Daniel Leven

Bio: Daniel Leven is an academic researcher from Tel Aviv University. The author has contributed to research in topics: Motion planning & 1-planar graph. The author has an hindex of 7, co-authored 7 publications receiving 1142 citations.

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Abstract: Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.

Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams

Abstract: AnO(n logn) algorithm for planning a purely translational motion for a simple convex object amidst polygonal barriers in two-dimensional space is given. The algorithm is based on a new generalization of Voronoi diagrams (similar to that proposed by Chew and Drysdale [1] and by Fortune [2]), and adapts and uses a recent technique of Yap for the efficient construction of these diagrams.

Linear time algorithms for visibility and shortest path problems inside simple polygons

Abstract: We present linear time algorithms for solving the following problems involving a simple planar polygon P: (i) Computing the collection of all shortest paths inside P from a given source vertex s to all the other vertices of P; (ii) Computing the subpolygon of P consisting of points that are visible from a segment within P; (iii) Preprocessing P so that for any query ray r emerging from some fixed edge e of P, we can find in logarithmic time the first intersection of r with the boundary of P; (iv) Preprocessing P so that for any query point x in P, we can find in logarithmic time the portion of the edge e that is visible from x; (v) Preprocessing P so that for any query point x inside P and direction u, we can find in logarithmic time the first point on the boundary of P hit by the ray at direction u from x; (vi) Calculating a hierarchical decomposition of P into smaller polygons by recursive polygon cutting, as in [Ch]. (vii) Calculating the (clockwise and counterclockwise) “convex ropes” (in the terminology of [PS]) from a fixed vertex s of P lying on its convex hull, to all other vertices of P. All these algorithms are based on a recent linear time algorithm of Tarjan and Van Wyk for triangulating a simple polygon, but use additional techniques to make all subsequent phases of these algorithms also linear.

On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space

Abstract: We show that the number of critical positions of a convex polygonal objectB moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle isO(kn?s(kn)) for somes≤6, wherek is the number of boundary segments ofB,n is the number of wall segments, and?s(q) is an almost linear function ofq yielding the maximal number of "breakpoints" along the lower envelope (i.e., pointwise minimum) of a set ofq continuous functions each pair of which intersect in at mosts points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is ?(k2n2), showing that in the worst case our upper bound is almost optimal.

Planning Algorithms: Introductory Material

Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

An optimal algorithm for approximate nearest neighbor searching fixed dimensions

Abstract: Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(cd, ϵ log n) time, where cd,ϵ≤d ⌈1 + 6d/ϵ⌉d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

A sweepline algorithm for Voronoi diagrams

Abstract: We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.

Davenport-Schinzel sequences and their geometric applications

Abstract: An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.