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Monotone Paths in Geometric Triangulations

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Abstract

(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of n points in the plane is O(1.7864n). This improves an earlier upper bound of O(1.8393n); the current best lower bound is Ω(1.7003n). (II) Given a planar geometric graph G with n vertices, we show that the number of monotone paths in G can be computed in O(n2) time.

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Notes

  1. Refer to the .c file and the A within the source at arXiv:http://arXiv.org/abs/1608.04812.

  2. The algorithm has been revised, as some of the ideas were implemented incorrectly in the earlier conference version.

  3. The argument of a vector \(\mathbf {u}\in \mathbb {R}^{2}\setminus \{\mathbf {0}\}\) is the angle measure in [0, 2π) of the minimum counterclockwise rotation that carries the positive x-axis to the ray spanned by u.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI, USA

    Adrian Dumitrescu & Ritankar Mandal

  2. Department of Mathematics, California State University, Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  3. Department of Computer Science, Tufts University, Medford, MA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
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  2. Ritankar Mandal
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  3. Csaba D. Tóth
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Corresponding author

Correspondence to Ritankar Mandal.

Additional information

This article is part of the Topical Collection on Special Issue on Combinatorial Algorithms

The work of Csaba D. Tóth was supported in part by the NSF awards CCF-1422311 and CCF-1423615.

An extended abstract of this paper appeared in the Proceedings of the 27th International Workshop on Combinatorial Algorithms (IWOCA 2016), LNCS 9843, pp. 411–422, Springer International Publishing, 2016.

Appendix A: Extremal configurations

Appendix A: Extremal configurations

The groups of 4 vertices with 12 and 11 patterns

There are exactly 4 groups with exactly 12 incidence patterns (modulo reflections about the x-axis); see Fig. 35.

  • I(B1): ∅, 1234, 123, 12, 134, 13, 234, 23, 2, 34, 3, 4.

  • I(B2): ∅, 1234, 123, 124, 134, 13, 234, 23, 24, 34, 3, 4.

  • I(B3): ∅, 1234, 123, 124, 12, 134, 13, 1, 23, 234, 24, 2.

  • I(B4): ∅, 1234, 123, 124, 12, 1, 23, 234, 24, 2, 34, 3.

Fig. 35
figure 35

B1–B4 are the only four groups with 12 incidence patterns

Full size image

There are exactly 20 groups with exactly 11 incidence patterns (modulo reflections about the x-axis); see Fig. 36.

  • I(C1) : ∅, 1234, 123, 134, 13, 1, 234, 23, 34, 3, 4.

  • I(C2) : ∅, 1234, 123, 12, 1, 234, 23, 2, 34, 3, 4.

  • I(C3) : ∅, 1234, 123, 12, 134, 13, 234, 23, 2, 34, 3.

  • I(C4) : ∅, 1234, 123, 134, 13, 14, 234, 23, 34, 3, 4.

  • I(C5) : ∅, 1234, 123, 12, 134, 13, 14, 234, 23, 2, 4.

  • I(C6) : ∅, 1234, 123, 134, 13, 14, 1, 234, 23, 34, 3.

  • I(C7) : ∅, 1234, 123, 12, 134, 13, 14, 1, 234, 23, 2.

  • I(C8) : ∅, 1234, 123, 124, 134, 13, 234, 23, 24, 34, 3.

  • I(C9) : ∅, 1234, 123, 124, 134, 13, 1, 234, 23, 24, 4.

  • I(C10) : ∅, 1234, 123, 124, 12, 1, 234, 23, 24, 2, 4.

  • I(C11) : ∅, 1234, 123, 124, 14, 234, 23, 24, 34, 3, 4.

  • I(C12) : ∅, 1234, 123, 124, 12, 14, 234, 23, 24, 2, 4.

  • I(C13) : ∅, 1234, 123, 124, 14, 1, 234, 23, 24, 34, 3.

  • I(C14) : ∅, 1234, 123, 124, 12, 14, 1, 234, 23, 24, 2.

  • I(C15) : ∅, 1234, 123, 12, 134, 13, 234, 23, 2, 34, 3.

  • I(C16) : ∅, 1234, 123, 124, 12, 234, 23, 24, 2, 34, 3.

  • I(C17) : ∅, 1234, 123, 124, 12, 234, 23, 24, 2, 34, 3.

  • I(C18) : ∅, 1234, 123, 124, 12, 234, 23, 24, 2, 34, 3.

  • I(C19) : ∅, 1234, 123, 124, 12, 134, 13, 234, 23, 24, 2.

  • I(C20) : ∅, 1234, 123, 12, 134, 13, 234, 23, 2, 34, 3.

Fig. 36
figure 36

C1–C20 are the only 20 groups with 11 incidence patterns

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Dumitrescu, A., Mandal, R. & Tóth, C.D. Monotone Paths in Geometric Triangulations. Theory Comput Syst 62, 1490–1524 (2018). https://doi.org/10.1007/s00224-018-9855-4

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