sum of absolute differences of a number in an array

Question

I want to calculate the sum of absolute differences of a number at index i with all integers up to index i-1 in o(n). But i am not able to think of any approach better than o(n^

Answer 1

Here's an Omega(n log n)-comparison lower bound in the linear decision tree model. This rules out the possibility of a "nice" o(n log n)-time algorithm (two now-deleted answers both were in this class).

There is a trivial reduction to this problem from the problem of computing

f(x1, ..., xn) = sum_i sum_j |xi - xj|.

The function f is totally differentiable at x1, ..., xn if and only if x1, ..., xn are pairwise distinct. The set where f is totally differentiable thus has n! connected components, of which each leaf of the decision tree can handle at most one.