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# Optimized Merge Sort For DSP

This implementation of the Merge Sort algorithm has been created as part of a short study on the optimization of sorting algorithms for Texas Instruments DSPs. If you have any suggestions for further optimizations please do not hesitate to give us feedback.

Refer to our study about Optimized Sort Algorithms For DSP for some benchmark results and comparison with other sort algorithms.

## Contents

# Sort Function[edit]

static int do_merge(unsigned int **inputs, int *sizes, int count, unsigned int *output) { int i; int remainder = count & 3; int iterations = (count - remainder)/4; unsigned int **current_inputs = inputs; int *current_sizes = sizes; int merge_count = 0, merge_size; unsigned int *current_output = output; if (iterations > 0 && remainder > 0 && remainder < 3) { remainder += 10; iterations--; } for (i=0;i<iterations;i++) { /* Do a fourWays merge. */ merge_size = current_sizes[0] + current_sizes[1] + current_sizes[2] + current_sizes[3]; Merge_fourWays32(current_inputs, current_sizes, current_output, (unsigned int *)0/*unused*/); inputs[i] = current_output; sizes[i] = merge_size; merge_count++; current_inputs +=4; current_sizes +=4; current_output += merge_size; } /* Tail of the merges. */ switch (remainder) { case 11: /* Do one threeWays and one twoWays merge. */ merge_size = current_sizes[0] + current_sizes[1] + current_sizes[2]; Merge_threeWays32(current_inputs, current_sizes, current_output, (unsigned int *)0/*unused*/); inputs[merge_count] = current_output; sizes[merge_count] = merge_size; merge_count++; current_inputs +=3; current_sizes +=3; current_output += merge_size; /* Intentional fall-through. */ case 2: /* Here is the twoWays merge. */ merge_size = current_sizes[0] + current_sizes[1]; Merge_twoWays32(current_inputs, current_sizes, current_output, (unsigned int *)0/*unused*/); inputs[merge_count] = current_output; sizes[merge_count] = merge_size; merge_count++; current_inputs +=2; current_sizes +=2; current_output += merge_size; break; case 12: /* Do two threeWays merges. */ /* First threeWays merge. */ merge_size = current_sizes[0] + current_sizes[1] + current_sizes[2]; Merge_threeWays32(current_inputs, current_sizes, current_output, (unsigned int *)0/*unused*/); inputs[merge_count] = current_output; sizes[merge_count] = merge_size; merge_count++; current_inputs +=3; current_sizes +=3; current_output += merge_size; /* Second threeWays merge. Fall through, no "break" statement. */ case 3: case 13: /* Do one threeWays merge. */ merge_size = current_sizes[0] + current_sizes[1] + current_sizes[2]; Merge_threeWays32(current_inputs, current_sizes, current_output, (unsigned int *)0/*unused*/); inputs[merge_count] = current_output; sizes[merge_count] = merge_size; merge_count++; current_inputs +=3; current_sizes +=3; current_output += merge_size; break; case 1: /* There is nothing to do in this case. Why has this function been called? */ default: break; } return merge_count; } void Sort_merge32(unsigned int * restrict array, int size, unsigned int * restrict output, unsigned int * restrict scratch) { unsigned int *inputs[128]; int sizes[128]; unsigned int i1, i2, i3, i4; int insert_1st = 0, insert_2nd = 0, insert_3rd = 0, insert_4th = 0; unsigned int i_temp; int i, remainder, iterations, list_count = 0, shift; unsigned int offset = 0; unsigned int *scratch_output; remainder = size & 3; iterations = (size - remainder)/4; if (size>16) { offset += 16;} if (size>64) { offset += 64;} if (size>256) { offset += 256;} if (size>1024) { offset += 1024;} /* FIXME: For now we don't support sorting more than 4K elements. */ if (size <= 4) { scratch_output = output; } else { scratch_output = scratch+offset; } _nassert((int)array % 8 == 0); _nassert((int)scratch_output % 8 == 0); _nassert((int)output % 8 == 0); for (i=0;i<iterations;i++) { /* Prepare the array for the merges. */ inputs[i] = scratch_output+4*i; sizes[i] = 4; list_count++; /* Sort_sortFour32(array+i, scratch+i); */ i1 = array[4*i]; i2 = array[4*i+1]; i3 = array[4*i+2]; i4 = array[4*i+3]; insert_2nd = 0; insert_3rd = 0; insert_4th = 0; i_temp = i3; /* First element to be inserted. */ if (i4 < i3) { /* We just have to swap the two inputs if required. */ i3 = i4; i4 = i_temp; } /* Second element to be inserted. */ i_temp = i2; if (i4 < i2) insert_3rd = 1; if (i3 < i2) insert_2nd = 1 & ~insert_3rd; if (insert_2nd) { /* We just have to swap the two inputs if required. */ i2 = i3; i3 = i_temp; } if (insert_3rd) { i2 = i3; i3 = i4; i4 = i_temp; } /* Last element to be inserted */ insert_2nd = 0; insert_3rd = 0; i_temp = i1; if (i4 < i1) insert_4th = 1; if (i3 < i1) insert_3rd = 1 & ~insert_4th; if (i2 < i1) insert_2nd = 1 & ~insert_3rd & ~insert_4th; if (insert_2nd) { /* We just have to swap the two inputs if required. */ i1 = i2; i2 = i_temp; } if (insert_3rd) { i1 = i2; i2 = i3; i3 = i_temp; } if (insert_4th) { i1 = i2; i2 = i3; i3 = i4; i4 = i_temp; } #ifdef _TMS320C6X _amem8((void *)&(scratch_output[4*i])) = _itoll(i2, i1); _amem8((void *)&(scratch_output[4*i+2])) = _itoll(i4, i3); #else scratch_output[4*i] = i1; scratch_output[4*i+1] = i2; scratch_output[4*i+2] = i3; scratch_output[4*i+3] = i4; #endif } /* Merge the remaining elements if any. */ if (remainder > 0) { inputs[i] = scratch_output+4*i; sizes[i] = remainder; list_count++; } switch(remainder) { case 0: break; case 1: scratch_output[4*i] = array[4*i]; break; case 2: i1 = array[4*i]; i2 = array[4*i+1]; if (i1 <= i2) { scratch_output[4*i] = i1; scratch_output[4*i+1] = i2; } else { scratch_output[4*i] = i2; scratch_output[4*i+1] = i1; } break; case 3: i1 = array[4*i]; i2 = array[4*i+1]; i3 = array[4*i+2]; /* Sort the three entries. */ if (i2 < i1) { i_temp = i1; i1 = i2; i2 = i_temp; } if (i3 < i2) { i_temp = i2; i2 = i3; i3 = i_temp; } if (i2 < i1) { i_temp = i1; i1 = i2; i2 = i_temp; } /* Write the output. */ scratch_output[4*i] = i1; scratch_output[4*i+1] = i2; scratch_output[4*i+2] = i3; break; } if (list_count < 2) return; /* All array elements are now sorted by groups of 4. We are about to start merging all the array elements. */ shift = 16; while (list_count > 4) { /* We merge into a temporary space. */ scratch_output -= shift; shift *= 4; list_count = do_merge(inputs, sizes, list_count, scratch_output); } /* This is the final merge that will produce the final output. */ do_merge(inputs, sizes, list_count, output); }

