Sorting the sums (x_i+y_j) in O(n²) comparisons

Abstract

Let (x_i)_1≤i≤n and (y_j)_1≤i≤n be two sequences of numbers. It was proved by M.L. Fredman in [1] that the n² sums (x_i+y_j)_1≤i,j≤n can be sorted in O(n²) comparisons, but until now, no explicit algorithm was known to do it. We present such an algorithm and generalize it to sort $(x_{i_1 }^1 + ... + x_{i_k }^k )1 \leqslant i_1 ,...i_k \leqslant n$ in O(n^k) comparisons.