From wikipedia
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the length of the input (the number of bits required to represent it) and the numeric value of the input (the largest integer present in the input). In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length.
From this SO post
An algorithm runs in pseudo-polynomial time if the runtime is some polynomial in the numeric value of the input, rather than in the number of bits required to represent it.
I have some questions about the wiki definition. To my knowledge, the definition provided by the SO post makes more sense. Shouldn't the wiki say pseudo-polynomial time is if the running time is
1) exponential (or something greater than polynomial) in the length of the input, and
2) polynomial in the numeric value of the input (largest number of input)
??
Assuming you have answered the question of definition above,
I am having trouble analyzing the running time of Counting Sort which is $O(N + K)$ with respect to $N, K$, the numeric values of the input.
Assuming we do not have the $k = \Theta(N)$ simplifying condition, the length of the input is $x = \Theta(N\log(k))$ bits.
The Wiki says "In general, the numeric value of the input is exponential in the input length". But $k = 2^{\Theta(\frac{x}{N})}$. With N potentially bringing down the magnitude of the exponent, I don't really see how $k$ is exponential in $x$.
In addition, the SO post I referenced claims counting sort runtime is exponential with respect to the length of the input, $x = \Theta(N\log(k))$ bits. How can $O(N + k)$ be transformed into an expression exponential in $n$ to demonstrate the claim? I got it down to $O(\frac{x}{\log(x)} + k)$ but got stuck there.
If I just think the length of the input is $\Theta(\log k)$ then I can see why the run-time is exponential but this doesn't seem to be consistent with how $x$ should be defined.