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Minimum Monotone Spanning Trees

Emilio Di Giacomo, Walter Didimo, Eleni Katsanou, Lena Schlipf, Antonios Symvonis, Alexander Wolff

TL;DR

This work provides a characterization of monotone spanning trees and describes algorithms that are much faster than those for the general case for the Euclidean minimum spanning tree, and shows that for every even integer $k, there exists a point set and a set of directions such that any minimum-length-monotone spanning tree of S_k has maximum vertex degree $2k.

Abstract

Computing a Euclidean minimum spanning tree of a set of points is a seminal problem in computational geometry and geometric graph theory. We combine it with another classical problem in graph drawing, namely computing a monotone geometric representation of a given graph. More formally, given a finite set $S$ of points in the plane and a finite set $\cal D$ of directions, a geometric spanning tree $T$ with vertex set $S$ is ${\cal D}$-monotone if, for every pair $\{u,v\}$ of vertices of $T$, there exists a direction $d \in \cal D$ for which the unique path from $u$ to $v$ in $T$ is monotone with respect to $d$. We provide a characterization of ${\cal D}$-monotone spanning trees. Based on it, we show that a ${\cal D}$-monotone spanning tree of minimum length can be computed in polynomial time if the number $k=|{\cal D}|$ of directions is fixed, both when (i) the set ${\cal D}$ of directions is prescribed and when (ii) the objective is to find a minimum-length ${\cal D}$-monotone spanning tree over all sets ${\cal D}$ of $k$ directions. For $k = 2$, we describe algorithms that are much faster than those for the general case. Furthermore, in contrast to the classical Euclidean minimum spanning tree, whose vertex degree is at most six, we show that for every even integer $k$, there exists a point set $S_k$ and a set $\cal D$ of $k$ directions such that any minimum-length $\cal D$-monotone spanning tree of $S_k$ has maximum vertex degree $2k$.

Minimum Monotone Spanning Trees

TL;DR

This work provides a characterization of monotone spanning trees and describes algorithms that are much faster than those for the general case for the Euclidean minimum spanning tree, and shows that for every even integer 2k.

Abstract

Computing a Euclidean minimum spanning tree of a set of points is a seminal problem in computational geometry and geometric graph theory. We combine it with another classical problem in graph drawing, namely computing a monotone geometric representation of a given graph. More formally, given a finite set of points in the plane and a finite set of directions, a geometric spanning tree with vertex set is -monotone if, for every pair of vertices of , there exists a direction for which the unique path from to in is monotone with respect to . We provide a characterization of -monotone spanning trees. Based on it, we show that a -monotone spanning tree of minimum length can be computed in polynomial time if the number of directions is fixed, both when (i) the set of directions is prescribed and when (ii) the objective is to find a minimum-length -monotone spanning tree over all sets of directions. For , we describe algorithms that are much faster than those for the general case. Furthermore, in contrast to the classical Euclidean minimum spanning tree, whose vertex degree is at most six, we show that for every even integer , there exists a point set and a set of directions such that any minimum-length -monotone spanning tree of has maximum vertex degree .

Paper Structure

This paper contains 12 sections, 37 theorems, 8 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Properties of Monotone Paths and Trees
  4. A Characterization of $\mathcal{D}$-Monotone Spanning Trees
  5. Algorithms for $\mathop{\mathrm{MMST}}\nolimits(S,\mathcal{D}\xspace)$
  6. Maximum Degree of the Minimum $k$-Directional MST
  7. Open Problems
  8. Additional Material for Section \ref{['se:properties-paths-trees']}
  9. Additional Material for Section \ref{['se:k-monotone']}
  10. Speed-Up for $|\mathcal{D}|=2$
  11. Algorithms for $\mathop{\mathrm{MMST}}\nolimits(S,k)$
  12. Additional Material for Section \ref{['se:maxdeg']}

Key Result

lemma 1

Let $S$ be a set of points, and let $P=\langle u,x,v \rangle$ be a geometric path on $S$. Let $d$ be a direction such that $S$ is in $d$-general position. If $u$ and $v$ lie in the same half-plane determined by $\overline{d}(x)$, then the path $P$ is not $d$-monotone.

Figures (20)

  • Figure 1: (a) A point set $S$ with its Delaunay triangulation, (b) MST of $S$, (c) $\mathop{\mathrm{MMST}}\nolimits$ of $S$ w.r.t. $\{\binom{1}{0},\binom{0}{1}\}$. The $v_2$--$v_3$ path is $x$-monotone; the $v_1$--$v_2$ and $v_1$--$v_3$ paths are $y$-monotone.
  • Figure 2: The set $\mathcal{W}_\mathcal{D}\xspace(p)$ for the point $p$ and the set $\mathcal{D}\xspace=\{d_1,d_2,d_3\}$.
  • Figure 3: (a) A directed geometric path $P$, (b) its sector of directions $\sec(P)$ (in dark gray) and (c) the wedge set $\mathcal{W}\xspace_P$ of path $P$ (in blue).
  • Figure 4: (a) A monotone tree and its sets of utilized wedges for each leaf path. (b) All sets of utilized wedges drawn on the same unit circle. Set $\mathcal{W}_{u \setminus v}\xspace$ (resp. $\mathcal{W}_{v \setminus u}\xspace$) consists of all wedges in the blue (resp. gray) region.
  • Figure 5: The different shapes of $R_{u,v}\xspace$ depending on $|\mathcal{W}\xspace_{B_{u,v}\xspace}|$.
  • ...and 15 more figures

Theorems & Definitions (59)

  • lemma 1: pr:monotone-half-plane
  • lemma 2: pr:diff_wedges
  • lemma 3: pr:max-degree-2k
  • lemma 4: AngeliniCBFP12
  • lemma 5: lemma:monotone-path-range
  • corollary 1
  • lemma 6: le:path-properties
  • lemma 7
  • lemma 8: le:k-directional-properties
  • lemma 9: le:branch-disjoint
  • ...and 49 more