Recounting the Rationals: Twice!
نویسندگان
چکیده
We derive an algorithm that enables the rationals to be efficiently enumerated in two different ways. One way is known and is credited to Moshe Newman; it corresponds to a deforestation of the so-called Calkin-Wilf tree of rationals. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a Stern-Brocot enumeration algorithm with the same time and space complexity as Newman’s algorithm.
منابع مشابه
Recounting the Rationals
It is well known (indeed, as Paul Erdős might have said, every child knows) that the rationals are countable. However, the standard presentations of this fact do not give an explicit enumeration; rather they show how to construct an enumeration. In this note we will explicitly describe a sequence b(n) with the property that every positive rational appears exactly once as b(n)/b(n+1). Moreover, ...
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