory locations printed into by instruction I₁.

The sets W_I and R_I are not disjoint as the similar locations are used for writing and reading by S_I.

The following are Bernstein Parallelism conditions which are determine to conclude whether the statements are parallel or not:

1) Position in R₁ from which S₁ reads and the Position W₂  onto which S₂ writes must be commonly exclusive. It  means S₁does'nt read from any memory location onto which S₂ writes. It can be indicated as:

2) Likewise, locations in R₂ from which S₁ writes and the locations W₁ onto which S₂ reads must be commonly exclusive.It means S₂  does not read from some memory location onto which S₁ writes. It can be indicate as:

3) The memory locations W₁ and W₂ onto which S₁ and S₂ write, should not be examine by S₁ and S₂. It means R₁ and R₂ should be free of W₁ and W₂.  It can be indicated as :

To demonstrate the operation of Bernstein's conditions, let consider the following instructions of sequential program:

I₁ : x = (a + b) / (a * b)

I₂ : y = (b + c) * d

I₃ : z = x2 + (a * e)

Now, the read set and write set of I1, I2 and I3 are as given:

R₁ = {a,b}                W1 = {x}

R₂ = {b,c,d}             W2 = {y}

R₃ = {x,a,e}             W3 = {z}

Now let us search out whether I₁ and I₂ are parallel or not