A Hierarchical Heuristic for Clustered Steiner Trees in the Plane with Obstacles

TL;DR

The paper addresses the challenge of computing multiple obstacle-avoiding Euclidean Steiner trees in the plane to support decentralized multi-agent networks. It proposes a hierarchical bundling framework that (i) clusters terminals with a dendrogram $Z$, (ii) computes $A^*$ shortest paths from cluster roots on visibility graphs, (iii) clusters these paths using a path-similarity metric $d(S_{i,u},S_{i,v})$ to form dendrograms $Z_i$, and (iv) bundles and concatenates component trees guided by $Z$ while optimizing Steiner points via a trust-region SQP search in a triangulated free space. The method balances total tree length and hub spread through a cost $F = w_l L_t + w_d L_d$, and demonstrates feasibility on 50 maps with 5 obstacles and 100 terminals, yielding multiple obstacle-avoiding trees that avoid clutter. The work offers a practical, scalable framework with potential for parallelization and new operators in obstacle-avoiding ESTs, enabling improved decentralized planning and network design in constrained 2D environments.

Abstract

Euclidean Steiner trees are relevant to model minimal networks in real-world applications ubiquitously. In this paper, we study the feasibility of a hierarchical approach embedded with bundling operations to compute multiple and mutually disjoint Euclidean Steiner trees that avoid clutter and overlapping with obstacles in the plane, which is significant to model the decentralized and the multipoint coordination of agents in constrained 2D domains. Our computational experiments using arbitrary obstacle configuration with convex and non-convex geometries show the feasibility and the attractive performance when computing multiple obstacle-avoiding Steiner trees in the plane. Our results offer the mechanisms to elucidate new operators for obstacle-avoiding Steiner trees.

Figures (10)

• Figure 1: Example of (a) the configuration of a plane with obstacles with either convex and non-convex geometries and terminal nodes located in the free space, (b)-(d) obstacle-avoiding multi-Euclidean Steiner trees on the plane with 5, 15, and 25 modules.
• Figure 2: Basic idea of a hierarchical bundling of shortest paths.
• Figure 3: Examples of the concatenation of multi-Steiner trees in the plane with obstacles. The colored regions indicate areas where concatenation of Steiner trees occur, following the sequence of arrows in orange color.
• Figure 4: Examples of the landscape of a cost function and the corresponding configuration multi-Steiner trees in the plane.
• Figure 5: The rendering of 50 arbitrary configurations of terminal nodes and obstacles in the plane.