Minimizing Vertical Length in Linked Bar Charts

TL;DR

This work investigates minimizing the total vertical length of links in linked bar charts with a fixed bar order, focusing on the distinction between dependent and independent links and the resulting impact on readability. It provides algorithmic frameworks that exploit structural properties: when the subgraph of dependent links forms a forest, a dynamic program yields an $O(nm)$ solution, and when non-adjacent dependent links form a forest, the bound becomes $O(n^4m)$; additionally, the problem is fixed-parameter tractable in the maximum degree of a bar. The authors also develop efficient preprocessing for L/R edge sets and link-type classification, enabling constant-time cost evaluations for placements and a component-wise optimization strategy. These results illuminate the tractability frontier for stacked-bar visualizations and offer practical, structure-aware methods for reducing vertical link length in cross-category visualizations.

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.

Figures (5)

• Figure 1: Two linked bar charts PairWiseUncertainty 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).
• Figure 2: Schematics of links, as determined by the interval of possible center coordinates (drawn ranges). Left: the three independent link cases. Right: three possible placements for a single dependent link.
• Figure 3: Left: $P(B, k)$ is the minimum total cost of all blocks in $B$ and its child subtrees, except that of block $l_{p^*}$ (hatched), when $l_{p^*}$ is placed with $k$ blocks of $R$ under it. Grey dots are independent links. Middle: block $l_{p^*}$ connects $B$ to its parent in $T$. When placing $k$ blocks of $R$ under $l_{p^*}$, $B$ is split (black line) into two parts that can be stacked independently. Right: $D_\downarrow(p, q)$ is the minimum cost of the first $p$ and $q$ blocks from $L$ and $R$, respectively.
• Figure 4: Left: illustration of $P(B, k, l, r)$ and ADLs $a_{\leftarrow, 1}, a_{\leftarrow, 2}, \dots$. The parent link (hatched) is excluded. Right: illustration of $D_\uparrow(p, q, x, y)$. There cannot be a leftward DL to a child of $B$.
• Figure 5: Left: illustration of $D_\downarrow(p, q, x, y)$. Right: the subproblems for computing $D_\downarrow(p, q, x, y)$ when dependent block $l_p$ is on top, with Cost \ref{["eq:D'"]} (purple) and Cost \ref{["eq:P'"]} (orange).

Theorems & Definitions (5)

• Lemma 0
• Lemma 0
• Theorem 1
• Lemma 1
• Theorem 2