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data structures - How to get the list of defined values for an indexed variable?


I would like a table with rational indexes - thus it would be practical to use a dictionary, which, in Mathematica are implemented with the indexed variables. I would like to be able to do:


...
If[a[1/2] is defined, a[1/2] = a[1/2] + 1, a[1/2] = 1]
...
If[a[4568/8746] is defined, a[4568/8746] = a[4568/8746] + 1, a[4568/8746] = 1]
...

and then further along:


ConvertToList[a] -> {... {1/2, 4}, {4568/8746, 9}, ... }


Had my indexes been integers, I could have used the built-in function Array. But they are not, thus at some point, I need to get a list of all the indexes for which ahas a value. I could hack a solution (for instance: implement my own dictionary structure, or keeping a set of all values I defined and use it to know where I should look), but this seems silly. Especially since Mathematica provides the functionality I desire as a meta-function:


?a

Unfortunately I can't seem to make use of it programmatically.


How can I do what I want? Am I wrong to think that indexed variables are the way to do it?



UPDATE: Thank you so much for your extremely informative answers---I admit I wish I were able to accept both of them! Especially as they both give complementary information.


To give a little bit more information on my usage, I have both been looking for this general functionality for some time (a generic hash table type structure), but the example I suggested is a real-world example not a minimal one :)


As a first approach (because I can't get an easy formula), I am computing the probability distribution function of the ratio of two discrete random variables. To do this, I am iterating over all values of the first random variable, and then iterating over all values the second one can take (which depend on the value of the first random variable). I then complete the table using as index the ratio of the two random variables. The problem is there are great many rationals and that 1/2 is the same as 2/4, etc., hence the reason for using a hash table.



I had already managed a clunky way of doing this, but now I have a very fast function, thanks to you two, that allows me to get the PDF of my distribution. It is then convenient to convert this to a CDF of the distribution to avoid the many fluctuations inherent to the discrete rational distribution.


You are both deeply thanked for your explanations.




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