153 research outputs found
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
Shortest Path Problems on a Polyhedral Surface
We develop algorithms to compute shortest path edge sequences, Voronoi diagrams, the Fréchet distance, and the diameter for a polyhedral surface
Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle Union
We revisit the classical problem of determining the largest copy of a simple polygon that can be placed into a simple polygon . Despite significant effort, known algorithms require high polynomial running times. (Barequet and Har-Peled, 2001) give a lower bound of under the 3SUM conjecture when and are (convex) polygons with vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized or . In this paper, we affirmatively answer these questions under the SUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, -translation, -translation, rotation): (1) Finding the largest copy of that can be -translated into requires time under the 3SUM conjecture. (2) Finding the largest copy of that can be arbitrarily translated into requires time under the 4SUM conjecture. (3) The above lower bounds are almost tight when one of the polygons is of constant size: we obtain an -time algorithm for orthogonal polygons with and vertices, respectively. (4) Finding the largest copy of that can be arbitrarily rotated and translated into requires time under the 5SUM conjecture. We are not aware of any other such natural degree of freedom -SUM hardness for a geometric optimization problem
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