Quadratic algorithm for 4-SUM

Question

The 4-SUM is as follows:

Given an array of N distinct integers find 4 integers a, b, c, d such that a+b+c+d = 0.

I could come up with a cubic algorithm using quadratic algorithm for 3-SUM problem. Can we do better than cubic for 4-SUM?

Answer 1

Yes you can. Go over all pairs of numbers and store their sum(and also store which numbers give that sum). After that for each sum check if its negation is found among the sums you have. Using a hash you can reach quadratic complexity, using std::map, you will reach O(n^2*log(n)).

EDIT: to make sure no number is used more than once it will better to store indices instead of the actual numbers for each sum. Also as a given sum may be formed by more than one pair, you will have to use a hash multimap. Having in mind the numbers are different for a sum X = a1 + a2 the sum -X may be formed at most once using a1 and once using a2 so for a given sum X you will have to iterate over at most 3 pairs giving -X as sum. This is still constant.