Maximum subset which has no sum of two divisible by K

Question

I am given the set {1, 2, 3, ... ,N}. I have to find the maximum size of a subset of the given set so that the sum of any 2 numbers from the subset is not divisible by a given n

Answer 1

first calculate all of the set elements mod k.and solve simple problem: find the maximum size of a subset of the given set so that the sum of any 2 numbers from the subset is not equal by a given number K. i divide this set to two sets (i and k-i) that you can not choose set(i) and set(k-i) Simultaneously.

int myset[]
int modclass[k]

for(int i=0; i< size of myset ;i++)
{
    modclass[(myset[i] mod k)] ++;
}

choose

for(int i=0; i< k/2 ;i++)
{ 
    if (modclass[i] > modclass[k-i])
    {
        choose all of the set elements that the element mod k equal i
    }
    else
    {
        choose all of the set elements that the element mod k equal k-i
    }
}

finally you can add one element from that the element mod k equal 0 or k/2.

this solution with an algorithm of complexity O(K).

you can improve this idea with dynamic array:

for(int i=0; i< size of myset ;i++)
{
    x= myset[i] mod k;
    set=false;
    for(int j=0; j< size of newset ;j++)
    {
        if(newset[j][1]==x or newset[j][2]==x)
        {
            if (x < k/2)
            {
                newset[j][1]++;
                set=true;
            }
            else
            {
                newset[j][2]++;
                set=true;
            }
        }
    }
    if(set==false)
    {
        if (x < k/2)
        {
            newset.add(1,0);
        }
        else
        {
            newset.add(0,1);
        }
    }
}

now you can choose with an algorithm of complexity O(myset.count).and your algorithm is more than O(myset.count) because you need O(myset.count) for read your set. complexity of this solution is O(myset.count^2),that you can choose algorithm depended your input.with compare between O(myset.count^2) and o(k). and for better solution you can sort myset based on mod k.

Answer 2

I'm assuming that the set of numbers is always 1 through N for some N.

Consider the first N-(N mod K) numbers. The form floor(N/K) sequences of K consecutive numbers, with reductions mod K from 0 through K-1. For each group, floor(K/2) have to be dropped for having a reduction mod K that is the negation mod K of another subset of floor(K/2). You can keep ceiling(K/2) from each set of K consecutive numbers.

Now consider the remaining N mod K numbers. They have reductions mod K starting at 1. I have not worked out the exact limits, but if N mod K is less than about K/2 you will be able to keep all of them. If not, you will be able to keep about the first ceiling(K/2) of them.

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I believe the concept here is correct, but I have not yet worked out all the details.

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Here is my analysis of the problem and answer. In what follows |x| is floor(x). This solution is similar to the one in @Constantine's answer, but differs in a few cases.

Consider the first K*|N/K| elements. They consist of |N/K| repeats of the reductions modulo K.

In general, we can include |N/K| elements that are k modulo K subject to the following limits:

If (k+k)%K is zero, we can include only one element that is k modulo K. That is the case for k=0 and k=(K/2)%K, which can only happen for even K.

That means we get |N/K| * |(K-1)/2| elements from the repeats.

We need to correct for the omitted elements. If N >= K we need to add 1 for the 0 mod K elements. If K is even and N>=K/2 we also need to add 1 for the (K/2)%K elements.

Finally, if M(N)!=0 we need to add a partial or complete copy of the repeat elements, min(N%K,|(K-1)/2|).

The final formula is:

|N/K| * |(K-1)/2| +
(N>=K ? 1 : 0) +
((N>=K/2 && (K%2)==0) ? 1 : 0) +
min(N%K,|(K-1)/2|)

This differs from @Constantine's version in some cases involving even K. For example, consider N=4, K=6. The correct answer is 3, the size of the set {1, 2, 3}. @Constantine's formula gives |(6-1)/2| = |5/2| = 2. The formula above gets 0 for each of the first two lines, 1 from the third line, and 2 from the final line, giving the correct answer.