# Merge Functions[edit]

The large majority of the CPU time used by our Merge Sort implementation is spent merging sorted arrays.

Our implementation provides three functions that have been carefully written to perform the merge of 2, 3 and 4 sorted arrays.

## Two-Ways Merge[edit]

This function merges in one pass the content of two arrays that were previously sorted in ascending order. It does not require any scratch memory.

void Merge_twoWays32 (unsigned int **inputs, int *sizes, unsigned int * restrict out, unsigned int * restrict scratch) { /* In this optimized version we are using restricts and replacing the original loop which had a complex condition in it with two loops. */ int i; int count1 = 0, count2 = 0; unsigned int * restrict in1 = inputs[0], * restrict in2 = inputs[1]; int size1 = sizes[0], size2 = sizes[1]; unsigned int val1 , val2; _nassert((int)in1 % 8 == 0); _nassert((int)in2 % 8 == 0); _nassert((int)out % 8 == 0); _nassert((int)scratch % 8 == 0); if (in1[size1-1] < in2[size2-1]) { /* Just swap the two lists. */ unsigned int *in_temp = in1; int size_temp = size1; in1 = in2; in2 = in_temp; size1 = size2; size2 = size_temp; } val1 = *in1; val2 = *in2; for (i=0; count2 < size2; i++) { /* Merge until 'in2' is exhausted first. */ if (val1 < val2) { *out++ = val1; val1 = *++in1; count1++; } else { *out++ = val2; val2 = *++in2; count2++; } } for (; count1 < size1; i++) { /* Just copy the remaining elements in 'in1' to 'out'. */ *out++ = *in1++; count1++; } }

## Three-Ways Merge[edit]

This function merges in one pass the content of three arrays that were previously sorted in ascending order. It does not require any scratch memory.

