Loop Constructs

Loop Exit Conditions

Loop exit conditions determine the number of iterations a loop executes. For example, fixed indexes for loops determine the iterations. The loop iterations must be countable; in other words, the number of iterations must be expressed as one of the following:

• A constant.

• A loop invariant term.

• A linear function of outermost loop indices.

In the case where a loops exit depends on computation, the loops are not countable. The examples below show loop constructs that are countable and non-countable.

void cnt1(float a[], float b[], float c[],
          int n, int lb) { 
// Exit condition specified by "N-1b+1"
  int cnt=n, i=0;
  while (cnt >= lb) { 
// lb is not affected within loop.
    a[i] = b[i] * c[i];
    cnt--;
    i++;
  } 
}

The following example demonstrates a different countable loop construct.

void cnt2(float a[], float b[], float c[],
          int m, int n) 
{ 
// Number of iterations is "(n-m+2)/2".
  int i=0, l;
  for (l=m; l<n; l+=2) {
    a[i] = b[i] * c[i];
    i++;
  } 
}

The following examples demonstrates a loop construct that is non-countable due to dependency loop variant count value.

void no_cnt(float a[], float b[], float c[]) {
  int i=0; 
// Iterations dependent on a[i].
  while (a[i]>0.0) {
    a[i] = b[i] * c[i];
    i++;
  } 
}

Strip-mining and Cleanup

Strip-mining, also known as loop sectioning, is a loop transformation technique for enabling SIMD-encodings of loops, as well as a means of improving memory performance. By fragmenting a large loop into smaller segments or strips, this technique transforms the loop structure in two ways:

• By increasing the temporal and spatial locality in the data cache if the data are reusable in different passes of an algorithm.

• By reducing the number of iterations of the loop by a factor of the length of each vector, or number of operations being performed per SIMD operation. In the case of Streaming SIMD Extensions, this vector or strip-length is reduced by four times: four floating-point data items per single Intel® SSE single-precision floating-point SIMD operation are processed.

First introduced for vectorizers, this technique consists of the generation of code when each vector operation is done for a size less than or equal to the maximum vector length on a given vector machine.

The compiler automatically strip-mines your loop and generates a cleanup loop. For example, assume the compiler attempts to strip-mine the following loop:

i=0; 
while(i<n) {
  // Original loop code 
  a[i]=b[i]+c[i];
  ++i; 
}

The compiler might handle the strip mining and loop cleaning by restructuring the loop in the following manner:

// The vectorizer generates the following two loops 
i=0; 
while(i<(n-n%4)) {
  // Vector strip-mined loop
  // Subscript [i:i+3] denotes SIMD execution
  a[i:i+3]=b[i:i+3]+c[i:i+3];
  i=i+4; 
} 
while(i<n) {
  // Scalar clean-up loop
  a[i]=b[i]+c[i];
  ++i; 
}

Loop Blocking

It is possible to treat loop blocking as strip-mining in two or more dimensions. Loop blocking is a useful technique for memory performance optimization. The main purpose of loop blocking is to eliminate as many cache misses as possible. This technique transforms the memory domain into smaller chunks rather than sequentially traversing through the entire memory domain. Each chunk should be small enough to fit all the data for a given computation into the cache, thereby maximizing data reuse.

Consider the following example, loop blocking allows arrays A and B to be blocked into smaller rectangular chunks so that the total combined size of two blocked (A and B) chunks is smaller than cache size, which can improve data reuse.

#include <time.h> 
#include <stdio.h> 
#define MAX 7000 

void add(int a[][MAX], int b[][MAX]); 
int main() { 
int i, j; 
int A[MAX][MAX]; 
int B[MAX][MAX]; 
time_t start, elaspe; 
int sec; 

//Initialize array 
for(i=0;i<MAX;i++) {
  for(j=0;j<MAX; j++) {
    A[i][j]=j;
    B[i][j]=j;
  }
}

 start= time(NULL);
 add(A, B);
 elaspe=time(NULL);
 sec = elaspe - start;
 printf("Time %d",sec); //List time taken to complete add function 
} 

void add(int a[][MAX], int b[][MAX]) {
 int i, j;
 for(i=0;i<MAX;i++) {
  for(j=0; j<MAX;j++ {
     a[i][j] = a[i][j] + b[j][i]; //Adds two matrices
    }
  } 
}

The following example illustrates loop blocking the add function (from the previous example). In order to benefit from this optimization you might have to increase the cache size.

#include <stdio.h> 
#include <time.h> 
#define MAX 7000 
void add(int a[][MAX], int b[][MAX]); 

int main() { 
  #define BS 8  //Block size is selected as the loop-blocking factor. 
  int i, j; 
  int A[MAX][MAX]; 
  int B[MAX][MAX]; 
  time_t start, elaspe; 
  int sec; 

//initialize array 
for(i=0;i<MAX;i++) {
  for(j=0;j<MAX;j++) {
    A[i][j]=j;
    B[i][j]=j;
  }
} 
start= time(NULL);

add(A, B); 
elaspe=time(NULL); 
sec = elaspe - start; 
printf("Time %d",sec); //Display time taken to complete loopBlocking function 
} 

void add(int a[][MAX], int b[][MAX]) { 
  int i, j, ii, jj; 
  for(i=0;i<MAX;i+=BS) {
   for(j=0; j<MAX;j+=BS) {
     for(ii=i; ii<i+BS; ii++) {   //outer loop
       for(jj=j;jj<j+BS; jj++) {  //Array B experiences one cache miss
                                  //for every iteration of outer loop
         a[ii][jj] = a[ii][jj] + b[jj][ii]; //Add the two arrays
        }
      }
    }
  } 
}

Loop Interchange and Subscripts: Matrix Multiply

Loop interchange is often used for improving memory access patterns. Matrix multiplication is commonly written as shown in the following example:

void matmul_slow(float *a[], float *b[], float *c[]) {
  int N = 100;
  for (int i = 0; i < N; i++)
    for (int j = 0; j < N; j++)
      for (int k = 0; k < N; k++)
        c[i][j] = c[i][j] + a[i][k] * b[k][j]; 
}

The use of B(K,J) is not a stride-1 reference and therefore will not be vectorized efficiently.

If the loops are interchanged, however, all the references will become stride-1 as shown in the following example.

void matmul_fast(float *a[], float *b[], float *c[]) {
  int N = 100;
  for (int i = 0; i < N; i++)
    for (int k = 0; k < N; k++)
      for (int j = 0; j < N; j++)
        c[i][j] = c[i][j] + a[i][k] * b[k][j]; 
}

Interchanging is not always possible because of dependencies, which can lead to different results.