Generate n pairs of random integers, whose 0 <= pairwise_sum <= k

I want to generate n pairs of integers {(x_i, y_i)}, such that x_i + y_i <= k. One way I can think of is to generate one pair at a time

pairs = zeros(n,2)
for i=1:n
  pairs(i,1) = randi(0,k);
  pairs(i,2) = randi(0,k - pairs(i,1));
end

But, certainly, for loop is not favorable in matlab. Also, I'm not sure if that guarantees randomness either. I wonder if there's any existing function or a better way to do that?

Variation: consider a constraint that we want m% of generated pairs have pairwise_sum <= h ( h is some number < k). My approach won't work in this case. I'd love to get some hints to solve that as well.

Many thanks,

Answer 1

[José-Luis]

num_pairs = 100;
max_val = 10;

tot_sum = randi(max_val + 1,[num_pairs,1]) - 1; %Need to include 0
result = rand(num_pairs,2);
result = bsxfun(@rdivide,result, sum(result,2));
result = round(bsxfun(@times,result,tot_sum));

plot(result)

I don't understand your second constraint.

[Khanh]

Thank you so much, Jose. The use of bsxfun is so cool. Just want to make sure I understand it right, the last 2 lines are equivalent to the following, right?

result = result ./ sum(result,2);
result = round(result .* tot_sum)

For the constraint, for example with max_val = 10 in your example, what if I want to assure that 70% of the pairs generated have sum <= 7 (h = 7 in this case). How can I do that?

Thank you,

[José-Luis]

Yes, it is equivalent, except that it applies it to both columns.

Two ways to do the second constraint pop to my mind:

1. Generate 70 samples with that constraint and add 30 with one such that 7 < h < 10
2. Trim the generated sample until you reach your condition (get rid of some of the rows)

Please accept the answer that best solves your problem.

[Khanh]

That makes sense. Thanks.