arXiv subjects

Tim Dwyer

Publications and source records attributed to Tim Dwyer.

44 records · Page 3Linked to original sources

arXiv2019

Maps and Globes in Virtual Reality

This paper explores different ways to render world-wide geographic maps in virtual reality (VR). We compare: (a) a 3D exocentric globe, where the user's viewpoint is outside the globe; (b) a flat map (rendered to a plane in VR); (c) an egocentric 3D globe, with the viewpoint inside the globe; and (d) a curved map, created by projecting the map onto a section of a sphere which curves around the user. In all four visualisations the geographic centre can be smoothly adjusted with a standard handheld VR controller and the user, through a head-tracked headset, can physically move around the visualisation. For distance comparison, exocentric globe is more accurate than egocentric globe and flat map. For area comparison, more time is required with exocentric and egocentric globes than with flat and curved maps. For direction estimation, the exocentric globe is more accurate and faster than the other visual presentations. Our study participants had a weak preference for the exocentric globe. Generally, the curved map had benefits over the flat map. In almost all cases the egocentric globe was found to be the least effective visualisation. Overall, our results provide support for the use of exocentric globes for geographic visualisation in mixed-reality.

cs.HC↗

arXiv2018

Exploring the Limits of Complexity: A Survey of Empirical Studies on Graph Visualisation

For decades, researchers in information visualisation and graph drawing have focused on developing techniques for the layout and display of very large and complex networks. Experiments involving human participants have also explored the readability of different styles of layout and representations for such networks. In both bodies of literature, networks are frequently referred to as being 'large' or 'complex', yet these terms are relative. From a human-centred, experiment point-of-view, what constitutes 'large' (for example) depends on several factors, such as data complexity, visual complexity, and the technology used. In this paper, we survey the literature on human-centred experiments to understand how, in practice, different features and characteristics of node-link diagrams affect visual complexity.

cs.HC↗

arXiv2018

Wilf equivalence relations for consecutive patterns

Two permutations $π$ and $τ$ are c-Wilf equivalent if, for each $n$, the number of permutations in $S_n$ avoiding $π$ as a consecutive pattern (i.e., in adjacent positions) is the same as the number of those avoiding $τ$. In addition, $π$ and $τ$ are strongly c-Wilf equivalent if, for each $n$ and $k$, the number of permutations in $S_n$ containing $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In this paper we introduce a third, more restrictive equivalence relation, defining $π$ and $τ$ to be super-strongly c-Wilf equivalent if the above condition holds for any set of prescribed positions for the $k$ occurrences. We show that, when restricted to non-overlapping permutations, these three equivalence relations coincide. We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if $π,τ$ in $S_m$ are strongly c-Wilf equivalent, then $|π_m-π_1|=|τ_m-τ_1|$. In the special case of non-overlapping permutations $π$ and $τ$, this proves a weaker version of a conjecture of the second author stating that $π$ and $τ$ are c-Wilf equivalent if and only if $π_1=τ_1$ and $π_m=τ_m$, up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.

math.CO↗

arXiv2014

Stress-Minimizing Orthogonal Layout of Data Flow Diagrams with Ports

We present a fundamentally different approach to orthogonal layout of data flow diagrams with ports. This is based on extending constrained stress majorization to cater for ports and flow layout. Because we are minimizing stress we are able to better display global structure, as measured by several criteria such as stress, edge-length variance, and aspect ratio. Compared to the layered approach, our layouts tend to exhibit symmetries, and eliminate inter-layer whitespace, making the diagrams more compact.

cs.OH↗

arXiv2013

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.

cs.CG↗

arXiv2013

Incremental Grid-like Layout Using Soft and Hard Constraints

We explore various techniques to incorporate grid-like layout conventions into a force-directed, constraint-based graph layout framework. In doing so we are able to provide high-quality layout---with predominantly axis-aligned edges---that is more flexible than previous grid-like layout methods and which can capture layout conventions in notations such as SBGN (Systems Biology Graphical Notation). Furthermore, the layout is easily able to respect user-defined constraints and adapt to interaction in online systems and diagram editors such as Dunnart.

cs.HC↗

arXiv2012

On 021-Avoiding Ascent Sequences

Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev in their study of $(\bf{2+2})$-free posets. An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_{1}x_{2}... x_{n}$ such that $x_{1}=0$ and $x_{i}\leq \asc(x_{1}x_{2}...x_{i-1})+1$ for all $1<i\leq n$, where $\asc(x_{1}x_{2}...x_{i-1})$ is the number of ascents in the sequence $x_{1}x_{2}... x_{i-1}$. We let $\cA_n$ stand for the set of such sequences and use $\cA_n(p)$ for the subset of sequences avoiding a pattern $p$. Similarly, we let $S_{n}(τ)$ be the set of $τ$-avoiding permutations in the symmetric group $S_{n}$. Duncan and Steingr\'ımsson have shown that the ascent statistic has the same distribution over $\cA_n(021)$ as over $S_n(132)$. Furthermore, they conjectured that the pair $(\asc, \rlm)$ is equidistributed over $\cA_n(021)$ and $S_n(132)$ where $\rlm$ is the right-to-left minima statistic. We prove this conjecture by constructing a bistatistic-preserving bijection.

math.CO↗

arXiv2012

Permutation patterns and statistics

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.

math.CO↗