On Searching a Table Consistent with Division Poset

TL;DR

This paper studies the problem of efficiently searching for a value in a table consistent with divisibility order, proposing a near-optimal search algorithm and establishing a lower bound on the number of comparisons needed.

Contribution

It introduces a new search algorithm with complexity close to the theoretical lower bound for tables consistent with division posets.

Findings

Proposed a search algorithm with complexity rac{55n}{72}+O(\u2219ln^2 n)

Established a lower bound of rac{3/4+17/2160}{n}+O(1) comparisons

Demonstrated the near-optimality of the algorithm via adversary argument

Abstract

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.