20 research outputs found
Drawing graphs with vertices and edges in convex position
A graph has strong convex dimension , if it admits a straight-line drawing
in the plane such that its vertices are in convex position and the midpoints of
its edges are also in convex position. Halman, Onn, and Rothblum conjectured
that graphs of strong convex dimension are planar and therefore have at
most edges. We prove that all such graphs have at most edges
while on the other hand we present a class of non-planar graphs of strong
convex dimension . We also give lower bounds on the maximum number of edges
a graph of strong convex dimension can have and discuss variants of this
graph class. We apply our results to questions about large convexly independent
sets in Minkowski sums of planar point sets, that have been of interest in
recent years.Comment: 15 pages, 12 figures, improved expositio
Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems inComputational Biology Efficiently
In this paper, we introduce the notion of a constrained Minkowski sum: for two (finite) point-sets P,Q⊆ℝ2 and a set of k inequalities Ax≥b, it is defined as the point-set (P ⊕ Q) Ax≥b ={x=p+q∣p∈P,q∈Q,Ax≥b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(Nlog N), where N=|P|+|Q| if k is fixed. For the special case where P and Q consist of points with integer x 1-coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solve
Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently
In this paper, we introduce the notion of a constrained Minkowski sum: for two (finite) point-sets P, Q subset of R-2 and a set of k inequalities Ax >= b, it is defined as the point-set (P circle plus Q)(Ax >= b) = {x = p + q vertical bar p is an element of P, q is an element of Q, Ax >= b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O (N log N), where N = vertical bar P vertical bar + vertical bar Q vertical bar if k is fixed. For the special case (P circle plus Q)(x1 >=beta) where P and Q consist of points with integer x(1)-coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solved
Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls
A real valued function defined on a convex is anemconvex function iff
it satisfies A thorough study of
approximately convex functions is made. The principal results are a sharp
universal upper bound for lower semi-continuous approximately convex functions
that vanish on the vertices of a simplex and an explicit description of the
unique largest bounded approximately convex function~ vanishing on the
vertices of a simplex.
A set in a normed space is an approximately convex set iff for all
the distance of the midpoint to is . The bounds
on approximately convex functions are used to show that in with the
Euclidean norm, for any approximately convex set , any point of the
convex hull of is at a distance of at most
from . Examples are given to show
this is the sharp bound. Bounds for general norms on are also given.Comment: 39 pages. See also http://www.math.sc.edu/~howard
Sampling Based Motion Planning with Reachable Volumes
Motion planning for constrained systems is a version of the motion planning problem in which the motion of a robot is limited by constraints. For example, one can require that a humanoid robot such as a PR2 remain upright by constraining its torso to be above its base or require that an object such as a bucket of water remain upright by constraining the vertices of the object to be parallel to the robot’s base. Grasping can be modeled by requiring that the end effectors of the robot be located at specified handle positions. Constraints might require that the robot remain in contact with a surface, or that certain joints of the robot remain in contact with each other (e.g., closed chains). Such problems are particularly difficult because the constraints form a manifold in C-space, and planning must be restricted to this manifold. High degree of freedom motion planning and motion planning for constrained systems has applications in parallel robotics, grasping and manipulation, computational biology and molecular simulations, and animation.
In this work, we introduce a new concept, reachable volumes, that are a geometric representation of the regions the joints and end effectors of a robot can reach, and use it to define a new planning space, called RV-space, where all points automatically satisfy a problem’s constraints. Visualizations of reachable volumes can enable operators to see the regions of workspace that different parts of the robot can reach. Samples and paths generated in RV-space naturally conform to constraints, making planning for constrained systems no more difficult than planning for unconstrained systems. Consequently, constrained motion planning problems that were previously difficult or unsolvable become manageable and in many cases trivial.
We provide tools and techniques to extend the state of the art sampling based motion planning algorithms to RV-space. We define a reachable volume sampler, a reachable volume local planner and a reachable volume distance metric. We showcase the effectiveness of RV-space by applying these tools to motion planning problems for robots with constraints on the end effectors and/or internal joints of the robot. We show that RV-based planners are more efficient than existing methods, particularly for higher dimensional problems, solving problems with 1000+ degrees of freedom for multi-loop, and tree-like linkages