Mapping two integers to one, in a unique and deterministic way

Imagine two positive integers A and B. I want to combine these two into a single integer C.

There can be no other integers D and E which combine to C. So combining them with the addition operator doesn't work. Eg 30 + 10 = 40 = 40 + 0 = 39 + 1 Neither does concatination work. Eg "31" + "2" = 312 = "3" + "12"

This combination operation should also be deterministic (always yield the same result with the same inputs) and should always yield an integer on either the positive or the negative side of integers.

Solution 1:

You're looking for a bijective NxN -> N mapping. These are used for e.g. dovetailing. Have a look at this PDF for an introduction to so-called pairing functions. Wikipedia introduces a specific pairing function, namely the Cantor pairing function:

Three remarks:

• As others have made clear, if you plan to implement a pairing function, you may soon find you need arbitrarily large integers (bignums).
• If you don't want to make a distinction between the pairs (a, b) and (b, a), then sort a and b before applying the pairing function.
• Actually I lied. You are looking for a bijective ZxZ -> N mapping. Cantor's function only works on non-negative numbers. This is not a problem however, because it's easy to define a bijection f : Z -> N, like so:
  • f(n) = n * 2 if n >= 0
  • f(n) = -n * 2 - 1 if n < 0

Solution 2:

Cantor pairing function is really one of the better ones out there considering its simple, fast and space efficient, but there is something even better published at Wolfram by Matthew Szudzik, here. The limitation of Cantor pairing function (relatively) is that the range of encoded results doesn't always stay within the limits of a 2N bit integer if the inputs are two N bit integers. That is, if my inputs are two 16 bit integers ranging from 0 to 2^16 -1, then there are 2^16 * (2^16 -1) combinations of inputs possible, so by the obvious Pigeonhole Principle, we need an output of size at least 2^16 * (2^16 -1), which is equal to 2^32 - 2^16, or in other words, a map of 32 bit numbers should be feasible ideally. This may not be of little practical importance in programming world.

Cantor pairing function:

(a + b) * (a + b + 1) / 2 + a; where a, b >= 0

The mapping for two maximum most 16 bit integers (65535, 65535) will be 8589803520 which as you see cannot be fit into 32 bits.

Enter Szudzik's function:

a >= b ? a * a + a + b : a + b * b;  where a, b >= 0

The mapping for (65535, 65535) will now be 4294967295 which as you see is a 32 bit (0 to 2^32 -1) integer. This is where this solution is ideal, it simply utilizes every single point in that space, so nothing can get more space efficient.

Now considering the fact that we typically deal with the signed implementations of numbers of various sizes in languages/frameworks, let's consider signed 16 bit integers ranging from -(2^15) to 2^15 -1 (later we'll see how to extend even the ouput to span over signed range). Since a and b have to be positive they range from 0 to 2^15 - 1.

Cantor pairing function:

The mapping for two maximum most 16 bit signed integers (32767, 32767) will be 2147418112 which is just short of maximum value for signed 32 bit integer.

Now Szudzik's function:

(32767, 32767) => 1073741823, much smaller..

Let's account for negative integers. That's beyond the original question I know, but just elaborating to help future visitors.

Cantor pairing function:

A = a >= 0 ? 2 * a : -2 * a - 1;
B = b >= 0 ? 2 * b : -2 * b - 1;
(A + B) * (A + B + 1) / 2 + A;

(-32768, -32768) => 8589803520 which is Int64. 64 bit output for 16 bit inputs may be so unpardonable!!

Szudzik's function:

A = a >= 0 ? 2 * a : -2 * a - 1;
B = b >= 0 ? 2 * b : -2 * b - 1;
A >= B ? A * A + A + B : A + B * B;

(-32768, -32768) => 4294967295 which is 32 bit for unsigned range or 64 bit for signed range, but still better.

Now all this while the output has always been positive. In signed world, it will be even more space saving if we could transfer half the output to negative axis. You could do it like this for Szudzik's:

A = a >= 0 ? 2 * a : -2 * a - 1;
B = b >= 0 ? 2 * b : -2 * b - 1;
C = (A >= B ? A * A + A + B : A + B * B) / 2;
a < 0 && b < 0 || a >= 0 && b >= 0 ? C : -C - 1;

(-32768, 32767) => -2147483648

(32767, -32768) => -2147450880

(0, 0) => 0 

(32767, 32767) => 2147418112

(-32768, -32768) => 2147483647

What I do: After applying a weight of 2 to the the inputs and going through the function, I then divide the ouput by two and take some of them to negative axis by multiplying by -1.

See the results, for any input in the range of a signed 16 bit number, the output lies within the limits of a signed 32 bit integer which is cool. I'm not sure how to go about the same way for Cantor pairing function but didn't try as much as its not as efficient. Furthermore, more calculations involved in Cantor pairing function means its slower too.

Here is a C# implementation.

public static long PerfectlyHashThem(int a, int b)
{
    var A = (ulong)(a >= 0 ? 2 * (long)a : -2 * (long)a - 1);
    var B = (ulong)(b >= 0 ? 2 * (long)b : -2 * (long)b - 1);
    var C = (long)((A >= B ? A * A + A + B : A + B * B) / 2);
    return a < 0 && b < 0 || a >= 0 && b >= 0 ? C : -C - 1;
}

public static int PerfectlyHashThem(short a, short b)
{
    var A = (uint)(a >= 0 ? 2 * a : -2 * a - 1);
    var B = (uint)(b >= 0 ? 2 * b : -2 * b - 1);
    var C = (int)((A >= B ? A * A + A + B : A + B * B) / 2);
    return a < 0 && b < 0 || a >= 0 && b >= 0 ? C : -C - 1;
}

Since the intermediate calculations can exceed limits of 2N signed integer, I have used 4N integer type (the last division by 2 brings back the result to 2N).

The link I have provided on alternate solution nicely depicts a graph of the function utilizing every single point in space. Its amazing to see that you could uniquely encode a pair of coordinates to a single number reversibly! Magic world of numbers!!