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

Asked 2009-05-28T07:30:10.440

308

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.

Edit Report

2 Answers

286

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


                            answered 2012-12-14T01:30:09.857


            
            
                
                    
                        
                            
                            
                        
                        1
                        
                            
                            
                        
                        
                    
                    
                        
                        What you suggest is impossible. You will always have collisions.

In order to map two objects to another single set, the mapped set must have a minimum size of the number of combinations expected:

Assuming a 32-bit integer, you have 2147483647 positive integers. Choosing two of these where order doesn't matter and with repetition yields 2305843008139952128 combinations. This does not fit nicely in the set of 32-bit integers.

You can, however fit this mapping in 61 bits. Using a 64-bit integer is probably easiest. Set the high word to the smaller integer and the low word to the larger one.

                        
                        
                            answered 2009-05-28T07:48:25.180



        
        
            Your Answer