Minimizing Vertical Length in Linked Bar Charts

2025-11-20

AI Summary

This paper addresses the problem of minimizing the total vertical length of orthogonal inter-block connecting lines in linked bar charts under a fixed bar ordering, where dependencies among links necessitate joint optimization rather than independent line routing. We propose a dynamic programming approach grounded in combinatorial optimization and graph theory: for dependency forests, we devise an optimal $O(nm)$-time algorithm; for general dependency graphs, we introduce the first fixed-parameter tractable (FPT) formulation and present an exact $O(n^4m)$ algorithm. Theoretical analysis guarantees global optimality, while empirical evaluation confirms computational feasibility and measurable improvements in chart readability. Our primary contribution is overcoming the fundamental bottleneck of dependency-aware joint link optimization, establishing the first systematic solution framework that simultaneously provides rigorous theoretical guarantees and practical efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Minimizing vertical link length in linked bar charts
• Optimizing block order to improve visual legibility
• Solving algorithmic complexity for dependent link arrangements

Innovation

• Minimizes vertical length of dependent links
• Optimizes link layout for fixed bar order
• Uses forest-based algorithms for efficient computation