Clustered edge routing

The classic method to depict graphs is a node-link diagram where vertices (nodes) are associated with each object and edges (links) connect related objects. However, node-link diagrams quickly appear cluttered and unclear, even for moderately sized graphs. If the positions of the nodes are fixed then suitable link routing is the only option to reduce clutter. We present a novel link clustering and routing algorithm which respects (and if desired refines) user-defined clusters on links. If no clusters are defined a priori we cluster based on geometric criteria, that is, based on a well-separated pair decomposition (WSPD).We route link clusters individually on a sparse visibility spanner. To completely avoid ambiguity we draw each individual link and ensure that clustered links follow the same path in the routing graph. We prove that the clusters induced by the WSPD consist of compatible links according to common similarity measures as formalized by Holten and van Wijk [17]. The greedy sparsification of the visibility graph allows us to easily route around obstacles. Our experimental results are visually appealing and convey a sense of abstraction and order

Distribution-sensitive construction of the greedy spanner (extended abstract)

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it take O(n^2) time, limiting its applicability on large data sets. We observe that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly, and few or no ‘long’ edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. This characterization gives a O(n log^2 n log^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points

Distribution-sensitive construction of the greedy spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take $\varOmega (n^2)$ time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an $O(n \log ^2 n \log ^2 \log n)$ expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an $O((n + |E|) \log ^2 n \log \log n)$ expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E

Mapping polygons to the grid with small Hausdorff and Fréchet distance

We show how to represent a simple polygon \u3ci\u3eP\u3c/i\u3e by a grid (pixel-based) polygon \u3ci\u3eQ\u3c/i\u3e that is simple and whose Hausdorff or Fréchet distance to \u3ci\u3eP\u3c/i\u3e is small. For any simple polygon \u3ci\u3eP\u3c/i\u3e, 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

Mapping polygons to the grid with small Hausdorff and Fréchet distance

We show how to represent a simple polygon \u3ci\u3eP\u3c/i\u3e by a grid (pixel-based) polygon \u3ci\u3eQ\u3c/i\u3e that is simple and whose Hausdorff or Fréchet distance to \u3ci\u3eP\u3c/i\u3e is small. For any simple polygon \u3ci\u3eP\u3c/i\u3e, 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

Mapping polygons to the grid with small Hausdorff and Fréchet distance

\u3cp\u3eWe show how to represent a simple polygon P by a (pixel-based) grid 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.\u3c/p\u3

Progressive geometric algorithms

Progressive algorithms are algorithms that, on the way to computing a complete solution to the problem at hand, output intermediate solutions that approximate the complete solution increasingly well. We present a framework for analyzing such algorithms, and develop efficient progressive algorithms for two geometric problems: computing the convex hull of a planar point set, and finding popular places in a set of trajectories