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Minimum spanning blob-trees

Katharina Klost, Marc van Kreveld, Daniel Perz, Günter Rote, Josef Tkadlec

TL;DR

This work introduces the minimum blob-tree, a structure that couples convex blobs with tree-edges to connect a planar point set. The authors prove structural properties showing that optimal blob-trees arise from the MST by replacing certain subtrees with convex hull blobs, and they develop a dynamic-programming framework anchored on a rooted MST to compute the optimum in $O(n^3)$ time. The core contributions include a detailed taxonomy of geometric components (valid chords, walls, and triangles/digons) and a decomposition into edge, chord, and wall subproblems, with preprocessing to support efficient DP transitions. This provides a scalable approach that unifies convex-hull and MST concepts, enabling practical computation of near- or exact-optimal blob-trees for geometric networks and visualizations.

Abstract

We investigate blob-trees, a new way of connecting a set of points, by a mixture of enclosing them by cycles (as in the convex hull) and connecting them by edges (as in a spanning tree). We show that a minimum-cost blob-tree for $n$ points can be computed in $O(n^3)$ time.

Minimum spanning blob-trees

TL;DR

This work introduces the minimum blob-tree, a structure that couples convex blobs with tree-edges to connect a planar point set. The authors prove structural properties showing that optimal blob-trees arise from the MST by replacing certain subtrees with convex hull blobs, and they develop a dynamic-programming framework anchored on a rooted MST to compute the optimum in time. The core contributions include a detailed taxonomy of geometric components (valid chords, walls, and triangles/digons) and a decomposition into edge, chord, and wall subproblems, with preprocessing to support efficient DP transitions. This provides a scalable approach that unifies convex-hull and MST concepts, enabling practical computation of near- or exact-optimal blob-trees for geometric networks and visualizations.

Abstract

We investigate blob-trees, a new way of connecting a set of points, by a mixture of enclosing them by cycles (as in the convex hull) and connecting them by edges (as in a spanning tree). We show that a minimum-cost blob-tree for points can be computed in time.

Paper Structure

This paper contains 27 sections, 11 theorems, 1 equation, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural insights
  4. Sidedness.
  5. Valid chords.
  6. Walls.
  7. Triangles and Digons.
  8. The algorithm
  9. Edge problems.
  10. Chord problems.
  11. Wall problems.
  12. Relation between subproblems.
  13. Preprocessing.
  14. Chord problems.
  15. Edge problems.
  16. ...and 12 more sections

Key Result

Lemma 1

In any optimal solution: all blobs are convex polygons with vertices at $P$; any two blobs are disjoint; and when contracting the blobs to vertices, the tree-edges form a tree.

Figures (11)

  • Figure 1: The optimal blob-trees on two point sets (computed by our implementation repo). Blob edges are blue, tree-edges are green. The cost of the solution is the total length of blue and green. MST edges are black (and green, if they are part of the blob-tree).
  • Figure 2: Three variants of blob-trees for the same point set, and their costs.
  • Figure 3: A minimum spanning tree, a potential subset of non-tree edges (fat blue), and the resulting blob-tree.
  • Figure 4: A rooted, directed blob tree, two types of exits, and a bottom-vertex triangulation of a blob with a dynamic programming order through it towards the root.
  • Figure 5: A blob is divided into triangles and digons.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • theorem 1
  • proof
  • proof
  • ...and 11 more