Abstract
In this article, we study a variant of the geometric minimum spanning tree (MST) problem. Given a set \(\mathcal{S}\) of n disjoint line segments in 
, we need to find a tree spanning one endpoint from each of the segments in \(\mathcal{S}\). Note that, we have \(2^n\) possible choices of such a set of endpoints, each being referred as an instance. Thus, our objective is to choose one among those instances such that the sum of the lengths of all the edges of the tree spanning the points of that instance is minimum. We show that finding such a spanning tree is NP-complete in general, and propose a \(O(\log ^2 n)\)-factor approximation algorithm for the same.
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Notes
- 1.
Its vertices are the 2n segment endpoints; two vertices a and b are connected by an edge, if and only if the corresponding line segment ab is either in \(\mathcal{S}\) or if the open segment ab does not intersect any (closed) segment from \(\mathcal{S}\).
- 2.
A variable gadget may be connected with multiple literal gadgets.
- 3.
This segment corresponds to the binary relation or.
- 4.
Only one endpoint of a segment can participate in the tree.
- 5.
The maximum possible distance between a pair of points.
- 6.
In each iteration at least half of the segments are deleted from \(\mathcal{S}\).
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Dey, S., Jallu, R.K., Nandy, S.C. (2018). Minimum Spanning Tree of Line Segments. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_44
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