hash function providing unique uint from an integer coordinate pair

Question

The problem in general: I have a big 2d point space, sparsely populated with dots. Think of it as a big white canvas sprinkled with black dots. I have to it

Answer 1

Your "ideal" is impossible.

You want a mapping (x, y) -> i where x, y, and i are all 32-bit quantities, which is guaranteed not to generate duplicate values of i.

Here's why: suppose there is a function hash() so that hash(x, y) gives different integer values. There are 2^32 (about 4 billion) values for x, and 2^32 values of y. So hash(x, y) has 2^64 (about 16 million trillion) possible results. But there are only 2^32 possible values in a 32-bit int, so the result of hash() won't fit in a 32-bit int.

See also http://en.wikipedia.org/wiki/Counting_argument

Generally, you should always design your data structures to deal with collisions. (Unless your hashes are very long (at least 128 bit), very good (use cryptographic hash functions), and you're feeling lucky).