Complexity of Radix Sort
We typically analyze algorithms in the so-called &quot;RAM machine&quot; or &quot;transdichotomous model&quot;. In this model, operations on machine words take $O(1)$, and a machine word has length $O(\log n)$ bits, where $n$ is the size of the input (in bits), or any other polynomially related quantity.
If $R$ is polynomial in $n$ (in this case, the length of the array), then an integer in the range $\{1,\ldots,R\}$ can be stored in a single machine word, and so operations on such integers take constant time in this model.