Finding combinations in a set of numbers efficiently

Question

I am currently working on a java project, and I need one of part of it to give me a list of combination of numbers that fullfils
1<= A<=B<=C<=D<=E<=F<=N where N is any integer as small as 75 which will be taken as the input. ABCDEF can be any integer as long as it fulfills the equality .

I know that I can just simply just go through every combination using brute force but it takes to long. What I want to try to do is split the equality into two separate equalities in a way that still fulfills the original, but it would cut the run in almost half.

Answer 1

Assuming you want a list of all possible combinations of A, B, C, D, E, F that satisfy the specified condition, you can't get more efficient than doing a brute force backtracking search .

For each acceptable value of A : find all acceptable values of B , then for each of those, find for C ... and so on.

You'll get equivalent run time with:

• divide and conquer
• dynamic programming
• greedy (which just reduces to backtracking)

(but these algorithms aren't suitable for this problem and would require contrived implementations)

Answer 2

The following will only generate valid valid combinations. It might take a while for large values of N, but that is because there will be a large number of combinations that satisfy your inequality.

N = 75; 300500200 combinations

N = 100; 1609344100 combinations

    for(int A = 1; A <= N ; ++A) {
        for(int B = A; B <= N; ++B) {
            for(int C = B; C <= N; ++C) {
                for(int D = C; D <= N; ++D) {
                    for(int E = D; E <= N; ++E) {
                        for(int F = E; F <=N; ++F) {
                            // do whatever you want with the combo
                        }
                    }
                }
            }
        }
    }