연결 막대 차트의 수직 선 길이 최소화

📝 Abstract

A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with ``dependent’’ links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in $O(nm)$ time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an $O(n^4m) $-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar.

💡 Analysis

A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with ``dependent’’ links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in $O(nm)$ time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an $O(n^4m) $-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar.

📄 Content

Minimizing Vertical Length in Linked Bar Charts Steven van den Broek # TU Eindhoven, the Netherlands Marc van Kreveld # Utrecht University, the Netherlands Wouter Meulemans # TU Eindhoven, the Netherlands Arjen Simons # TU Eindhoven, the Netherlands Abstract A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with dependent links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in O(nm) time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an O(n4m)-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar. 2012 ACM Subject Classification Theory of computation →Design and analysis of algorithms; Human-centered computing →Graph drawings Keywords and phrases Graph drawing, bar chart, length minimization, dynamic programming, fixed-parameter tractability Funding Wouter Meulemans: Partially supported by the Dutch Research Council (NWO) under project number VI.Vidi.223.137. Arjen Simons: Supported by the Dutch Research Council (NWO) under project number VI.Vidi.223.137. Acknowledgements Research on the topic of this paper was initiated at the 8th Workshop on Applied Geometric Algorithms (AGA 2024) in Otterlo, The Netherlands. 1 Introduction Bar charts are a ubiquitous tool for visualizing scalar values across categories. Stacked bar charts, in particular, allow different quantities to be aggregated in a single column. In their traditional form, they primarily show single-category values: values that are uniquely attributable to a specific category. In many settings, however, certain quantities are not uniquely attributable to a single category but instead relate to multiple categories. Such cross-category values may arise in different forms. They may represent shared quantities, for example, in a bar chart encoding total communication per account: the communication between two accounts is present in both bars. They may also represent pairwise uncertainties, that is, quantities that may belong to one of two categories. For example, in election poll results, groups of voters may hesitate between two political parties, or in the analysis of pollution, factories near country borders may contribute to the pollution of either country, though the exact distribution may not be known. As standard bar charts cannot directly visualize such cross-category values, linked bar charts were recently introduced [17]: the single- and cross-category values partition each bar into blocks of appropriate height, such that the total bar height reflects the aggregate value of arXiv:2511.16812v1 [cs.CG] 20 Nov 2025 2 Minimizing Vertical Length in Linked Bar Charts size Figure 1 Two linked bar charts [17] that show the same data using different vertical orderings, with cross-category scalar values between linked blocks (pink) and single-category values drawn as unlinked blocks (gray). the category. Each cross-category value is then visualized through a link: a polyline between the blocks that passes over intermediate bars (Figure 1). We refer to blocks as unlinked or linked blocks, for single-category and cross-category values, respectively, that is, depending on whether the block is linked to another. Note that linked bar charts effectively show a weighted graph: the bars are vertices, weighted by their single-category values, and linked blocks are edges, weighted by their cross-category values. The quality of the resulting drawing readily depends on the order of the bars, as well as the order in which the blocks are stacked on top of each other. Arising from (orthogonal) graph-drawing literature, there are various natural measures [8], such as the number of crossings [5, 13, 14], the length of the links [15, 16] and the number of bends [2, 10, 13, 15]. Vertical distance between elements has also been considered as a quality measure for other visualizations, such as storylines [9] and parallel coordinate plots [7]. For quality measures that rely solely on the bar order without considering the stacking order of the blocks, the problem is effectively that of drawing the (unweighted) graph with a one-page book embedding. The outerplanar graphs are exactly the family of graphs that can be drawn without edge crossings in this style [1]. For such graphs, minimizing total and maximum (horizontal) edge length, cutwidth, or bandwidth is polynomial-time solvable [11]. These problems are NP-hard for general graphs [12]. In contrast, the stacking order of the blocks

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