Node Labeling in Line Diagrams of Ordered Sets

2026-07-25

AI Summary

This study addresses the challenge of label overlap with nodes, edges, and other labels in ordered parallel coordinates plots by proposing a two-stage flexible labeling algorithm that balances layout readability with minimal connection distance. The method first pre-selects fixed-position labels based on readability criteria; for overflow labels, it leverages both internal and external plot space to connect them to their corresponding nodes via non-crossing straight lines. An initial placement is computed using a sparse grid-based cost function, followed by refinement through a force-directed model. Notably, this approach introduces the first strategy specifically tailored to the vertical-edge characteristics of ordered parallel coordinates, flexibly accommodating the dual-region constraints inherent in formal concept analysis and requiring only minor adjustments to suit concept lattice labeling. Experimental results demonstrate that the algorithm efficiently produces non-crossing, highly readable label layouts and has been successfully applied to concept lattice visualization.

Abstract

We propose a flexible, two-phase algorithm for labeling line diagrams of ordered sets, in which the nodes of direct neighbors in the order relation are connected by a straight, upward-pointing line. In contrast to the labeling of diagrams of arbitrary graphs, we benefit from the fact that all edges in line diagrams of ordered sets are more or less vertical. In this paper, we study the placement of all labels such that they do not intersect with any nodes, lines, or other labels while minimizing the distances between the nodes and their labels. Our approach starts by filtering the fixed-position model using line diagram-specific readability criteria. For labels that cannot be placed adjacent to their node (overflow labels), we exploit the free space in the graph's interior or the infinite space surrounding the drawing and link the labels with their respective node by straight binding lines that should not cross other nodes or labels if possible. To balance quality and runtime, we derive an initial placement of the overflow labels using a cost function over a sparse grid of candidates, followed by a force-based refinement step to fine-tune the layout. Furthermore, we demonstrate the flexibility of this approach by applying it to the visual constraints of line diagrams in the field of Formal Concept Analysis (FCA), where certain labels have to be placed above their node and others below. This special version of the algorithm shows that a pre-filtering in the first phase and minimal adjustments for the cost function and force-based model are sufficient to handle the dual labeling requirements of concept lattices.

Problem

• node labeling
• line diagrams
• ordered sets
• label placement
• graph drawing

Innovation

• line diagrams
• node labeling
• ordered sets
• force-directed refinement
• Formal Concept Analysis