Terrain Guarding Problem(TGP), which is known to be NP-complete, asks to find a smallest set of guard locations on a terrain T such that every point on T is visible by a guard. Here, we study this problem on 1.5D orthogonal terrains where the edges are bound to be horizontal or vertical. We propose a 2-approximation algorithm that runs in O(nlog m) time, where n and m are the sizes of input and output, respectively. This is an improvement over the previous best algorithm, which is a 2-approximation with O(n 2 ) running time.
For orthogonal terrains, Katz and Roisman @cite_9 gave a 2-approximation algorithm that runs in O( @math ) time, by computing a minimum clique cover in chordal graphs. Recently, Durocher . @cite_5 studied the orthogonal terrain guarding problem under where two vertices @math are considered to see each other only if the interior of the segment @math is strictly above the terrain. Under this restricted definition, no reflex vertex of the input terrain @math can see convex vertices both on its left and right side. This property simplifies the problem, and leads to a linear time greedy exact algorithm. Under standard visibility, Durocher . @cite_5 also observed that the hardness result for TGP in @cite_6 does not apply for orthogonal terrains, leaving the complexity of OTGP open.
@cite_5 @cite_9 @cite_6
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A 1.5-dimensional terrain T with n vertices is an xmonotone polygonal chain in the plane. A point guard p on T guards a point q of T if the line segment connecting p to q lies on or above T ; p is a vertex guard if it is a vertex of T . In the Optimal Terrain Guarding (OTG) problem on T , the objective is to guard the vertices of T by the minimum number of vertex guards. King and Krohn [9] showed that the OTG problem is NP-hard on arbitrary terrains, and [6] gave a PTAS for this problem. In this paper, we introduce directed visibility in which the visibility is directed only at adjacent vertices. We give an O(n)-time algorithm that solves the OTG problem exactly on orthogonal terrains under directed visibility. A set @math of points on a terrain, also known as an @math -monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in @math . Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. In the continuous 1.5-dimensional terrain guarding problem we are given an x-monotone chain (the terrain T ) and ask for the minimum number of point guards (located anywhere on T ), such that all points of T are covered by at least one guard. It has been shown that the 1.5-dimensional terrain guarding problem is NPhard. The currently best known approximation algorithm achieves a factor of 4. For the discrete problem version with a nite set of guard candidates and a nite set of points on the terrain that need to be monitored, a polynomial time approximation scheme (PTAS) has been presented [11]. We show that for the general problem we can construct nite guard and witness sets, G and W , such that there exists an optimal guard cover G G that covers T , and when these guards monitor all points in W the entire terrain is guarded. This leads to a PTAS as well as an (exact) IP formulation for the continuous terrain guarding problem. We present the first constant-factor approximation algorithm for a nontrivial instance of the optimal guarding (coverage) problem in polygons. In particular, we give an @math -approximation algorithm for placing the fewest point guards on a 1.5D terrain, so that every point of the terrain is seen by at least one guard. While polylogarithmic-factor approximations follow from set cover results, our new results exploit the geometric structure of terrains to obtain a substantially improved approximation algorithm. A mobile robot represented by a point moving in the plane has to explore an unknown flat terrain with impassable obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point p of the terrain explores (sees) all points q of the terrain for which the segment pq belongs to the terrain, and the limited vision, when we require additionally that the distance between p and q is at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm, called its complexity, is measured by the length of the trajectory of the robot. For unlimited vision we show an exploration algorithm with complexity O(P+Dk), where P is the total perimeter of the terrain (including perimeters of obstacles), D is the diameter of the convex hull of the terrain, and k is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity O(P+A+Ak), where A is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains (if one of the parameters A or k is known) or for c-fat terrains, where c is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain T with obstacles is c-fat if R r= We prove that guarding the vertices of a rectilinear polygon P, whether by guards lying at vertices of P, or by guards lying on the boundary of P, or by guards lying anywhere in P, is NP-hard. For the first two proofs (i.e., vertex guards and boundary guards), we construct a reduction from minimum piercing of 2-intervals. The third proof is somewhat simpler; it is obtained by adapting a known reduction from minimum line cover. We also consider the problem of guarding the vertices of a 1.5D rectilinear terrain. We establish an interesting connection between this problem and the problem of computing a minimum clique cover in chordal graphs. This connection yields a 2-approximation algorithm for the guarding problem. We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled [2] and Mustafa and Ray [15]. Our key contribution is to show the existence of a planar graph that appropriately relates the local and global optimum. We give a nearly optimal sublinear-time algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the ith neighbor of v. Letting VCopt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate [EQUATION] such that [EQUATION], where e is a given additive approximation parameter. We refer to such an estimate as a (2, e)-estimate. The query complexity and running time of the algorithm are O([EQUATION] · poly(1 e)), where d denotes the average vertex degree in the graph. The best previously known sublinear algorithm, of (STOC 2009), has query complexity and running time O(d4 e2), where d is the maximum degree in the graph. Given the lower bound of Ω(d) (for constant e) for obtaining such an estimate (with any constant multiplicative factor) due to Parnas and Ron (TCS 2007), our result is nearly optimal. In the case that the graph is dense, that is, the number of edges is Θ(n2), we consider another model, in which the algorithm may ask, for any pair of vertices u and v, whether there is an edge between u and v. We show how to adapt the algorithm that uses neighbor queries to this model and obtain an algorithm that outputs a (2, e)-estimate of the size of a minimum vertex cover whose query complexity and running time are O(n) · poly(1 