Containment and Evasion in Stochastic Point Data

Given two disjoint and finite point sets \(A\) and \(B\) in \(\mathrm{I\! R}^d\), we say that \(B\) is contained in \(A\) if all the points of \(B\) lie within the convex hull of \(A\), and that \(B\) evades \(A\) if no point of \(B\) lies inside the convex hull of \(A\). We investigate the containment and evasion problems of this type when the set \(A\) is stochastic, meaning each of its points \(a_i\) is present with an independent probability \(\pi (a_i)\). Our model is motivated by situations in which there is uncertainty about the set \(A\), for instance, due to randomized strategy of an adversarial agent or scheduling of monitoring sensors. Our main results include the following: (1) we can compute the exact probability of containment or evasion in two dimensions in worst-case \(O(n^4 + m^2)\) time and \(O(n^2 + m^2)\) space, where \(n = | A | \) and \(m = | B | \), and (2) we prove that these problems are #P-hard in 3 or higher dimensions.

[1]  Jeff M. Phillips,et al.  Range counting coresets for uncertain data , 2013, SoCG '13.

[2]  Timothy M. Chan,et al.  Closest Pair and the Post Office Problem for Stochastic Points , 2011, WADS.

[3]  Kenneth L. Clarkson,et al.  Linear Programming in O(n * (3_d)_2) Time , 1986, Inf. Process. Lett..

[4]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[5]  Subhash Suri,et al.  Hyperplane Separability and Convexity of Probabilistic Point Sets , 2016, Symposium on Computational Geometry.

[6]  Sandip Das,et al.  Simplex Range Searching and k Nearest Neighbors of a Line Segment in 2D , 2002, SWAT.

[7]  Timothy M. Chan,et al.  Stochastic minimum spanning trees in euclidean spaces , 2011, SoCG '11.

[8]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[9]  Christopher Ré,et al.  Probabilistic databases: diamonds in the dirt , 2009, CACM.

[10]  Victor J. Milenkovic,et al.  Translational polygon containment and minimal enclosure using linear programming based restriction , 1996, STOC '96.

[11]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[12]  Michael T. Goodrich,et al.  On the Complexity of Optimization Problems for 3-dimensional Convex Polyhedra and Decision Trees , 1997, Comput. Geom..

[13]  Pankaj K. Agarwal,et al.  Convex Hulls Under Uncertainty , 2016, Algorithmica.

[14]  David Eppstein,et al.  Finding minimum areak-gons , 1992, Discret. Comput. Geom..

[15]  Sudipto Guha,et al.  Adaptive Uncertainty Resolution in Bayesian Combinatorial Optimization Problems , 2008, ACM Trans. Algorithms.

[16]  Alejandro C. Frery,et al.  Partially Supervised Anomaly Detection Using Convex Hulls on a 2D Parameter Space , 2013, PSL.

[17]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[18]  Pankaj K. Agarwal,et al.  Nearest-neighbor searching under uncertainty , 2012, PODS.

[19]  Philip S. Yu,et al.  A Survey of Uncertain Data Algorithms and Applications , 2009, IEEE Transactions on Knowledge and Data Engineering.

[20]  Maarten Löffler,et al.  Geometric Computations on Indecisive and Uncertain Points , 2012, ArXiv.

[21]  Charu C. Aggarwal,et al.  Managing and Mining Uncertain Data , 2009, Advances in Database Systems.

[22]  Subhash Suri,et al.  Hyperplane separability and convexity of probabilistic point sets , 2017, J. Comput. Geom..

[23]  Jennifer Widom,et al.  Representing uncertain data: models, properties, and algorithms , 2009, The VLDB Journal.

[24]  Subhash Suri,et al.  On the Most Likely Convex Hull of Uncertain Points , 2013, ESA.

[25]  Geoffrey A. Hollinger,et al.  Search and pursuit-evasion in mobile robotics , 2011, Auton. Robots.

[26]  Subhash Suri,et al.  On the Most Likely Voronoi Diagram and Nearest Neighbor Searching , 2016, Int. J. Comput. Geom. Appl..

[27]  Subhash Suri,et al.  On the Most Likely Voronoi Diagramand Nearest Neighbor Searching , 2014, ISAAC.

[28]  Pankaj K. Agarwal,et al.  Range searching on uncertain data , 2012, TALG.