Minimum spanning blob-trees

2025-03-04

AI Summary

This paper introduces *blob-tree*, a novel hybrid structure for modeling point-set connectivity that unifies convex-hull-like cyclic enclosures with spanning-tree-like edge connections, aiming to compute a minimum-cost blob-tree. We formally define the blob-tree concept for the first time, bridging geometric enclosure and graph-theoretic connectivity paradigms. We propose the first exact algorithm, leveraging computational-geometric analysis, dynamic programming, and nested-cycle structural modeling to solve instances of $n$ points in $O(n^3)$ time. The algorithm is theoretically sound—guaranteeing optimality—and geometrically interpretable, offering explicit spatial semantics for each structural component. This work establishes blob-trees as a new polynomial-time solvable model for structured point-set representation, advancing both combinatorial geometry and network design theory.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Develops blob-trees for connecting points
• Combines cycles and edges for connectivity
• Computes minimum-cost blob-tree in O(n^3) time

Innovation

• Combines cycles and edges for point connection
• Introduces blob-trees for efficient point linking
• Computes minimum-cost blob-trees in O(n^3) time