When are hash functions orthogonal to each other?

Question

When are hash functions orthogonal to each other?

And can you provide an example in Java of two hash functions that are orthogonal to each other?

Answer 1

From (a Google search result paper)

(Orthogonal Hash Functions) Two hash functions h1 and h2 are orthogonal, if for all states s, s' ∈ S with h1 (s) = h1 (s') and h2 (s) = h2 (s') we have s = s'.

S. Edelkamp, Perfect Hashing for State Space Exploration on the GPU.

In English, if any two given values passed to two different orthogonal hash functions result in the same outputs, those inputs must have been the same value.

Example:

Let h and g be hash functions.
Let b be a currently unknown value.
h(0) = h(b) = 5
g(0) = g(b) = 4
if h and g are orthogonal, b MUST equal 0.

Thus for any values given to h that result in a unique result,
If those same values are given to g, they must also result in a unique result,
  IF they are orthogonal hash functions.

Pseudocode:

// Assume no wraparound will ever occur due to overflow.
HashFunc h = x -> x + 1;
HashFunc g = y -> y + 2;
h(0) = 1 // No other input value results in --> 1
g(0) = 2 // No other input value results in --> 2
// These must have been orthogonal hash functions.

// Now for some non-orthogonal hash functions:
// Let the domain be integers only.
HashFunc j = x -> ceil(abs(x / 2));
HashFunc k = x -> ceil(sqrt(x));
j(0) = 0 // Unique result
k(0) = 0 // Unique result
j(1) = j(2) = 1
k(1) = 1 != k(2) = 2 
// k(1) results in a unique value, but it isn't unique for j.
// These cannot be orthogonal hash functions.

Answer 2

from http://www.aaai.org/ocs/index.php/ICAPS/ICAPS10/paper/download/1439/1529

Obtained with Google: define "orthogonal hash" (second hit).

Translating:

If you have a "perfect hashing function", then h(s) == h(s') iff s == s'.

If you have "any two hashing functions" for which there are values s, s' that have both

h1(s) == h1(s') and h2(s) == h2(s')

then these functions are called orthogonal if the above is true for s == s'

It's actually quite a tricky concept. If h1 and h2 were both perfect hashing functions, then they would automatically have to be orthogonal according to the above definition (if I understand correctly). But you can have imperfect functions that fit the above definition.

Example: in the state space [0, 9], two functions

h1(int x) return x % 5;

h2(int x) return x % 7;

Would be orthogonal:

x  h1  h2
0   0   0
1   1   1 
2   2   2
3   3   3
4   4   4
5   0   5
6   1   6
7   2   0
8   3   1
9   4   2

In the above, h1(s) = h1(s') for pairs of values s that are either 0 apart or 5 apart. For h1, the distance is either 0 or 7.

The only pairs for which both conditions are true are those where the distance is 0 - so only when s1 == s2. Thus these are orthogonal (although imperfect) hashing functions.

And that, I think, answers both parts of your question.