ABSTRACT
A set A of n points in the plane has to be stored in such a way that for any query triangle t the number of points of A inside t can be computed efficiently. For this problem a solution is presented with Ο(√n log n) query time, Ο (n log n) space and Ο(n3/2 log n) preprocessing time. The constants in the asymptotic bounds are small, and the method is easy to implement.
- Br.K.Q.Brown, Geometric Transforms for fast Geometric Algorithms, PhD thesis, Report CMU-CS-80-101, Carnegie-Mellon University (1980). Google ScholarDigital Library
- CGL.B.Chazelle, L.J.Guibas & D.T.Lee, The power of geometric duality, in "Proc. 25th IEEE Symp. on Foundations of Computer Science" (1983) 83-91.Google Scholar
- CW.B.Chazelle & E.Welzl, Range searching and VC-dimension: a characterization of efficiency, Discrete Comput. Geom. (1989), to appear.Google Scholar
- CSSS.R.Cole, J.Salowe, W.L.Steiger & E.Szeme&di, Optimal slope selection, manuscript.Google Scholar
- E.H.Edelsbrunner, Algorithms in corn&n&orial geometry, Springer Verlag (1987) Berlin. Google ScholarDigital Library
- EGHSSSW.H.Edelsbrunner, J.Hershberger, L.J.Guibas, R.Seidel, M.Sharir, J.Snoeyink & E.Welzl, Implicitly representing arrangements of lines or segments, in "Proc. 4th Ann. ACM Symp. Comput. Geom." (1988) 56-69. Google ScholarDigital Library
- EKM.H.Edelsbrunner, D.G. Kirkpatrick & H.A. Maurer, Polygon intersection searching, Inform. Process. Lett. 14 (1982) 74-79.Google ScholarCross Ref
- EW.H.Edelsbrunner & E. Welzl, Halfplanar range search in linear space and O(n0*6s5) query time, Inform. Process. Lett. 23 (1986) 289-293. Google ScholarDigital Library
- ES.P.Erd~s & G.Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935) 463-470.Google Scholar
- HW.D.Haussler & E.Welzl, Epsilon-nets and simplex range queries, Disc&e Comput. Geom. 2 (1987) 237-256.Google Scholar
- M1.J.MatouSek, Line arrangements and range search, Inform. Process. Lett. 27 (1988) 275- 280. Google ScholarDigital Library
- M2.J.MatouSek, Spanning trees with low stabbing number, manuscript.Google Scholar
- Wi.D.E.Willard, Polygon retrieval, SIAM J. Comput. 11 (1982) 149-165.Google ScholarCross Ref
Index Terms
- Good splitters for counting points in triangles
Recommendations
Threesomes, Degenerates, and Love Triangles
Distributed Computing, Cryptography, Distributed Computing, Cryptography, Coding Theory, Automata Theory, Complexity Theory, Programming Languages, Algorithms, Invited Paper Foreword and DatabasesThe 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial O(n2)-time algorithm is optimal on the Real RAM, and optimal even in the nonuniform linear decision tree model. Over the ...
Approximately Counting Triangles in Sublinear Time
FOCS '15: Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sub linear-time algorithm for ...
Adaptive and Approximate Orthogonal Range Counting
We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model.
—It is well known that there are linear-space data structures for 2-D orthogonal ...
Comments