Rapid Sampling for Visualizations with Ordering Guarantees

Visualizations are frequently used as a means to understand trends and gather insights from datasets, but often take a long time to generate. In this paper, we focus on the problem of rapidly generating approximate visualizations while preserving crucial visual proper- ties of interest to analysts. Our primary focus will be on sampling algorithms that preserve the visual property of ordering; our techniques will also apply to some other visual properties. For instance, our algorithms can be used to generate an approximate visualization of a bar chart very rapidly, where the comparisons between any two bars are correct. We formally show that our sampling algorithms are generally applicable and provably optimal in theory, in that they do not take more samples than necessary to generate the visualizations with ordering guarantees. They also work well in practice, correctly ordering output groups while taking orders of magnitude fewer samples and much less time than conventional sampling schemes.Comment: Tech Report. 17 pages. Condensed version to appear in VLDB Vol. 8 No.

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - \epsilon})$ for some constant $\epsilon > 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time