Question

I am implementing a collision detection algorithm stores the distance between all the objects in a single octree node. For instance if there are 4 objects in the node, there is a distance between objects 1&2, 1&3, 1&4, 2&3, 2&4 and 3&4. The formula for the total number of pairs is t = n * (n-1) / 2, where t is the total number of pairs and n is the number of objects in a node.

My question is, how do I convert from a position in the list to a pair of objects. For instance, using the above list of pairs, 3 would return the pair 2&3.

To save space in memory, the list is just a list of floats for the distance instead of containing distance and pointers to 2 objects.

I am unsure how to mathematically convert the single list index to a pair of numbers. Any help would be great. I am hoping to be able to break this down to 2 functions, the first returns the first object in the pair and the second returns the second, both the functions taking 2 variables, one being the index and the other being the total objects in the node. If possible I would like to make a function without any looping or having a recursive function because this will be run in real time for my collision detection algorithm.

Answer 1

Based on my understanding of the question, one way to get a pair a&b (1-based, 2&3 in your example) from the index (0-based, 3 in your example) and the number of objects n (4 in your example) is:

t = n * (n - 1) / 2;
a = n - floor((1 + sqrt(1 + 8 * (t - index - 1))) / 2);
b = index + (n - a) * (n - a + 1) / 2 - t + a + 1;

Some credits to http://oeis.org/A002024

Generalized algorithms (for tuples rather than pairs) can be found at Calculate Combination based on position and http://saliu.com/bbs/messages/348.html, but they seem to involve calculating combinations in a loop.

Edit: a nicer formula for a (from the same source):

a = n - floor(0.5 + sqrt(2 * (t - index)));