Distribute elements between equivalent arrays to achieve balanced sums - c

Question

I am given a set of elements from, say, 10 to 21 (always sequential),
I generate arrays of the same size, where size is determined runtime.
Example of 3 generated arrays (arrays # is dynamic as well as # of elements in all arrays, where some elements can be 0s - not used):
A1 = [10, 11, 12, 13]
A2 = [14, 15, 16, 17]
A3 = [18, 19, 20, 21]
these generated arrays will be given to different processes to to do some computations on the elements. My aim is to balance the load for every process that will get an array. What I mean is:
With given example, there are
A1 = 46
A2 = 62
A3 = 78
potential iterations over elements given for each thread.
I want to rearrange initial arrays to give equal amount of work for each process, so for example:
A1 = [21, 11, 12, 13] = 57
A2 = [14, 15, 16, 17] = 62
A3 = [18, 19, 20, 10] = 67
(Not an equal distribution, but more fair than initial). Distributions can be different, as long as they approach some optimal distribution and are better than the worst (initial) case of 1st and last arrays. As I see it, different distributions can be achieved using different indexing [where the split of arrays is made {can be uneven}]
This works fine for given example, but there may be weird cases..
So, I see this as a reflection problem (due to the lack of knowledge of proper definition), where arrays should be seen with a diagonal through them, like:
10|111213
1415|1617
181920|21
And then an obvious substitution can be done..
I tried to implement like:
if(rest == 0)
payload_size = (upper-lower)/(processes-1);
else
payload_size = (upper-lower)/(processes-1) + 1;
//printf("payload size: %d\n", payload_size);
long payload[payload_size];
int m = 0;
int k = payload_size/2;
int added = 0; //track what been added so far (to skip over already added elements)
int added2 = 0; // same as 'added'
int p = 0;
for (i = lower; i <= upper; i=i+payload_size){
for(j = i; j<(i+payload_size); j++){
if(j <= upper){
if((j-i) > k){
if(added2 > j){
added = j;
payload[(j-i)] = j;
printf("1 adding data: %d at location: %d\n", payload[(j-i)], (j-i));
}else{
printf("else..\n");
}
}else{
if(added < upper - (m+1)){
payload[(j-i)] = upper - (p*payload_size) - (m++);
added2 = payload[(j-i)];
printf("2 adding data: %d at location: %d\n", payload[(j-i)], (j-i));
}else{
payload[(j-i)] = j;
printf("2.5 adding data: %d at location: %d\n", payload[(j-i)], (j-i));
}
}
}else{ payload[(j-i)] = '\0'; }
}
p++;
k=k/2;
//printf("send to proc: %d\n", ((i)/payload_size)%(processes-1)+1);
}
..but failed horribly.
You definitely can see the problem in the implementation, because it is poorly scalable, not complete, messy, badly written and so on, and on, and on, ...
So, I need help either with the implementation or with an idea of a better approach to do what I want to achieve, given the description.
P.S. I need the solution to be as 'in-liney' as possible (avoid loop nesting) - that is why I am using bunch of flags and global indexes.
Surely this can be done with extra loops and unnecessary iterations. I invite people that can and appreciate t̲h̲e̲ ̲a̲r̲t̲ ̲o̲f̲ ̲i̲n̲d̲e̲x̲i̲n̲g̲ when it comes to arrays.
I am sure there is a solution somewhere out there, but I just cannot make an appropriate Google query to find it.
Hint? I thought of using index % size_of_my_data to achieve this task..
P.S. Application: described here

Answer 1

Here is an O(n) solution I wrote using deque (double-ended queue, a deque is not necessary and a simple array can be used, but a deque makes the code clean because of popRight and popLeft). The code is Python, not pseudocode, but it should be pretty to understand (because it's Python).:
def balancingSumProblem(seqStart = None, seqStop = None, numberOfArrays = None):
from random import randint
from collections import deque
seq = deque(xrange(seqStart or randint(1, 10),
seqStop and seqStop + 1 or randint(11,30)))
arrays = [[] for _ in xrange(numberOfArrays or randint(1,6))]
print "# of elements: {}".format(len(seq))
print "# of arrays: {}".format(len(arrays))
averageNumElements = float(len(seq)) / len(arrays)
print "average number of elements per array: {}".format(averageNumElements)
oddIteration = True
try:
while seq:
for array in arrays:
if len(array) < averageNumElements and oddIteration:
array.append(seq.pop()) # pop() is like popright()
elif len(array) < averageNumElements:
array.append(seq.popleft())
oddIteration = not oddIteration
except IndexError:
pass
print arrays
print [sum(array) for array in arrays]
balancingSumProblem(10,21,3) # Given Example
print "\n---------\n"
balancingSumProblem() # Randomized Test
Basically, from iteration to iteration, it alternates between grabbing large elements and distributing them evenly in the arrays and grabbing small elements and distributing them evenly in the arrays. It goes from out to in (though you could go from in to out) and tries to use what should be the average number of elements per array to balance it out further.
It's not 100 percent accurate with all tests but it does a good job with most randomized tests. You can try running the code here: http://repl.it/cJg

