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. TCS 2012].
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 \(O(\frac{\log n}{\log \log n})\)-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.
B. Keszegh—Research supported by Hungarian National Science Fund (OTKA), under grant PD 108406 and under grant NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
B. Lidický—Research is partially supported by NSF grants DMS-1266016 and DMS-1600390.
D. Pálvölgyi—Research supported by Hungarian National Science Fund (OTKA), under grant PD 104386 and under grant NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
T. Valla—Supported by the Centre of Excellence – Inst. for Theor. Comp. Sci. (project P202/12/G061 of GA ČR).
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Notes
- 1.
For the sake of avoiding confusion between the input tree and the decision tree, we will reserve the term vertex for the elements of V and the term node for the vertices of the decision tree D.
- 2.
Recall that a centroid of a tree T is a vertex v such that any component of \(T- v\) has size at most |T| / 2.
- 3.
A node \(\nu \) is an ancestor of another node \(\nu '\) if \(\nu \) lies on the path connecting \(\nu '\) to the head.
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Acknowledgment
We are very grateful to Balázs Patkós for organizing \(5^{\mathrm {th}}\) Emléktábla Workshop where we collaborated on this paper.
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Appendix
Appendix
1.1 Figures
An example of the tree search problem, T is the input tree and D is a decision tree with \(cost(D) = 7 = cost^D(a) = cost^D(c)\). If the vertices of the tree T represent the parts of a device to assemble, the decision tree corresponds to the assembly procedure that at time 0 joins e with b; then at time 3 joins b with c and e with g. At time 4 the joining of d with c and e with f is started. Finally, at time 6 part a is joined with part c and the procedure ends by time 7.
An example of the tree S, the important set of vertices X and the auxiliary tree Y in the construction of Sect. 3
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Cicalese, F., Keszegh, B., Lidický, B., Pálvölgyi, D., Valla, T. (2016). On the Tree Search Problem with Non-uniform Costs. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_7
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