I would like to design a data structure and algorithm such that, given an array of elements, where each element has a weight according to [a,b], I can achieve constant time insertion and deletion. The deletion is performed randomly where the probability of an element being deleted is proportional to its weight.
I do not believe there is a deterministic algorithm that can achieve both operations in constant time, but I think there are there randomized algorithms that should be can accomplish this?
I don't know if O(1) worst-case time is impossible; I don't see any particular reason it should be. But it's definitely possible to have a simple data structure which achieves O(1) expected time.
The idea is to store a dynamic array of pairs (or two parallel arrays), where each item is paired with its weight; insertion is done by appending in O(1) amortised time, and an element can be removed by index by swapping it with the last element so that it can be removed from the end of the array in O(1) time. To sample a random element from the weighted distribution, choose a random index and generate a random number in the half-open interval [0, 2); if it is less than the element's weight, select the element at that index, otherwise repeat this process until an element is selected. The idea is that each index is equally likely to be chosen, and the probability it gets kept rather than rejected is proportional to its weight.
This is a Las Vegas algorithm, meaning it is expected to complete in a finite time, but with very low probability it can take arbitrarily long to complete. The number of iterations required to sample an element will be highest when every weight is exactly 1, in which case it follows a geometric distribution with parameter p = 1/2, so its expected value is 2, a constant which is independent of the number of elements in the data structure.
In general, if all weights are in an interval [a, b] for real numbers 0 < a <= b, then the expected number of iterations is at most b/a. This is always a constant, but it is potentially a large constant (i.e. it takes many iterations to select a single sample) if the lower bound a is small relative to b.