Answer 2

With a simple sequence to assign, you can just iteratively add the min and max elements to each list in turn. There are some termination details to fix up, but that's the general idea. Applied to your example the output would look like:
john-schultzs-macbook-pro:~ jschultz$ ./a.out
10 21 13 18 = 62
11 20 14 17 = 62
12 19 15 16 = 62
A simple reflection assignment like this will be optimal when num_procs evenly divides num_elems. It will be sub-optimal, but still decent, when it doesn't:
#include <stdio.h>
int compute_dist(int lower, int upper, int num_procs)
{
if (lower > upper || num_procs <= 0)
return -1;
int num_elems = upper - lower + 1;
int num_elems_per_proc_floor = num_elems / num_procs;
int num_elems_per_proc_ceil = num_elems_per_proc_floor + (num_elems % num_procs != 0);
int procs[num_procs][num_elems_per_proc_ceil];
int i, j, sum;
// assign pairs of (lower, upper) to each process until we can't anymore
for (i = 0; i + 2 <= num_elems_per_proc_floor; i += 2)
for (j = 0; j < num_procs; ++j)
{
procs[j][i] = lower++;
procs[j][i+1] = upper--;
}
// handle left overs similarly to the above
// NOTE: actually you could use just this loop alone if you set i = 0 here, but the above loop is more understandable
for (; i < num_elems_per_proc_ceil; ++i)
for (j = 0; j < num_procs; ++j)
if (lower <= upper)
procs[j][i] = ((0 == i % 2) ? lower++ : upper--);
else
procs[j][i] = 0;
// print assignment results
for (j = 0; j < num_procs; ++j)
{
for (i = 0, sum = 0; i < num_elems_per_proc_ceil; ++i)
{
printf("%d ", procs[j][i]);
sum += procs[j][i];
}
printf(" = %d\n", sum);
}
return 0;
}
int main()
{
compute_dist(10, 21, 3);
return 0;
}

Answer 3

I have used this implementation, which I mentioned in this report (Implementation works for cases I've used for testing (1-15K) (1-30K) and (1-100K) datasets. I am not saying that it will be valid for all the cases):
int aFunction(long lower, long upper, int payload_size, int processes)
{
long result, i, j;
MPI_Status status;
long payload[payload_size];
int m = 0;
int k = (payload_size/2)+(payload_size%2)+1;
int lastAdded1 = 0;
int lastAdded2 = 0;
int p = 0;
int substituted = 0;
int allowUpdate = 1;
int s;
int times = 1;
int times2 = 0;
for (i = lower; i <= upper; i=i+payload_size){
for(j = i; j<(i+payload_size); j++){
if(j <= upper){
if(k != 0){
if((j-i) >= k){
payload[(j-i)] = j- (m);
lastAdded2 = payload[(j-i)];
}else{
payload[(j-i)] = upper - (p*payload_size) - (m++) + (p*payload_size);
if(allowUpdate){
lastAdded1 = payload[(j-i)];
allowUpdate = 0;
}
}
}else{
int n;
int from = lastAdded1 > lastAdded2 ? lastAdded2 : lastAdded1;
from = from + 1;
int to = lastAdded1 > lastAdded2 ? lastAdded1 : lastAdded2;
int tempFrom = (to-from)/payload_size + ((to-from)%payload_size>0 ? 1 : 0);
for(s = 0; s < tempFrom; s++){
int restIndex = -1;
for(n = from; n < from+payload_size; n++){
restIndex = restIndex + 1;
payload[restIndex] = '\0';
if(n < to && n >= from){
payload[restIndex] = n;
}else{
payload[restIndex] = '\0';
}
}
from = from + payload_size;
}
return 0;
}
}else{ payload[(j-i)] = '\0'; }
}
p++;
k=(k/2)+(k%2)+1;
allowUpdate = 1;
}
return 0;
}