Approximating Polygons and Subdivisions with Minimum-Link Paths

International Journal of Computational Geometry & Applications, 1996

An algorithm for approximating a given open polygonal curve with a minimum number of biarcs is introduced. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to prove anything about the quality of these algorithms. Also, in robotics motion planning, smoothing polygonal paths will increase robot performance, because robots usually have difficulties dealing with sharp turns. We present an algorithm which allows us to build a directed graph of all possible biarcs and look for the shortest path from the start point to the end point of the polygonal curve. We can prove a runtime of O(n 2 log n) for an original polygonal chain with n vertices with given tangent directions.

BIT, 1987

We show that it is possible to find a diagonal partition of any n-vertex simple polygon into smaller polygons, each of at most m edges, minimizing the total length of the partitioning diagonals, in time O(n3m2). We derive the same asymptotic upper time-bound for minimum length diagonal partitions of simple polygons into exactly m-gons provided that the input polygon can be partitioned into m-gons. Also, in the latter case, if the input polygon is convex, we can reduce the upper timebound to O(n 3 logm).

Computational Geometry, 2013

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We assume that a simple n-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in O(n 2 ) time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow s words of working space. After a preprocessing of O(n 2 ) time, we are able to solve shortest path queries between any two points inside the polygon in O(n 2 /s) time.

1986

In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n log n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(log n)2)-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed. 9

ArXiv, 2015

Given a convex polygon $P$ with $n$ vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover $P$. We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius $r$ satisfying $r\leq2r_{opt}$ using $O(1)$ space, where $r_{opt}$ is the optimum solution. Next, we show that in non-streaming setup, we can improve the approximation factor by $r\leq 1.84 r_{opt}$, maintaining the time complexity of the algorithm to $O(n)$, and using $O(1)$ extra space in addition to the space required for storing the input.

2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2013

We present the first I/O-and practically-efficient algorithm for simplifying a planar subdivision, such that no point is moved more than a given distance ε xy and such that neighbor relations between faces (homotopy) are preserved. Under some practically realistic assumptions, our algorithm uses O(SORT(N)) I/Os, where N is the size of the decomposition and SORT(N) is the number of I/Os need to sort in the standard externalmemory model of computation. Previously, such an algorithm was only known for the special case of contour map simplification. Our algorithm is simple enough to be of practical interest. In fact, although more general, it is significantly simpler than the previous contour map simplification algorithm. We have implemented our algorithm and present results of experimenting with it on massive reallife data. The experiments confirm that the algorithm is efficient in practice. For example, for the contour map simplification problem it is significantly faster than the previous algorithm, while obtaining approximately the same simplification factor.

Algorithmica, 1992

For compact Euclidean bodies P, Q, we define 2(P, Q) to be the smallest ratio r/s where r > 0, s > 0 satisfy sQ' c p c rQ". Here sQ denotes a scaling of Q by the factor s, and Q', Q" are some translates of Q. This function 2 gives us a new distance function between bodies which, unlike previously studied measures, is invariant under afl]ne transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies are homothetic if one can be obtained from the other by scaling and translation.)

Journal of Algorithms, 1986

SIAM Journal on Computing, 1999

We show that any polygonal subdivision in the plane can be converted into an "mguillotine" subdivision whose length is at most (1 + c m) times that of the original subdivision, for a small constant c. "m-Guillotine" subdivisions have a simple recursive structure that allows one to search for the shortest of such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a simple polynomial-time approximation scheme for geometric instances of several network optimization problems, including the Steiner minimum spanning tree, the traveling salesperson problem (TSP), and the k-MST problem.

Computational Geometry, 2008

An algorithm for approximating a given open polygonal curve with a minimum number of circular arcs is introduced. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to prove anything about the quality of these algorithms. We present an algorithm which allows us to build a directed graph of all possible arcs and look for the shortest path from the start point to the end point of the polygonal curve. We can prove a runtime of O(n 2 log n) for an original polygonal chain with n vertices. Using the same approach, we can prove a runtime of O(n 2 log 2 n), for computing a tangent-continuous approximation with the minimum number of biarcs, for a sequence of points with given tangent directions.