Abstract
We consider minimum-cardinality Manhattan connected sets with arbitrary demands: Given a collection of points P in the plane, together with a subset of pairs of points in P (which we call demands), find a minimum-cardinality superset of P such that every demand pair is connected by a path whose length is the \(\ell _1\)-distance of the pair. This problem is a variant of three well-studied problems that have arisen in computational geometry, data structures, and network design: (i) It is a node-cost variant of the classical Manhattan network problem, (ii) it is an extension of the binary search tree problem to arbitrary demands, and (iii) it is a special case of the directed Steiner forest problem. Since the problem inherits basic structural properties from the context of binary search trees, an \(O(\mathrm {log}\;n)\)-approximation is trivial. We show that the problem is NP-hard and present an \(O(\sqrt{\mathrm {log}\;n})\)-approximation algorithm. Moreover, we provide an \(O(\mathrm {log}\;\mathrm {log}\;n)\)-approximation algorithm for complete k-partite demands as well as improved results for unit-disk demands and several generalizations. Our results crucially rely on a new lower bound on the optimal cost that could potentially be useful in the context of BSTs.
The full version of this paper [1] can be found at https://arxiv.org/abs/2010.14338.
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Notes
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Demaine et al. [10] prove NP-hardness for MinGMConn with uniform demands but allow the input to contain multiple points on the same row. Their result is incomparable to ours.
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Antoniadis, A. et al. (2021). On Minimum Generalized Manhattan Connections. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_7
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