Enumerating Denumerable Sets in Polynomial Time via the Schröder--Bernstein Theorem

Abstract

We give methods to develop efficiently computable bijections between the rational numbers and the positive integers. That is, given a rational number in the standard representation a/b, where a, b are integers, we can compute n, its position in the enumeration, in time polynomial in the bit lengths of a and b. Conversely, given a position n in the enumeration, we can compute the corresponding rational number a/b at that position in time polynomial in the bit length of n. This is not the first such bijection to have appeared in the literature. However, we submit that the method presented here, which uses König's proof of the Schröder-Bernstein Theorem, is relatively simple to understand, and has a broad application. It can be applied to enumerating other denumerable sets. As an example, we use it to give a polynomial-time bijection between the algebraic numbers and the positive integers.