AI 中文总结
研究地理系统发生树绘图问题，针对引导线可分页的情况，给出一般情况的整数线性规划公式和特殊情况的多项式时间算法，还提出特定设置下的\(\mathcal{O}(n^7)\)时间算法，实验展示算法性能及引导线交叉与页面数的权衡。
AI 中文摘要
地理系统发生树是一种常见的图表，用于在地理背景下可视化物种的进化历史。作为一个绘图问题，这些图表通常被建模为一个有根二叉“系统发生”树\(T\)，其中每个叶子都与矩形地图范围内的一个点特征（“站点”）相关联。树向下平面绘制，叶子放置在地图上边界的等距位置，每个叶子通过一条直线引导线连接到其对应的站点。先前的工作主要集中在最小化引导线交叉的数量，或者完全避免引导线。在本文中，我们探索分页地理系统发生树，其中引导线可以被划分为多个页面，只计算页面内的交叉。对于一般情况，即每个页面可以包含引导线的任意子集，我们提供了一个整数线性规划（ILP）公式来最小化页面数量，并为一个特殊情况（等同于单边纠结图）提供了一个多项式时间算法。我们认为，从可视化的角度来看，每个页面必须只包含单个子树的引导线，并提供了一个\(\mathcal{O}(n^7)\)时间算法来在这种设置下最小化页面数量——这也涉及改进用于测试无交叉标记存在的最快已知算法。与我们的最坏情况界限所暗示的相反，我们的实现可以在一秒内处理数百个站点的实例，正如我们在实验评估中所展示的，该评估进一步研究了引导线交叉和页面数量之间的权衡。
英文摘要
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.
CommentsAppears in the Proceedings of the 34th International Symposium on Graph Drawing and Network Visualization (GD 2026); 29 pages, 27 figures
