Sergey Bereg

Edge routing with ordered bundles

We propose a new approach to edge bundling. At the first stage we route the edge paths so as to minimize a weighted sum of the total length of the paths together with their ink. As this problem is NP-hard, we provide an efficient heuristic that finds an approximate solution. The second stage then separates edges belonging to the same bundle. To achieve this, we provide a new and efficient algorithm that solves a variant of the metro-line crossing minimization problem. The method creates aesthetically pleasing edge routes that give an overview of the global graph structure, while still drawing each edge separately, without intersecting graph nodes, and with few crossings.

Curvature-bounded traversals of narrow corridors

We consider the existence and efficient construction of bounded curvature paths traversing constant-width regions of the plane, called corridors. We make explicit a width threshold τ with the property that (a) all corridors of width at least τ admit a unit-curvature traversal and (b) for any width w < τ there exist corridors of width w with no such traversal. Applications to the design of short, but not necessarily shortest, and high clearance, but not necessarily maximum clearance, curvature-bounded paths in general polygonal domains, are also discussed.

Enumerating pseudo-triangulations in the plane

A pseudo-triangle is a simple polygon with exactly three convex vertices. A pseudo-triangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudo-triangles whose vertices are points of S. A pointed pseudo-triangulation is one which has the least number of pseudo-triangles. We study the graph G whose vertices represent the pointed pseudo-triangulations and whose edges represent flips. We present an algorithm for enumerating pointed pseudo-triangulations in O(log n) time per pseudo-triangulation.

Approximating Barrier Resilience in Wireless Sensor Networks

Barrier coverage in a sensor network has the goal of ensuring that all paths through the surveillance domain joining points in some start region S to some target region T will intersect the coverage region associated with at least one sensor. In this paper, we revisit a notion of redundant barrier coverage known as k-barrier coverage. We describe two different notions of width, or impermeability, of the barrier provided by the sensors in

RNA multiple structural alignment with longest common subsequences

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Moving coins

to paths joining two arbitrary regions S to T. The first, what we refer to as the thickness of the barrier, counts the minimum number of sensor region intersections, over all paths from S to T. The second, what we refer to as the resilience of the barrier, counts the minimum number of sensors whose removal permits a path from S to T with no sensor region intersections. Of course, a configuration of sensors with resilience k has thickness at least k and constitutes a k-barrier for S and T. Our result demonstrates that any (Euclidean) shortest path from S to T that intersects a fixed number of distinct sensors, never intersects any one sensor more than three times. It follows that the resilience of

Transforming pseudo-triangulations

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Finding Nearest Larger Neighbors

(with respect to S and T) is at least one-third the thickness of

Certifying and constructing minimally rigid graphs in the plane

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Monadic Decomposition

(with respect to S and T). (Furthermore, if points in S and T are moderately separated (relative to the radius of individual sensor coverage) then no shortest path intersects any one sensor more than two times, and hence the resilience of