e). Given n points in a circular region C in the plane, we study the problem of moving these points to the boundary of C to form a regular n-gon such that the maximum of the Euclidean distances traveled by the points is minimized. These problems find applications in mobile sensor barrier coverage of wireless sensor networks. The problem further has two versions: the decision version and optimization version. In this paper, we present an O(nlog2 n) time algorithm for the decision version and an O(nlog3 n) time algorithm for the optimization version. The previously best algorithms for these two problem versions take O(n 3.5) time and O(n 3.5logn) time, respectively. A by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm performs each vertex insertion or deletion on the graph in O(log2 n) time. This result may be interesting in its own right. Given n points in a circular region C in the plane, we study the problems of moving the n points to the boundary of G to form a regular n-gon such that the maximum (min-max) or the sum (min-sum) of the Euclidean distances traveled by the points is minimized. These problems have applications, e.g., in mobile sensor barrier coverage of wireless sensor networks. The min-max problem further has two versions: the decision version and the optimization version. For the min-max problem, we present an O(nlog2n) time algorithm for the decision version and an O(nlog3n) time algorithm for the optimization version. The previously best algorithms for the two problem versions take O(n3.5) time and O(n3.5logn) time, respectively. For the min-sum problem we show that a special case with all points initially lying on the boundary of the circular region can be solved in O(n2) time, improving a previous O(n4) time solution. For the general min-sum problem, we present a 3-approximation O(n2) time algorithm. In addition, a by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm can handle each vertex insertion or deletion on the graph in O(log2n) time. This result may be interesting in its own right. In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved. We provide an O(log log opt)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building e-nets of size @math for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this e-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon’s vertices. If a finite set of potential guard locations is not specified, e.g., when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm’s running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log opt)-approximation algorithms that use generic net finders for set systems of finite VC-dimension. We present an O(nlogn)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G=(V,E) with nonnegative edge lengths d:E->R^>=^0 and a stretchk>=1, a subgraph H=(V,E"H) is a k-spanner of G if for every edge (s,t)@?E, the graph H contains a path from s to t of length at most [email protected]?d(s,t). The previous best approximation ratio was [email protected]?(n^2^ ^3), due to Dinitz and Krauthgamer (STOC @?11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths from [email protected]?(n) to O(n^1^ ^3logn). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS @?10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and [email protected]?s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n^1^ ^3logn)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n^2^ ^3^+^@e) for any constant @e>0, which improves the O(n^@[email protected]?min(n^4^ ^5,m^2^ ^3)) ratio due to Feldman, Kortsarz and Nutov ([email protected]?12). Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT 2). We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . Two mobile agents, modeled as points starting at different locations of an unknown terrain, have to meet. The terrain is a polygon with polygonal holes. We consider two versions of this rendezvous problem: exact RV, when the points representing the agents have to coincide at some time, and ?-RV, when these points have to get at distance less than ? in the terrain. In any terrain, each agent chooses its trajectory, but the movements of the agent on this trajectory are controlled by an adversary that may, e.g., speed up or slow down the agent. Agents have bounded memory: their computational power is that of finite state machines. Our aim is to compare the feasibility of exact and of ?-RV when agents are anonymous vs. when they are labeled. We show classes of polygonal terrains which distinguish all the studied scenarios from the point of view of feasibility of rendezvous. The features which influence the feasibility of rendezvous include symmetries present in the terrains, boundedness of their diameter, and the number of vertices of polygons in the terrains. We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an @math -sided polygon in @math time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm, we prove that the LMAPs interleave each other, thus the number of LMAPs is bounded by @math . We discuss applications of our result to, among others, the problem of computing the maximum area centrally-symmetric convex body (MAC) inside a convex polygon, and the simplest case of the Heilbronn Triangle Problem. In this paper we consider a fundamental problem in the area of viral marketing, called T ARGET S ET S ELECTION problem. In a a viral marketing setting, social networks are modeled by graphs with potential customers of a new product as vertices and friend relationships as edges, where each vertex @math is assigned a threshold value @math . The thresholds represent the different latent tendencies of customers (vertices) to buy the new product when their friend (neighbors) do. Consider a repetitive process on social network @math where each vertex @math is associated with two states, active and inactive, which indicate whether @math is persuaded into buying the new product. Suppose we are given a target set @math . Initially, all vertices in @math are inactive. At time step 0, we choose all vertices in @math to become active. Then, at every time step @math , all vertices that were active in time step @math remain active, and we activate any vertex @math if at least @math of its neighbors were active at time step @math . The activation process terminates when no more vertices can get activated. We are interested in the following optimization problem, called T ARGET S ET S ELECTION : Finding a target set @math of smallest possible size that activates all vertices of @math . There is an important and well-studied threshold called strict majority threshold, where for every vertex @math in @math we have @math and @math is the degree of @math in @math . In this paper, we consider the T ARGET S ET S ELECTION problem under strict majority thresholds and focus on three popular regular network structures: cycle permutation graphs, generalized Petersen graphs and torus cordalis. We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs.