On the tree search problem with non-uniform costs

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2016-09-27
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Cicalese, Ferdinando
Keszegh, Balázs
Pálvölgyid, Dömötör
Valla, Tomáš
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Lidicky, Bernard
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Mathematics
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Abstract

Searching in partially ordered structures has been considered in the context of information retrieval and efficient tree-like indices, as well as in hierarchy based knowledge representation. In this paper we focus on tree-like partial orders and consider the problem of identifying an initially unknown vertex in a tree by asking edge queries: an edge query e returns the component of T - e containing the vertex sought for, while incurring some known cost c(e). The Tree Search Problem with Non-Uniform Cost is the following: given a tree T on n vertices, each edge having an associated cost, construct a strategy that minimizes the total cost of the identification in the worst case.

Finding the strategy guaranteeing the minimum possible cost is an NP-complete problem already for input trees of degree 3 or diameter 6. The best known approximation guarantee was an O (log n/log log log n)-approximation algorithm of Cicalese et al. (2012) [4].

We improve upon the above results both from the algorithmic and the computational complexity point of view: We provide a novel algorithm that provides an O (log n/log log n)-approximation of the cost of the optimal strategy. In addition, we show that finding an optimal strategy is NP-hard even when the input tree is a spider of diameter 6, i.e., at most one vertex has degree larger than 2.

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This is a manuscript of an article published as Cicalese, Ferdinando, Balázs Keszegh, Bernard Lidický, Dömötör Pálvölgyi, and Tomáš Valla. "On the tree search problem with non-uniform costs." Theoretical Computer Science 647 (2016): 22-32. doi:10.1016/j.tcs.2016.07.019. Posted with permission.

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Fri Jan 01 00:00:00 UTC 2016
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