Omrit Filtser

A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

Consider a sliding camera that travels back and forth along an orthogonal line segment s inside an orthogonal polygon P with n vertices. The camera can see a point p inside P if and only if there exists a line segment containing p that crosses s at a right angle and is completely contained in P. In the minimum sliding cameras (MSC) problem, the objective is to guard P with the minimum number of sliding cameras. In this paper, we give an O(n 5/2)-time (7/2)-approximation algorithm to the MSC problem on any simple orthogonal polygon with n vertices, answering a question posed by Katz and Morgenstern (2011). To the best of our knowledge, this is the first constant-factor approximation algorithm for this problem.

The Discrete and Semicontinuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection

The <i>Fréchet distance</i> is a well-studied similarity measure between curves. The <i>discrete Fréchet distance</i> is an analogous similarity measure, defined for two sequences of <i>m</i> and <i>n</i> points, where the points are usually sampled from input curves. We consider a variant, called the <i>discrete Fréchet distance with shortcuts</i>, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in <i>O</i>((<i>m</i> + <i>n</i>)^{6/5 + ϵ}) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an <i>O</i>((<i>m</i>^2/3<i>n</i>^2/3 + <i>m</i> + <i>n</i>)log ³(<i>m</i> + <i>n</i>))-time deterministic algorithm. We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of <i>m</i> points and a polygonal curve of <i>n</i> edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time <i>O</i>((<i>m</i> + <i>n</i>)^2/3<i>m</i>^2/3<i>n</i>^1/3log (<i>m</i> + <i>n</i>)). Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points <i>A</i> and <i>B</i> in the plane and an interval <i>I</i>, we develop an algorithm that decides whether the number of pairs (<i>x</i>, <i>y</i>) ∈ <i>A</i> × <i>B</i> whose distance dist(<i>x</i>, <i>y</i>) is in <i>I</i> is less than some given threshold <i>L</i>. The running time of this algorithm decreases as <i>L</i> increases. In case there are more than <i>L</i> pairs of points whose distance is in <i>I</i>, we can get a small sample of pairs that contain a pair at approximate median distance (i.e., we can approximately “bisect” <i>I</i>). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.

Guarding orthogonal art galleries with sliding cameras

Abstract Let P be an orthogonal polygon with n vertices. A sliding camera travels back and forth along an orthogonal line segment s ⊆ P corresponding to its trajectory. The camera sees a point p ∈ P if there is a point q ∈ s such that p q ‾ is a line segment normal to s that is completely contained in P. In the Minimum-Cardinality Sliding Cameras (MCSC) problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S), while in the Minimum-Length Sliding Cameras (MLSC) problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we answer questions posed by Katz and Morgenstern (2011) by presenting the following results: (i) the MLSC problem is polynomially tractable even for orthogonal polygons with holes, (ii) the MCSC problem is NP -complete when P is allowed to have holes, and (iii) an O ( n 3 log ⁡ n ) -time 2-approximation algorithm for the MCSC problem on [NE]-star-shaped orthogonal polygons with n vertices (similarly, [NW]-, [SE]-, or [SW]-star-shaped orthogonal polygons).

Algorithms for the Discrete Fréchet Distance Under Translation

The (discrete) Frechet distance (DFD) is a popular similarity measure for curves. Often the input curves are not aligned, so one of them must undergo some transformation for the distance computation to be meaningful. Ben Avraham et al. [Rinat Ben Avraham et al., 2015] presented an O(m^3n^2(1+log(n/m))log(m+n))-time algorithm for DFD between two sequences of points of sizes m and n in the plane under translation. In this paper we consider two variants of DFD, both under translation. For DFD with shortcuts in the plane, we present an O(m^2n^2 log^2(m+n))-time algorithm, by presenting a dynamic data structure for reachability queries in the underlying directed graph. In 1D, we show how to avoid the use of parametric search and remove a logarithmic factor from the running time of (the 1D versions of) these algorithms and of an algorithm for the weak discrete Frechet distance; the resulting running times are thus O(m^2n(1+log(n/m))), for the discrete Frechet distance, and O(mn log(m+n)), for its two variants. Our 1D algorithms follow a general scheme introduced by Martello et al. [Martello et al., 1984] for the Balanced Optimization Problem (BOP), which is especially useful when an efficient dynamic version of the feasibility decider is available. We present an alternative scheme for BOP, whose advantage is that it yields efficient algorithms quite easily, without having to devise a specially tailored dynamic version of the feasibility decider. We demonstrate our scheme on the most uniform path problem (significantly improving the known bound), and observe that the weak DFD under translation in 1D is a special case of it.

Universal approximate simplification under the discrete Fréchet distance

Abstract The problem of simplifying a polygonal curve or chain is well studied and has many applications. The discrete Frechet distance is a useful similarity measure for curves, which has been utilized for many real-world applications. When the curves are huge, a simplification algorithm is needed in order to reduce running times. In this paper we adapt some of the techniques of Driemel and Har-Peled [5] (for the continuous Frechet distance) to obtain a universal approximate simplification of a given polygonal curve, under the discrete Frechet distance.