Equation of relation that has mulitple y points

[Learner]

Hi, well I need something close to curve fitting. But I am not sure if it could serve the purpose. The problem is as below:
Suppose, I have the following relation X r Y. Here X is a natural number set. Suppose we have "k" sets,D1,D2,..Dk of selected datapoints over natural number, such that the intersection of all Di's is null set. Let Y be a set of all Di's.

The relation relates every x, to a set of possible data point sets in Y.

As an example see this, here we have k=3.

X Y
1 {2, 4, 5, 10, 12, 13 }
2 { 3,7,8,9,11,14}
3 {1,6, 15,17, 20, 22}
4 {2, 4, 5, 10, 12, 13 }
5 { 3,7,8,9,11,14}

The question is to find the Smallest equation/function F that Exactly fits the given data in such a way that F(x) just outputs anyone of the datapoint from the Di. I do not need the whole Di.Just anyone of the value from Di.

As an example, the F may be

X Y
1 2
2 3
3 1
4 2
5 3

I need to find the relation F :)




Well, I am looking for the smallest degree function. There is no restriction that it should be polynomial even.( May be aCos(x)+bSin(x) etc. ). What I am looking for is a technique that can take advantage of the fact that it has multiple candidate y points for each x. Many traditional curve fitting techniques tend to fit the curve around one y point.

My hypothesis is that since we have multiple candidate y points for each x, one can exploit this redundancy, and might be able to get a function much smaller than (n-1) polynomial. How true is this ?


If you know that I have chosen the wrong forum, please point me to the right place. Thanks

 

[HallsofIvy]

n points can always be exactly fitted with an n-1 degree polynomial. See "Lagrange's interpolation formula":
http://en.wikipedia.org/wiki/Lagrange's_formula

 

[Learner]

Well, I am looking for the smallest degree function. There is no restriction that it should be polynomial even.( May be aCos(x)+bSin(x) etc. ). What I am looking for is a technique that can take advantage of the fact that it has multiple candidate y points for each x. Many traditional curve fitting techniques tend to fit the curve around one y point.

My hypothesis is that since we have multiple candidate y points for each x, one can exploit this redundancy, and might be able to get a function much smaller than (n-1) polynomial. How true is this?