Paged Geophylogenies: A Coloring Approach to External Labeling with Tree Constraints

2026-07-26

AI Summary

This study addresses the problem of excessive leader-line crossings in phylogeographic maps, which severely impair visual clarity. To mitigate this issue, the authors propose a pagination strategy that distributes leaders across multiple pages to minimize intra-page crossings. They formulate a pagination model incorporating subtree constraints and develop tailored optimization algorithms for both the general case and the constrained scenario. The general approach combines integer linear programming with graph coloring techniques, while the subtree-constrained variant yields the first exact algorithm with O(n⁷) time complexity and improves upon the fastest known method for detecting crossing-free labelings. Experimental results demonstrate that the proposed algorithms efficiently handle instances with hundreds of sites within one second, achieving a favorable trade-off between the number of leader crossings and the number of pages required.

Abstract

Geophylogenies are a common type of diagram for visualizing the evolutionary history of species in a geographic context. As a drawing problem, these diagrams are commonly modeled as a rooted binary ''phylogenetic'' tree $T$ where every leaf is associated with a point feature (''site'') in a rectangular map range. The tree is drawn downward planar, its leaves are placed at equidistant positions on the upper boundary of the map, and each leaf is connected to its corresponding site by a straight-line leader. Prior work focuses on minimizing the number of leader crossings, or avoiding leaders altogether. In this paper, we explore paged geophylogenies, where the leaders can be partitioned into multiple pages where only crossings within a page are counted. For the general case, where each page can contain an arbitrary subset of the leaders, we provide an integer linear programming (ILP) formulation for minimizing the number of pages, and a polynomial-time algorithm for a special case that is equivalent to a one-sided tanglegram. We argue that, from a visualization perspective, instead each page must contain only leaders for a single subtree, and provide an $\mathcal{O}(n^7)$ time algorithm for minimizing pages in this setting - which also involves improving the fastest known algorithm for testing the existence of a crossing-free labeling. Counter to what our worst case bound suggests, our implementation can handle instances with hundreds of sites within a second, as we show in our experimental evaluation, which further investigates the trade-offs between leader crossings and the number of pages.

Problem

• geophylogeny
• external labeling
• leader crossing
• tree constraints
• paged layout

Innovation

• paged geophylogenies
• external labeling
• tree constraints
• integer linear programming
• crossing minimization