Paper by Erik D. Demaine

Reference:

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth, “Minimum Plane Bichromatic Spanning Trees”, in Proceedings of the 35th International Symposium on Algorithms and Computation (ISAAC 2024), LIPIcs, volume 322, Sydney, Australia, December 8–11, 2024, 4:1–4:14.

Abstract:

For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings.

Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case.

Comments:

This paper is also available from LIPIcs and as arXiv:2409.11614.

Length:

The paper is 14 pages.

Availability:

The paper is available in PDF (776k).

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