Binary Proportional Pairing Functions

Abstract: A pairing function for the non-negative integers is said to be binary perfect if the binary representation of the output is of length 2k or less whenever each input has length k or less. Pairing functions with square shells, such as the Rosenberg-Strong pairing function, are binary perfect. Many well-known discrete space-filling curves, including the discrete Hilbert curve, are also binary perfect. The concept of a binary proportional pairing function generalizes the concept of a binary perfect pairing function. Binary proportional pairing functions may be useful in applications where a pairing function is used, and where the function's inputs have lengths differing by a fixed proportion. In this article, a general technique for constructing a pairing function from any non-decreasing unbounded function is described. This technique is used to construct a binary proportional pairing function and its inverse.