How to design a data structure that allows one to search, insert and delete an integer X in O(1) time

Question

Here is an exercise (3-15) in the book \"Algorithm Design Manual\".

Design a data structure that allows one to search, insert, and delete an integer X

Answer 1

I first saw the idea in Cameron's answer in Jon Bentley Programming Pearls.

The idea is pretty simple but it's not straightforward to see why the initial random values that may be on the uninitialized arrays does not matter. This link explains pretty well the insertion and search operations. Deletion is left as an exercise, but is answered by one of the commenters:

remove-member(i):
    if not is-member(i): return
    j = dense[n-1];
    dense[sparse[i]] = j;
    sparse[j] = sparse[i];
    n = n - 1