void Merge_threeWays32(unsigned int **inputs, int *sizes, unsigned int * restrict out, unsigned int * restrict scratch) { unsigned int * restrict in1 = inputs[0], * restrict in2 = inputs[1], * restrict in3 = inputs[2]; int size1 = sizes[0], size2 = sizes[1], size3 = sizes[2]; int count1 = 0, count2 = 0, count3 = 0; unsigned int value1 = *in1, value2 = *in2, value3 = *in3; int insert_2nd = 0, insert_3rd = 0; unsigned int *in_temp; unsigned int value_temp; int size_temp, count_temp; /* As the merge is progressing the various inputs are exhausted and the code deals differently with each stage. For the 'Four inputs' and 'Three inputs' stages we basically keep a 'list' of the inputs sorted (ascending) by the value of their current first value. */ /* PHASE #0: Sort the inputs one first time, sort is ascending. */ { insert_2nd = 0; insert_3rd = 0; in_temp = in2; value_temp = value2; size_temp = size2; /* First element to be inserted. */ if (value3 < value2) { size2 = size3; in2 = in3; value2 = value3; size3 = size_temp; in3 = in_temp; value3 = value_temp; } /* Second element to be inserted. */ in_temp = in1; value_temp = value1; size_temp = size1; if (value3 < value1) insert_3rd = 1; if (value2 < value1) insert_2nd = 1 & ~insert_3rd; if (insert_2nd) { /* We just have to swap the two inputs if required. */ size1 = size2; in1 = in2; value1 = value2; size2 = size_temp; in2 = in_temp; value2 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ size1 = size2; in1 = in2; value1 = value2; size2 = size3; in2 = in3; value2 = value3; size3 = size_temp; in3 = in_temp; value3 = value_temp; } } /* PHASE #1: Three inputs processing. */ /* The list of inputs is already sorted the first time. */ *out++ = value1; value1 = *++in1; count1++; for (;count1<size1;) { insert_2nd = 0; insert_3rd = 0; /* Sort the inputs. The nice thing is that we just have insert 'value1' in the right position because the rest of the list (2 other elements) is already sorted. */ if (value3 < value1) insert_3rd = 1; if (value2 < value1) insert_2nd = 1 & ~insert_3rd; if (insert_2nd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count_temp; size2 = size_temp; in2 = in_temp; value2 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; count3 = count_temp; size3 = size_temp; in3 = in_temp; value3 = value_temp; } /* Add the first element of 'in1' to the 'out'. */ *out++ = value1; value1 = *++in1; count1++; } /* Get rid of the first input, and shift the other two into position. */ count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; /* PHASE #2: Two inputs processing. */ /* The list of inputs is already sorted the first time. */ *out++ = value1; value1 = *++in1; count1++; for (;count1<size1;) { /* Sort the inputs. The nice thing is that we just have insert 'value1' in the right position because the rest of the list (1 other element) is already sorted. */ if (value2 < value1) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count_temp; size2 = size_temp; in2 = in_temp; value2 = value_temp; } /* Add the first element of 'in1' to the 'out'. */ *out++ = value1; value1 = *++in1; count1++; } /* Get rid of the first input, and shift the other two into position. */ count1 = count2; size1 = size2; in1 = in2; value1 = value2; /* PHASE #3: One input left. */ for (; count1 < size1; ) { /* Just copy the remaining elements in 'in1' to 'out'. */ *out++ = value1; value1 = *++in1; count1++; } }

## Four-Ways Merge[edit]

This function merges in one pass the content of four arrays that were previously sorted in ascending order. It does not require any scratch memory.

