Context: Consider an ensemble $X = (x; A; P)$. Let's examine $X^N$, which represents the set of all possible strings of length $N $produced by $X$. The objective is to systematically derive sequences from this set based on their increasing information content without generating the entire set of sequences due to its potentially large size.
Objective:
- Design a function $g: \mathbb{N} \rightarrow Y$, where $Y \subseteq X^N$, such that for any two integers $i$ and $j$ with $i < j$:
- $I(g(i)) \leq I(g(j))$
- Every sequence in $Y$ should be reachable by some integer input to $g$.
- Formulate an inverse function $g^{-1}: Y \rightarrow \mathbb{N}$ to retrieve the index of any sequence in $Y$.
Where the information content $I(x_i)$ of a sequence $x_i$ from the ensemble $X$ is defined as: $I(x_i) = - \sum_{a \in A} p(a) \log_2 p(a)$
With:
- $A$ as the alphabet of $X$,
- $p(a)$ as the probability of the symbol $a$ appearing in sequence $x_i$.
Constraints:
- The function $g$ should ideally possess sublinear complexity with respect to the size of $X^N $for computational feasibility. This means we aim to derive sequences without having to generate or scrutinize every sequence in $X^N$.
- The function $g^{-1}$ should be computationally efficient for lookups.
Additional Notes: Recognizing that as $N$ grows, the computational challenge may increase considerably. So we are also on the lookout for potential methods offering approximate solutions efficiently for larger $N$ and precise solutions for more manageable $N$
Given these objectives and constraints, how might one develop a methodology or algorithm to create both functions $g$ and $g^{-1}$?