Optimal Orthogonal Drawings in Linear Time

2025-02-05

AI Summary

This paper addresses the orthogonal drawing problem for planar graphs with maximum degree at most three under variable embedding, aiming to simultaneously minimize total bend count and curve complexity (i.e., maximum bends per edge). This longstanding open problem had no known linear-time algorithm achieving optimality in both objectives. We present the first O(n)-time algorithm that attains both theoretical lower bounds—on total bend count and curve complexity—by leveraging a novel graph decomposition framework, customized data structures, and an efficient decomposition-tree rerooting strategy. Our approach is the first orthogonal drawing method for variable embedding that achieves dual optimality—both in total bends and curve complexity—while maintaining strict linear-time complexity. This resolves the classical open problem definitively.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A planar orthogonal drawing {Gamma} of a connected planar graph G is a geometric representation of G such that the vertices are drawn as distinct points of the plane, the edges are drawn as chains of horizontal and vertical segments, and no two edges intersect except at common end-points. A bend of {Gamma} is a point of an edge where a horizontal and a vertical segment meet. Drawing {Gamma} is bend-minimum if it has the minimum number of bends over all possible planar orthogonal drawings of G. Its curve complexity is the maximum number of bends per edge. In this paper we present a linear-time algorithm for the computation of planar orthogonal drawings of 3-graphs (i.e., graphs with vertex-degree at most three), that minimizes both the total number of bends and the curve complexity. The algorithm works in the so-called variable embedding setting, that is, it can choose among the exponentially many planar embeddings of the input graph. While the time complexity of minimizing the total number of bends of a planar orthogonal drawing of a 3-graph in the variable embedding settings is a long standing, widely studied, open question, the existence of an orthogonal drawing that is optimal both in the total number of bends and in the curve complexity was previously unknown. Our result combines several graph decomposition techniques, novel data-structures, and efficient approaches to re-rooting decomposition trees.

Problem

• minimize bends in drawings
• reduce curve complexity
• linear-time algorithm for 3-graphs

Innovation

• Linear-time orthogonal drawing algorithm
• Minimizes bends and curve complexity
• Variable embedding graph decomposition techniques