Linear-time recovery of hidden sorted sequences in the Real RAM

Ascertain whether a hidden sorted sequence X of n numbers (a sequence that contains, as a subsequence, all k distinct numbers in sorted order) can be processed in the Real RAM model to output the sorted order of the k distinct numbers in O(n) time without using universal hashing and without assuming any smoothness conditions on X.

Background

As a one-dimensional subroutine for the two-dimensional convex-hull problem, the authors consider reconstructing the sorted order from a supersequence of unit intervals. They define a hidden sorted sequence and show a greedy algorithm runs in expected linear time with universal hashing (word RAM), and in O(n log k) time without hashing. They further obtain linear-time reconstruction in the Real RAM for sequences that satisfy smoothness (distance and packing) bounds.

However, it remains unresolved whether linear-time reconstruction is possible in the Real RAM without any smoothness assumptions and without hashing, which would significantly strengthen the one-dimensional reconstruction component and clarify the computational model dependence.