Optimizing Symbol Visibility through Displacement

TL;DR

This work initiates algorithmic study of optimizing symbol visibility under constrained displacement: placing $n$ unit square symbols at fixed $y$-coordinates inside a width-$w$ strip ($1<w\le 2$) to maximize the minimum visible perimeter (gap) via both x-displacement and drawing order. It provides tight bounds and efficient constructions in two regimes: (i) when $w,h\le 2$, all squares are stabbed by a point and staircase layouts can achieve the supremum gap up to an arbitrarily small $\delta$ in $O(n\log n)$ time, with the gap characterized by a linear program solvable by a water-filling method; (ii) when $w\le 2$ with arbitrary height, a vertical-line stabbing enables a $2$-approximation in $O(n\log n)$ time using a squeezing algorithm that partitions by height and recombines, with a zigzag layout proven asymptotically optimal for uniformly spaced $y$-coordinates. These results advance understanding of controlled symbol displacement to improve readability in information visualization and point to several avenues for future work, including extending to rectangular or variably sized symbols and improving theoretical bounds in the uniform-spacing regime.

Abstract

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper, we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider closed unit square symbols that need to be placed at specified $y$-coordinates. We optimize the drawing order of the symbols as well as their $x$-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most $2$, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in $O(n\log n)$ time. If the width is at most $2$, there is a vertical line that stabs all squares, and in this case, we design a 2-approximation algorithm (assuming fixed container height) that runs in $O(n\log n)$ time. As it turns out that a minimum visible perimeter of 2 is always achievable with a generic construction, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

Figures (6)

• Figure 1: A pro and con vaccination discussion on Twitter from $2018$, capturing $823$ accounts. Accounts are placed in a categorical strip according to the most frequent sentiment expressed. Color indicates if an account mostly mentions accounts with a similar opinion (blue) or with a different opinion (red). Top: random jittering, bottom: our approximation algorithm described in Theorem \ref{['thm:main_approx']}. Random jittering hides many details, such as the pro-vaccination accounts that mention predominantly anti-vaccination accounts (red boundaries). Many accounts at the bottom sent the same number of posts and hence share a $y$-coordinate, which inevitably covers most of their horizontal edges.
• Figure 2: An instance with $y_2-y_1=2\varepsilon, y_3-y_2=\epsilon$, and a strip of width $w=1+\varepsilon$, where $\varepsilon\leq 0.2$, has no optimal layout.
• Figure 3: A staircase (far left), and three generalized staircases.
• Figure 4: The visible perimeter of the bottom square $s$ is at most two if it is contained in the bounding box of the other squares.
• Figure 9: Bad squares: at least two covered corners; A standard bad square ((a) and (b)): one vertical side is covered.

Theorems & Definitions (16)

• Lemma 1
• Definition 2
• Proposition 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 7
• Lemma 8
• Theorem 9
• Definition 10