Layered Area-Proportional Rectangle Contact Representations

TL;DR

This paper studies optimization problems for layered rectangle contact representations of planar graphs, aiming to maximize contacts or minimize area, with applications in visualizing related words or tags.

Contribution

It introduces new models and algorithms, including a network flow model, a linear-time algorithm for two-layer graphs, and an ILP for multi-layer graphs.

Findings

Network flow model effectively minimizes area.

Linear-time algorithm maximizes contacts in two-layer graphs.

ILP approach handles multi-layer graph contact maximization.

Abstract

We investigate two optimization problems on area-proportional rectangle contact representations for layered, embedded planar graphs. The vertices are represented as interior-disjoint unit-height rectangles of prescribed widths, grouped in one row per layer, and each edge is ideally realized as a rectangle contact of positive length. Such rectangle contact representations find applications in semantic word or tag cloud visualizations, where a collection of words is displayed such that pairs of semantically related words are close to each other. In this paper, we want to maximize the number of realized rectangle contacts or minimize the overall area of the rectangle contact representation, while avoiding any false adjacencies. We present a network flow model for area minimization, a linear-time algorithm for contact maximization of two-layer graphs, and an ILP model for maximizing contacts of $k$-layer graphs.