On Searching a Table Consistent with Division Poset

Suppose $P_n=\{1,2,...,n\}$ is a partially ordered set with the partial order defined by divisibility, that is, for any two distinct elements $i,j\in P_n$ satisfying $i$ divides $j$, $i<_{P_n} j$. A table $A_n=\{a_i|i=1,2,...,n\}$ of distinct real numbers is said to be \emph{consistent} with $P_n$, provided for any two distinct elements $i,j\in \{1,2,...,n\}$ satisfying $i$ divides $j$, $a_i< a_j$. Given an real number $x$, we want to determine whether $x\in A_n$, by comparing $x$ with as few entries of $A_n$ as possible. In this paper we investigate the complexity $\tau(n)$, measured in the number of comparisons, of the above search problem. We present a $\frac{55n}{72}+O(\ln^2 n)$ search algorithm for $A_n$ and prove a lower bound $({3/4}+{17/2160})n+O(1)$ on $\tau(n)$ by using an adversary argument.Comment: 16 pages, no figure; same results, representation improved, add reference

On Searching a Table Consistent with Division Poset ∗

Suppose Pn = {1, 2,..., n} is a partially ordered set with the partial order defined by divisibility, that is, for any two elements i, j ∈ Pn satisfying i divides j, we have i ≤Pn j. A table An = {ai|i = 1, 2,..., n} of real numbers is said to be consistent with Pn, provided for any two elements i, j ∈ {1, 2,..., n} satisfying i divides j, ai ≤ aj. Given an real number x, we want to determine whether x ∈ An, by comparing x with as few entries of An as possible. In this paper we investigate the complexity τ(n), measured in the number of comparisons, of the above search problem. We present a 55n 72 + O(ln2 n) search algorithm for An and prove a lower bound)n + O(1) on τ(n) by using an adversary argument