Drawing non-layered tidy trees in linear time

The well-known Reingold–Tilford algorithm produces tidy-layered drawings of trees: drawings where all nodes at the same depth are vertically aligned. However, when nodes have varying heights, layered drawing may use more vertical space than necessary. A non-layered drawing of a tree places children at a fixed distance from the parent, thereby giving a more vertically compact drawing. Moreover, non-layered drawings can also be used to draw trees where the vertical position of each node is given, by adding dummy nodes. In this paper, we present the first linear-time algorithm for producing non-layered drawings. Our algorithm is a modification of the Reingold–Tilford algorithm, but the original complexity proof of the Reingold–Tilford algorithm uses an invariant that does not hold for the non-layered case. We give an alternative proof of the algorithm and its extension to non-layered drawings. To improve drawings of trees of unbounded degree, extensions to the Reingold–Tilford algorithm have been proposed. These extensions also work in the non-layered case, but we show that they then cause a O(n2) run-time. We then propose a modification to these extensions that restores the O(n) run-time

Drawing non-layered tidy trees in linear time

The well-known Reingold–Tilford algorithm produces tidy-layered drawings of trees: drawings where all nodes at the same depth are vertically aligned. However, when nodes have varying heights, layered drawing may use more vertical space than necessary. A non-layered drawing of a tree places children at a fixed distance from the parent, thereby giving a more vertically compact drawing. Moreover, non-layered drawings can also be used to draw trees where the vertical position of each node is given, by adding dummy nodes. In this paper, we present the first linear-time algorithm for producing non-layered drawings. Our algorithm is a modification of the Reingold–Tilford algorithm, but the original complexity proof of the Reingold–Tilford algorithm uses an invariant that does not hold for the non-layered case. We give an alternative proof of the algorithm and its extension to non-layered drawings. To improve drawings of trees of unbounded degree, extensions to the Reingold–Tilford algorithm have been proposed. These extensions also work in the non-layered case, but we show that they then cause a O(n2) run-time. We then propose a modification to these extensions that restores the O(n) run-time

Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Efficient abstractions for visualization and interaction

Abstractions, such as functions and methods, are an essential tool for any programmer. Abstractions encapsulate the details of a computation: the programmer only needs to know what the abstraction achieves, not how it achieves it. However, using abstractions can come at a cost: the resulting program may be inefficient. This can lead to programmers not using some abstractions, instead writing the entire functionality from the ground up. In this thesis, we present several results that make this situation less likely when programming interactive visualizations. We present results that make abstractions more efficient in the areas of graphics, layout and events

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

Beyond The Hobbit: J.R.R. Tolkien’s Other Works for Children

John Ronald Reuel Tolkien is best known to the world as the author of the classic fantasies The Hobbit and The Lord of the Rings. In his professional life, he was a superb philologist, a skilled translator, the author of a seminal essay on Beowulf, and a contributor to the Oxford English Dictionary. But Tolkien was also a father who loved to make up stories for his four children, write them down, and in many cases, as we’ve seen in the exhibit at the Morgan, illustrate them himself. Tolkien was an enthusiastic amateur artist with a unique style, loved color and line and repetitive decoration, but he was rather better at depicting landscapes than people. He usually worked in pen and ink, chalk, or colored pencil. In addition to The Hobbit, widely considered a classic of children’s literature, he also wrote four shorter works specifically for children, two published during his lifetime and two posthumously, as well as many poems and a delightful collection of annual illustrated letters from Father Christmas