Given a number n, find two pairs that can represent the number as sum of two cubes. In other words, find two pairs (a, b) and (c, d) such that given number n can be expressed as
n = a^3 + b^3 = c^3 + d^3
where a, b, c and d are four distinct numbers.
Input: n = 1729 Output: (1, 12) and (9, 10) Explanation: 1729 = 1^3 + 12^3 = 9^3 + 10^3 Input: n = 4104 Output: (2, 16) and (9, 15) Explanation: 4104 = 2^3 + 16^3 = 9^3 + 15^3 Input: n = 13832 Output: (2, 24) and (18, 20) Explanation: 13832 = 2^3 + 24^3 = 18^3 + 20^3
We have discussed a O(n2/3) solution in below set 1.
In this post, a O(n1/3) solution is discussed.
Any number n that satisfies the constraint will have two distinct pairs (a, b) and (c, d) such that a, b, c and d are all less than n1/3. The idea is to create an auxiliary array of size n1/3. Each index i in the array will store value equal to cube of that index i.e. arr[i] = i^3. Now the problem reduces to finding pair of elements in an sorted array whose sum is equal to given number n. The problem is discussed in detail here.
Below is the implementation of above idea :
(10, 27) (19, 24)
Time Complexity of above solution is O(n^(1/3)).
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