Computational Geometry: Generalized (or Colored) Intersection Searching

Michiel Smid Carleton University, Ottawa 1.1 Geometric intersection searching problems . . . . . . . . 1-1 Generalized intersection searching 1.2 Summary of known results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Axes-parallel objects • Arbitrarily-oriented objects • Problems on the grid • Single-shot problems • External memory and word-RAM algorithms 1.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 A transformation-based approach • A sparsification-based approach • A persistence-based approach • A general approach for reporting problems • Adding range restrictions • Exploiting the output size • A reverse transformation 1.4 Conclusion and future directions . . . . . . . . . . . . . . . . . . . . 1-20 1.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20

[1]  George S. Lueker,et al.  Adding range restriction capability to dynamic data structures , 1985, JACM.

[2]  Jon Louis Bentley,et al.  Decomposable Searching Problems I: Static-to-Dynamic Transformation , 1980, J. Algorithms.

[3]  Panayiotis Bozanis,et al.  New Upper Bounds for Generalized Intersection Searching Problems , 1995, ICALP.

[4]  Benyun Shi,et al.  Approximate colored range and point enclosure queries , 2008, J. Discrete Algorithms.

[5]  Joseph JáJá,et al.  Optimal and near-optimal algorithms for generalized intersection reporting on pointer machines , 2005, Inf. Process. Lett..

[6]  Gonzalo Navarro,et al.  Spaces, Trees, and Colors , 2013, ACM Comput. Surv..

[7]  Torben Hagerup,et al.  Sorting and Searching on the Word RAM , 1998, STACS.

[8]  Leonidas J. Guibas,et al.  Fractional cascading: I. A data structuring technique , 1986, Algorithmica.

[9]  Michiel H. M. Smid,et al.  Efficient Algorithms for Counting and Reporting Pairwise Intersections Between Convex Polygons , 1999, Inf. Process. Lett..

[10]  Panayiotis Bozanis,et al.  New Results on Intersection Query Problems , 1997, Comput. J..

[11]  Jeffrey Scott Vitter,et al.  Categorical range maxima queries , 2014, PODS.

[12]  Kasper Green Larsen,et al.  Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model , 2012, SoCG '12.

[13]  Gonzalo Navarro,et al.  Top-k document retrieval in optimal time and linear space , 2012, SODA.

[14]  S. Muthukrishnan,et al.  Efficient algorithms for document retrieval problems , 2002, SODA '02.

[15]  Pankaj K. Agarwal,et al.  An optimal dynamic interval stabbing-max data structure? , 2005, SODA '05.

[16]  Otfried Cheong.  Ray Shooting in Convex Polytopes , 1992, Symposium on Computational Geometry.

[17]  Michiel H. M. Smid,et al.  A Technique for Adding Range Restrictions to Generalized Searching Problems , 1997, Inf. Process. Lett..

[18]  Yuan Li,et al.  Range search on tuples of points , 2015, J. Discrete Algorithms.

[19]  Jeffrey Scott Vitter,et al.  External memory algorithms and data structures: dealing with massive data , 2001, CSUR.

[20]  Panayiotis Bozanis,et al.  Red-Blue Intersection Reporting for Objects of Non-Constant Size , 1996, Comput. J..

[21]  Kasper Green Larsen,et al.  Near-Optimal Range Reporting Structures for Categorical Data , 2013, SODA.

[22]  Mark de Berg,et al.  Significant-Presence Range Queries in Categorical Data , 2003, WADS.

[23]  Haim Kaplan,et al.  Colored intersection searching via sparse rectangular matrix multiplication , 2006, SCG '06.

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

[25]  Jirí Matousek.  Range searching with efficient hierarchical cuttings , 1992, SCG '92.

[26]  李幼升,et al.  Ph , 1989 .

[27]  Proceedings of the ACM Symposium on Principles of Database Systems, March 29-31, 1982, Los Angeles, California, USA , 1982 .

[28]  Jirí Matousek,et al.  Cutting hyperplane arrangements , 1990, SCG '90.

[29]  Yufei Tao.  Stabbing horizontal segments with vertical rays , 2012, SoCG '12.

[30]  Yakov Nekrich.  Efficient range searching for categorical and plain data , 2014, TODS.

[31]  Sunil Prabhakar,et al.  Indexing Uncertain Data , 2018, Encyclopedia of Database Systems.

[32]  Michiel H. M. Smid,et al.  Algorithms for Generalized Halfspace Range Searching and Other Intersection Searching Problems , 1995, Comput. Geom..

[33]  Yakov Nekrich.  Space-efficient range reporting for categorical data , 2012, PODS '12.

[34]  Siu-Wing Cheng,et al.  Efficient Dynamic Algorithms for Some Geometric Intersection Problems , 1990, Inf. Process. Lett..

[35]  Michiel Smid,et al.  Algorithms for Some Intersection Searching Problems InvolvingCircular Objects , 1998 .

[36]  Leonidas J. Guibas,et al.  The power of geometric duality , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[37]  Bernard Chazelle,et al.  Filtering search: A new approach to query-answering , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[38]  Pankaj K. Agarwal,et al.  Range Searching in Categorical Data: Colored Range Searching on Grid , 2002, ESA.

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

[40]  Michiel H. M. Smid,et al.  Further Results on Generalized Intersection Searching Problems: Counting, Reporting, and Dynamization , 1995, J. Algorithms.

[41]  Amr Elmasry,et al.  Dynamic Range Majority Data Structures , 2011, ISAAC.

[42]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[43]  Timothy M. Chan,et al.  Optimal halfspace range reporting in three dimensions , 2009, SODA.

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

[45]  Micha Sharir,et al.  Reporting intersecting pairs of convex polytopes in two and three dimensions , 2002, Comput. Geom..

[46]  Edward M. McCreight,et al.  Priority Search Trees , 1985, SIAM J. Comput..

[47]  Alok Aggarwal,et al.  The input/output complexity of sorting and related problems , 1988, CACM.

[48]  Haim Kaplan,et al.  Counting colors in boxes , 2007, SODA '07.

[49]  Divesh Srivastava,et al.  Two-dimensional substring indexing , 2001, J. Comput. Syst. Sci..

[50]  Saladi Rahul,et al.  Improved Bounds for Orthogonal Point Enclosure Query and Point Location in Orthogonal Subdivisions in ℝ3 , 2015, SODA.

[51]  Mario A. López,et al.  Generalized intersection searching problems , 1993, Int. J. Comput. Geom. Appl..

[52]  Jirí Matousek,et al.  Efficient partition trees , 1991, SCG '91.

[53]  Siu-Wing Cheng,et al.  Algorithms for Ray-Shooting and Intersection Searching , 1992, J. Algorithms.