On the arrangement of stochastic lines in ℤ2

Abstract Consider a set of n stochastic lines in R 2 , where the existence probability of each line is determined by a fixed probability distribution. For a fixed x-coordinate q, the n lines from top to bottom can be represented by an ordered n-element sequence. Consider all the ( n k ) k-element sub-sequences of that n-element sequence. Each k-element sub-sequence has an associated likelihood to be the true k-topmost lines at x-coordinate q, and the one with the largest probability is defined as the most likely k-topmost lines at q. This paper studies the most likely k-topmost lines of the arrangement of n lines taken over all the x-coordinates. Let cnt be the total number of distinct sequences of the most likely k-topmost lines over all x-coordinates. The main result established is that the expected value of cnt is O ( k n ) , which implies that it is possible to store all the distinct most likely k-topmost lines in O ( k 2 n ) expected space. An example is given showing that cnt, in the worst case, can be Θ ( n 2 ) even when k = 1 . This highlights the value of the expected bound. An algorithm is also given to compute the most likely k-topmost lines of the arrangement. Applications of this result to the stochastic Voronoi Diagram in R 1 and to the stochastic preference top-k query in R 2 are discussed.

[1]  Géza Tóth.  Point Sets with Many k-Sets , 2001, Discret. Comput. Geom..

[2]  Noga Alon,et al.  The number of small semispaces of a finite set of points in the plane , 1986, J. Comb. Theory, Ser. A.

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

[4]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

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

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

[7]  Timothy M. Chan.  On Levels in Arrangements of Surfaces in Three Dimensions , 2005, SODA '05.

[8]  Ketan Mulmuley,et al.  On levels in arrangements and voronoi diagrams , 1991, Discret. Comput. Geom..

[9]  Robert E. Tarjan,et al.  Making data structures persistent , 1986, STOC '86.

[10]  Maarten Löffler,et al.  Largest and Smallest Convex Hulls for Imprecise Points , 2010, Algorithmica.

[11]  Pankaj K. Agarwal,et al.  Geometric Range Searching and Its Relatives , 2007 .

[12]  P. Erdös,et al.  Dissection Graphs of Planar Point Sets , 1973 .

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

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

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

[16]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[17]  Micha Sharir,et al.  On Levels in Arrangements of Lines, Segments, Planes, and Triangles% , 1998, Discret. Comput. Geom..

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

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

[20]  Micha Sharir,et al.  On K-Sets in Arrangements of Curves and Surfaces , 2018 .

[21]  Timothy M. Chan,et al.  Closest pair and the post office problem for stochastic points , 2011, Comput. Geom..

[22]  Bernard Chazelle,et al.  The power of geometric duality , 1985, BIT Comput. Sci. Sect..

[23]  Yufei Tao,et al.  Indexing uncertain data , 2009, PODS.

[24]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[25]  William S. Evans,et al.  Guaranteed Voronoi Diagrams of Uncertain Sites , 2008, CCCG.

[26]  Maarten Löffler,et al.  Data Imprecision in Computational Geometry , 2009 .

[27]  Ihab F. Ilyas,et al.  A survey of top-k query processing techniques in relational database systems , 2008, CSUR.

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