Area-preserving C-oriented schematization

We define an edge-move operation for polygons and prove that every simple non-convex polygon P has a non-conflicting pair of complementary edge-moves that reduces the number of edges of P while preserving its area. We use this result to generate area-preserving C-oriented schematizations of polygons

Four Soviets walk the dog, with an application to Alt's conjecture

Given two polygonal curves in the plane, there are several ways to define a measure of similarity between them. One measure that has been extremely popular in the past is the Frechet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been described. However, even 20 years later, the original O(n^2 log n) algorithm by Alt and Godau for computing the Frechet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Frechet distance, where we consider sequences of points instead of polygonal curves. Building on their work, we give an algorithm to compute the Frechet distance between two polygonal curves in time O(n^2 (log n)^(1/2) (\log\log n)^(3/2)) on a pointer machine and in time O(n^2 (loglog n)^2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the Frechet problem of depth O(n^(2-epsilon)), for some epsilon > 0. This provides evidence that computing the Frechet distance may not be 3SUM-hard after all and reveals an intriguing new aspect of this well-studied problem

Four Soviets walk the dog, with an application to Alt's conjecture

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n^2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard.In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n^2 \sqrt log n (log log n)^{3/2}) on a pointer machine and in time O(n^2 (log log n)^2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n^{2¿}), for some ¿ > 0. This provides evidence that the decision problem may not be 3SUM-hard after all and reveals an intriguing new aspect of this well-studied problem

A framework for algorithm stability

We say that an algorithm is stable if small changes in the input result in small changes in the output. Algorithm stability plays an important role when analyzing and visualizing time-varying data. However, so far, there are only few theoretical results on the stability of algorithms, possibly due to a lack of theoretical analysis tools. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees

Exploring curved schematization

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of geographic outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of 'curviness': we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. We conducted an online user study investigating the effectiveness of the curved schematizations compared to straight-line schematizations of equivalent complexity. While the visual complexity of the curved shapes was judged higher than those using straight lines, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents

Small Multiples with Gaps

Small multiples enable comparison by providing different views of a single data set in a dense and aligned manner. A common frame defines each view, which varies based upon values of a conditioning variable. An increasingly popular use of this technique is to project two-dimensional locations into a gridded space (e.g. grid maps), using the underlying distribution both as the conditioning variable and to determine the grid layout. Using whitespace in this layout has the potential to carry information, especially in a geographic context. Yet, the effects of doing so on the spatial properties of the original units are not understood. We explore the design space offered by such small multiples with gaps. We do so by constructing a comprehensive suite of metrics that capture properties of the layout used to arrange the small multiples for comparison (e.g. compactness and alignment) and the preservation of the original data (e.g. distance, topology and shape). We study these metrics in geographic data sets with varying properties and numbers of gaps. We use simulated annealing to optimize for each metric and measure the effects on the others. To explore these effects systematically, we take a new approach, developing a system to visualize this design space using a set of interactive matrices. We find that adding small amounts of whitespace to small multiple arrays improves some of the characteristics of 2D layouts, such as shape, distance and direction. This comes at the cost of other metrics, such as the retention of topology. Effects vary according to the input maps, with degree of variation in size of input regions found to be a factor. Optima exist for particular metrics in many cases, but at different amounts of whitespace for different maps. We suggest multiple metrics be used in optimized layouts, finding topology to be a primary factor in existing manually-crafted solutions, followed by a trade-off between shape and displacement. But the rich range of possible optimized layouts leads us to challenge single-solution thinking; we suggest to consider alternative optimized layouts for small multiples with gaps. Key to our work is the systematic, quantified and visual approach to exploring design spaces when facing a trade-off between many competing criteria—an approach likely to be of value to the analysis of other design spaces

Mapping polygons to the grid with small Hausdorff and Fréchet distance

We show how to represent a simple polygon P by a grid (pixel-based) polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output