Example of true dependence

S1: int a, b;
S2: a = 2;
S3: b = a + 40;

S2 ->T S3, meaning that S2 has a true dependence on S3 because S2 writes to the variable a, which S3 reads from.

Example of anti-dependence

S1: int a, b = 40;
S2: a = b - 38;
S3: b = -1;

S2 ->A S3, meaning that S2 has an anti-dependence on S3 because S2 reads from the variable b before S3 writes to it.

Example of output-dependence

S1: int a, b = 40;
S2: a = b - 38;
S3: a = 2;

S2 ->O S3, meaning that S2 has an output dependence on S3 because both write to the variable a.

Example of input-dependence

S1: int a, b, c = 2;
S2: a = c - 1;
S3: b = c + 1;

S2 ->I S3, meaning that S2 has an input dependence on S3 because S2 and S3 both read from variable c.

Loop-carried vs loop-independent dependence

Loops can have two types of dependence:

• Loop-carried dependence
• Loop-independent dependence

In loop-independent dependence, loops have inter-iteration dependence, but do not have dependence between iterations. Each iteration may be treated as a block and performed in parallel without other synchronization efforts.

In the following example code used for swapping the values of two array of length n, there is a loop-independent dependence of S1 ->T S3.

for (int i = 1; i < n; ++i) {
    S1: tmp = a[i];
    S2: a[i] = b[i];
    S3: b[i] = tmp;
}

In loop-carried dependence, statements in an iteration of a loop depend on statements in another iteration of the loop. Loop-Carried Dependence uses a modified version of the dependence notation seen earlier.

Example of loop-carried dependence where S1[i] ->T S1[i + 1], where i indicates the current iteration, and i + 1 indicates the next iteration.

for (int i = 1; i < n; ++i) {
    S1: a[i] = a[i-1] + 1;
}

Loop carried dependence graph

A Loop-carried dependence graph graphically shows the loop-carried dependencies between iterations. Each iteration is listed as a node on the graph, and directed edges show the true, anti, and output dependencies between each iteration.

DISTRIBUTED loop

When a loop has a loop-carried dependence, one way to parallelize it is to distribute the loop into several different loops. Statements that are not dependent on each other are separated so that these distributed loops can be executed in parallel. For example, consider the following code.

for (int i = 1; i < n; ++i) {
    S1: a[i] = a[i-1] + b[i];
    S2: c[i] += d[i];
}

The loop has a loop carried dependence S1[i] ->T S1[i+1] but S2 and S1 do not have a loop-independent dependence so we can rewrite the code as follows.

loop1: for (int i = 1; i < n; ++i) {
    S1: a[i] = a[i-1] + b[i];
}
loop2: for (int i = 1; i < n; ++i) {
    S2: c[i] += d[i];
}

Note that now loop1 and loop2 can be executed in parallel. Instead of single instruction being performed in parallel on different data as in data level parallelism, here different loops perform different tasks on different data. Let's say the time of execution of S1 and S2 be ${\displaystyle T_{S_{1}}}$ and ${\displaystyle T_{S_{2}}}$ then the execution time for sequential form of above code is ${\displaystyle n*(T_{S_{1}}+T_{S_{2}})}$, Now because we split the two statements and put them in two different loops, gives us an execution time of ${\displaystyle n*T_{S_{1}}+T_{S_{2}}}$. We call this type of parallelism either function or task parallelism.

DOALL parallelism

DOALL parallelism exists when statements within a loop can be executed independently (situations where there is no loop-carried dependence).^[1] For example, the following code does not read from the array a, and does not update the arrays b, c. No iterations have a dependence on any other iteration.

for (int i = 0; i < n; ++i) {
    S1: a[i] = b[i] + c[i];
}

Let's say the time of one execution of S1 be ${\displaystyle T_{S_{1}}}$ then the execution time for sequential form of above code is ${\displaystyle n*T_{S_{1}}}$, Now because DOALL Parallelism exists when all iterations are independent, speed-up may be achieved by executing all iterations in parallel which gives us an execution time of ${\displaystyle T_{S_{1}}}$, which is the time taken for one iteration in sequential execution.

The following example, using a simplified pseudo code, shows how a loop might be parallelized to execute each iteration independently.

begin_parallelism();
for (int i = 0; i < n; ++i) {
    S1: a[i] = b[i] + c[i];
    end_parallelism();
}
block();

DOACROSS parallelism

DOACROSS Parallelism exists where iterations of a loop are parallelized by extracting calculations that can be performed independently and running them simultaneously.^[5]

Synchronization exists to enforce loop-carried dependence.

Consider the following, synchronous loop with dependence S1[i] ->T S1[i+1].

for (int i = 1; i < n; ++i) {
    a[i] = a[i-1] + b[i] + 1;
}

Each loop iteration performs two actions

• Calculate a[i-1] + b[i] + 1
• Assign the value to a[i]

Calculating the value a[i-1] + b[i] + 1, and then performing the assignment can be decomposed into two lines(statements S1 and S2):

S1: int tmp = b[i] + 1;
S2: a[i] = a[i-1] + tmp;

The first line, int tmp = b[i] + 1;, has no loop-carried dependence. The loop can then be parallelized by computing the temp value in parallel, and then synchronizing the assignment to a[i].

post(0);
for (int i = 1; i < n; ++i) {

    S1: int tmp = b[i] + 1;
    wait(i-1);

    S2: a[i] = a[i-1] + tmp;
    post(i);
}

Let's say the time of execution of S1 and S2 be ${\displaystyle T_{S_{1}}}$ and ${\displaystyle T_{S_{2}}}$ then the execution time for sequential form of above code is ${\displaystyle n*(T_{S_{1}}+T_{S_{2}})}$, Now because DOACROSS Parallelism exists, speed-up may be achieved by executing iterations in a pipelined fashion which gives us an execution time of ${\displaystyle T_{S_{1}}+n*T_{S_{2}}}$.

DOPIPE parallelism

DOPIPE Parallelism implements pipelined parallelism for loop-carried dependence where a loop iteration is distributed over multiple, synchronized loops.^[1] The goal of DOPIPE is to act like an assembly line, where one stage is started as soon as there is sufficient data available for it from the previous stage.^[6]

Consider the following, synchronous code with dependence S1[i] ->T S1[i+1].

for (int i = 1; i < n; ++i) {
    S1: a[i] = a[i-1] + b[i];
    S2: c[i] += a[i];
}

S1 must be executed sequentially, but S2 has no loop-carried dependence. S2 could be executed in parallel using DOALL Parallelism after performing all calculations needed by S1 in series. However, the speedup is limited if this is done. A better approach is to parallelize such that the S2 corresponding to each S1 executes when said S1 is finished.

Implementing pipelined parallelism results in the following set of loops, where the second loop may execute for an index as soon as the first loop has finished its corresponding index.

for (int i = 1; i < n; ++i) {
    S1: a[i] = a[i-1] + b[i];
        post(i);
}

for (int i = 1; i < n; i++) {
        wait(i);
    S2: c[i] += a[i];
}

Let's say the time of execution of S1 and S2 be ${\displaystyle T_{S_{1}}}$ and ${\displaystyle T_{S_{2}}}$ then the execution time for sequential form of above code is ${\displaystyle n*(T_{S_{1}}+T_{S_{2}})}$, Now because DOPIPE Parallelism exists, speed-up may be achieved by executing iterations in a pipelined fashion which gives us an execution time of ${\displaystyle n*T_{S_{1}}+(n/p)*T_{S_{2}}}$, where p is the number of processor in parallel.