Searching On Division Partially Ordered Set

Let P = {pi|i = 1, 2, . . . , n} be a set of n elements with partial order defined as follows: pi ≤ pj if and only if i divides j. Assume real numbers are assigned to all members of P in an order preserving manner, this results in a set A of n real numbers. Given an unknown x ∈ R, we want to determine whether x is belong to A or not, by comparing x with as few elements of A as possible. We present two kinds of algorithms searching on A, in which binary search and searching on two-chain [2] are involved, respectively. Also, we provide a method to obtain lower bounds on the number of necessary comparisons in the worst case.