void Merge_fourWays32(unsigned int **inputs, int *sizes, unsigned int * restrict out, unsigned int * restrict scratch) { unsigned int * restrict in1 = inputs[0], * restrict in2 = inputs[1], * restrict in3 = inputs[2], * restrict in4 = inputs[3]; int size1 = sizes[0], size2 = sizes[1], size3 = sizes[2], size4 = sizes[3]; int count1 = 0, count2 = 0, count3 = 0, count4 = 0; unsigned int value1 = *in1, value2 = *in2, value3 = *in3, value4 = *in4; int insert_2nd = 0, insert_3rd = 0, insert_4th = 0; unsigned int *in_temp; unsigned int value_temp; int size_temp, count_temp; /* As the merge is progressing the various inputs are exhausted and the code deals differently with each stage. For the 'Four inputs' and 'Three inputs' stages we basically keep a 'list' of the inputs sorted (ascending) by the value of their current first value. */ /* PHASE #0: Sort the inputs one first time, sort is ascending. */ { insert_2nd = 0; insert_3rd = 0; insert_4th = 0; in_temp = in3; value_temp = value3; size_temp = size3; /* First element to be inserted. */ if (value4 < value3) { size3 = size4; in3 = in4; value3 = value4; size4 = size_temp; in4 = in_temp; value4 = value_temp; } /* Second element to be inserted. */ in_temp = in2; value_temp = value2; size_temp = size2; if (value4 < value2) insert_3rd = 1; if (value3 < value2) insert_2nd = 1 & ~insert_3rd; if (insert_2nd) { /* We just have to swap the two inputs if required. */ size2 = size3; in2 = in3; value2 = value3; size3 = size_temp; in3 = in_temp; value3 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ size2 = size3; in2 = in3; value2 = value3; size3 = size4; in3 = in4; value3 = value4; size4 = size_temp; in4 = in_temp; value4 = value_temp; } /* Last element to be inserted */ insert_2nd = 0; insert_3rd = 0; insert_4th = 0; in_temp = in1; value_temp = value1; size_temp = size1; if (value4 < value1) insert_4th = 1; if (value3 < value1) insert_3rd = 1 & ~insert_4th; if (value2 < value1) insert_2nd = 1 & ~insert_3rd & ~insert_4th; if (insert_2nd) { /* We just have to swap the two inputs if required. */ size1 = size2; in1 = in2; value1 = value2; size2 = size_temp; in2 = in_temp; value2 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ size1 = size2; in1 = in2; value1 = value2; size2 = size3; in2 = in3; value2 = value3; size3 = size_temp; in3 = in_temp; value3 = value_temp; } if (insert_4th) { /* We just have to swap the two inputs if required. */ size1 = size2; in1 = in2; value1 = value2; size2 = size3; in2 = in3; value2 = value3; size3 = size4; in3 = in4; value3 = value4; size4 = size_temp; in4 = in_temp; value4 = value_temp; } } /* PHASE #1: Four inputs processing. */ /* The list of inputs is already sorted the first time. */ *out++ = value1; value1 = *++in1; count1++; for (;count1<size1;) { insert_2nd = 0; insert_3rd = 0; insert_4th = 0; /* Sort the inputs. The good thing is that we just have insert 'value1' in the right position because the rest of the list (3 other elements) is already sorted. */ if (value4 < value1) insert_4th = 1; if (value3 < value1) insert_3rd = 1 & ~insert_4th; if (value2 < value1) insert_2nd = 1 & ~insert_3rd & ~insert_4th; if (insert_2nd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count_temp; size2 = size_temp; in2 = in_temp; value2 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; count3 = count_temp; size3 = size_temp; in3 = in_temp; value3 = value_temp; } if (insert_4th) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; count3 = count4; size3 = size4; in3 = in4; value3 = value4; count4 = count_temp; size4 = size_temp; in4 = in_temp; value4 = value_temp; } /* Add the first element of 'in1' to the 'out'. */ *out++ = value1; value1 = *++in1; count1++; } /* Get rid of the first input, and shift the other three into position. */ count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; count3 = count4; size3 = size4; in3 = in4; value3 = value4; /* PHASE #2: Three inputs processing. */ /* The list of inputs is already sorted the first time. */ *out++ = value1; value1 = *++in1; count1++; for (;count1<size1;) { insert_2nd = 0; insert_3rd = 0; /* Sort the inputs. The nice thing is that we just have insert 'value1' in the right position because the rest of the list (2 other elements) is already sorted. */ if (value3 < value1) insert_3rd = 1; if (value2 < value1) insert_2nd = 1 & ~insert_3rd; if (insert_2nd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count_temp; size2 = size_temp; in2 = in_temp; value2 = value_temp; } if (insert_3rd) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; count3 = count_temp; size3 = size_temp; in3 = in_temp; value3 = value_temp; } /* Add the first element of 'in1' to the 'out'. */ *out++ = value1; value1 = *++in1; count1++; } /* Get rid of the first input, and shift the other two into position. */ count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count3; size2 = size3; in2 = in3; value2 = value3; /* PHASE #3: Two inputs processing. */ /* The list of inputs is already sorted the first time. */ *out++ = value1; value1 = *++in1; count1++; for (;count1<size1;) { /* Sort the inputs. The nice thing is that we just have insert 'value1' in the right position because the rest of the list (1 other element) is already sorted. */ if (value2 < value1) { /* We just have to swap the two inputs if required. */ in_temp = in1; value_temp = value1; count_temp = count1; size_temp = size1; count1 = count2; size1 = size2; in1 = in2; value1 = value2; count2 = count_temp; size2 = size_temp; in2 = in_temp; value2 = value_temp; } /* Add the first element of 'in1' to the 'out'. */ *out++ = value1; value1 = *++in1; count1++; } /* Get rid of the first input, and shift the other two into position. */ count1 = count2; size1 = size2; in1 = in2; value1 = value2; /* PHASE #4: One input left. */ for (; count1 < size1; ) { /* Just copy the remaining elements in 'in1' to 'out'. */ *out++ = value1; value1 = *++in1; count1++; } }