Searching in random partially ordered sets

Abstract

We consider the problem of searching for a given element in a partially ordered set. More precisely, we address the problem of computing e ciently near-optimal search strategies for typical partial orders under two classical models for random partial orders, the random graph model and the uniform model. We shall show that the problem of determining an optimal strategy is NP-hard, but there are simple, fast algorithms able to produce near-optimal search strategies for typical partial orders under the two models of random partial orders that we consider. We present a (1 + o(1))approximation algorithm for typical partial orders under the random graph model (constant p) and present a 6.34-approximation algorithm for typical partial orders under the uniform model. Both algorithms run in polynomial time.

Deterministic and probabilistic binary search in graphs

“…The algorithmic problem of identifying a target by edge queries in DAGs was initiated by Linial and Saks [30], who studied it for several specific classes of graphs. In general, the problem is known to be NP-hard [11,15]. As a result, several papers have studied it specifically on trees.…”

Searching in Dynamic Tree-Like Partial Orders

“…Thus, for our second experiment, we took the /usr directory of an Ubuntu 10.04 Linux distribution as our universe U and independently sampled 1000 sets of size n = 100, n = 1000, and n = 10000 from U respectively. The /usr directory contains 23,328 nodes, 1 In keeping with the uniform model for general partial orders defined in [2], we assume P(n) is the set of all rooted, labeled, oriented trees on 1, . .…”

“…For example, when S is totally ordered, the optimal minimum-height solution is a standard binary search tree. In contrast to the totally ordered case, finding a minimum height static search tree for an arbitrary partial order is NP-hard [2]. Because of this, most recent work has focused on partial orders that can be described by rooted, oriented trees.…”

An Approximation Algorithm for Binary Searching in Trees

“…Recent advances [21,23] have reduced the time complexity to O(n 3 ) and then O(n). In contrast, the more general version of the worst case minimization where the input is a poset instead of a tree is NP-hard [5].…”

“…Internal nodes correspond to arcs of T and leaves to nodes of T of the desired element. This can be generalized to searching in more general structures which have only a partial order for their elements instead of a total order [4,5,20,21,23].…”