Creating Unique Solutions Beyond Existing Standards

[Dirklin]

Solutions must not be based on existing standards or rules. As unique as possible.

Mathematical description:
1. X is a given set of discrete values where 0 <= X <= 65535
2. The set of values is ordered and it goes from 0 to a N (where the length N, is some known, finite number)
3. The applied algorithm should work for a list of any length.
4. The operation must be lossless (the information must be retained 100%)
5. The operation must be reversible (we need to get back the original set of values).
6. The generated functions need to be represented in compact way that will require less space than it contains the initial set of values.
(For example. If there are 10 (16bit points = 2by/point) = 20 bytes we need a function that has coefficients which total size are less than 20 bytes.
Basically we need some knowledge or expert in the field of mathematics that is dealing with function analysis.
In general the set of numbers can be anything. Our problem is contained in the range of values from X € [0,65535] we tried several methods like polynomial interpolation, linear regression or polynomial regression.
We tried to interpolate with polynomials and got results but then the coefficients are floating point numbers and to save them
It requires 4 bytes. We are losing a compression and in addition floating point numbers have some kind of error representation.
There are some numbers that can’t be represented 100% by the floating point notation. This can lead to wrong reverse operations.
To summarize we need a way to construct any function, polynomial or even a set of mathematical series or even sequences of number that can describe the initial set of values. The initial set can be split on the least number of functions, polynomials, series or sequences that can meet upper 6 conditions.

 

[Nono713]

So basically you're asking for a lossless compression algorithm. An efficient algorithm will depend on the distribution of your input 16-bit words; if they have a certain structure, you can get big savings by exploiting this, so you need to look into that (simple example, if your input data consists of sine waves, all you need to store is the frequency and amplitude and you're done). If they are uniformly random with no structure whatsoever, you are out of luck as no such algorithm exists, for any reversible function must preserve the entropy of its input.