Binary Proportional Pairing Functions

TL;DR

This paper introduces a general method for constructing binary proportional pairing functions from any non-decreasing unbounded function, expanding the class of pairing functions with potential applications in discrete space-filling curves.

Contribution

It presents a novel technique to create binary proportional pairing functions from arbitrary non-decreasing unbounded functions, generalizing binary perfect pairing functions.

Findings

Constructed a binary proportional pairing function and its inverse.

Showed that many known space-filling curves are binary perfect.

Provided a general construction method for 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.