How to find all taxicab numbers less than N?

Question

A taxicab number is an integer that can be expressed as the sum of two cubes of integers in two different ways: a^3+b^3 = c^3+d^3. Design an algorithm to find all t

Answer 1

The answers given by Novneet Nov and user3017842 are both correct ideas for finding the taxicab numbers with storage O(N) using minHeap. Just a little bit more explanation why the minHeap of size N works. First, if you had all the sums (O(N^2)) and could sort them (O(N^2lgN)) you would just pick the duplicates as you traverse the sorted array. Well, in our case using a minHeap we can traverse in-order all the sums: we just need to ensure that the minHeap always contains the minimum unprocessed sum.

Now, we have a huge number of sums (O(N^2)). But, notice that this number can be split into N groups each of which has an easily defined minimum! (fix a, change b from 0 to N-1 => here are your N groups. The sum in one group with a smaller b is smaller than one with a bigger b in the same group - because a is the same).

The minimum of union of these groups is in the union of mins of these groups. Therefore, if you keep all minimums of these groups in the minHeap you are guaranteed to have the total minimum in the minHeap.

Now, when you extract Min from the heap, you just add next smallest element from the group of this extracted min (so if you extracted (a, b) you add (a, b+1)) and you are guaranteed that your minHeap still contains the next unprocessed min of all the sums.