Bijection on the integers below x

i'm working on image processing, a开发者_如何学编程nd i'm writing a parallel algorithm that iterates over all the pixels in an image, and changes the surrounding pixels based on it's value. In this algorithm, minor non-deterministic is acceptable, but i'd rather minimize it by only querying distant pixels simultaneously. Could someone give me an algorithm that bijectively maps the integers below n to the integers below n, in a fast and simple manner, such that two integers that are close to each other before mapping are likely to be far apart after application.

---

For simplicity let's say n is a power of two. Could you simply reverse the order of the least significant log2(n) bits of the number?

---

Considering the pixels to be a one dimentional array you could use a hash function j = i*p % n where n is the zero based index of the last pixel and p is a prime number chosen to place the pixel far enough away at each step. % is the remainder operator in C, mathematically I'd write j(i) = i p (mod n).

So if you want to jump at least 10 rows at each iteration, choose p > 10 * w where w is the screen width. You'll want to have a lookup table for p as a function of n and w of course.

Note that j hits every pixel as i goes from 0 to n.

CORRECTION: Use (mod (n + 1)), not (mod n). The last index is n, which cannot be reached using mod n since n (mod n) == 0.

---

Apart from reverting the bit order, you can use modulo. Say N is a prime number (like 521), so for all x = 0..520 you define a function:

f(x) = x * fac mod N

which is bijection on 0..520. fac is arbitrary number different from 0 and 1. For example for N = 521 and fac = 122 you get the following mapping:

which as you can see is quite uniform and not many numbers are near the diagonal - there are some, but it is a small proportion.