Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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A Parallel Algorithm for Constructing the Delaunay Triangulation in the$L_\infty(L_1)$ Metric

$L_\infty(L_1)$디루니 삼각분할의 병렬처리 알고리즘

  • 위영철 (아주대학교 정보 및 컴퓨터공학부)
  • Published : 2001.04.01
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Abstract

본 논문은 영역별 근접 그래프 (geographic nearest neighbor graph)와 레인지 트리 (range tree)를 이용하여 평면 위의 n 개의 점에 대한 L$_{\infty}$ (L$_1$) 거리 (metric) 상의 디루니 삼각분할 (Delaunay triangulation)을 구축하는 방법을 소개한다. 이 방법은 L$_{\infty}$ (L$_1$) 거리 상에서 디루니 삼각분할에 있는 각 삼각형의 최소한 한 선분이 영역별 근접 그래프에 포함됨을 이용하여 레인지 트리 방법으로 디루니 삼각분할을 구축한다. 본 방법은 0(nlogn)의 순차계산 시간에 L$_{\infty}$ (L$_1$) 디루니 삼각분할을 구축하며, CREW-PRAM (Concurrent Read Exclusive Write Parallel Random Access Machine)에서 0(n)의 프로세서로 0(logn)의 병렬처리 시간에 L$_{\infty}$ (L$_1$) 디루니 삼각분할을 구축한다. 또한, 이 방법은 직선간의 교차점 계산 대신 거리비교를 하기 때문에 수치오차가 적고 구현이 용이하다.

Keywords

References

  1. M.J. Atallah, R. Cole, and M.T. Goodrich, Divide-and-Conquer: A Technique for Designing Parallel Algorithms, 28th IEEE Symp. on Foundations of Computer Science, Tech. Rep. 665, Dept. of Comp. Sci., Purdue Univ., 151-160, 1987
  2. A. Aggarwal, B. Chazelle, L. Guibas, C. O'Dunlaing and C. Yap, Parallel Comutational Geometry, Algorithmica, 293-327, 1988 https://doi.org/10.1007/BF01762120
  3. R. Cole, M.T. Goodrich, Optimal Parallel Algorithms for Polygon and Point-Set Problems, Proc. of the Fourth Ann. Symp. on Computational Geometry, 201-210, 1988 https://doi.org/10.1145/73393.73414
  4. K. Clarkson, Approximation Problems for Shortest Path Motion Planning, Proc. 19th Ann. ACM Symp. Theory of Computing, 56-65, 1987
  5. H.N. Gabow, J.L. Bentley and R. E. Tarjan, Scaling and Related Techniques for Geometry Problems, Proc. 16st Ann Symp. Theory of Computing, 135-143, 1984 https://doi.org/10.1145/800057.808675
  6. L.J. Guibas and J. Stolfi, On Computing all North-east Nearest Neighbors in the $L_1$ Metric, Info. Process. Lett. 17, 219-223, 1983 https://doi.org/10.1016/0020-0190(83)90045-5
  7. J. Katajainen, The Region Approach for Computing Relative Neighborhood Graphs in the $L_p$ Metric, Computing 40, 147-161, 1988 https://doi.org/10.1007/BF02247943
  8. D.T. Lee, Two-dimensional Voronoi Diagrams in the $L_p$-Metric, J.ACM 27(4), 604-618, 1980 https://doi.org/10.1145/322217.322219
  9. D.T. Lee and C. K. Wong, Voronoi Diagrams in $L_1(L_{\infty})$ Metrics with 2-Dimensional Storage Applications, SIAM J. Comput. 9(1), 200-211, 1980 https://doi.org/10.1137/0209017
  10. E. Papadopoulou and D.T. Lee, Critical Area Computation via Voronoi Diagrams, IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, 18(4), 463-474, 1999 https://doi.org/10.1109/43.752929
  11. W. Preilowski and W. Mumbeck, A Time-Optimal parallel Algorithm for the Computing of Voronoi diagrams, Lecture Notes in Computer Science, Springer Verlag, 424-433, 1988 https://doi.org/10.1007/3-540-50728-0_60
  12. P. Su, R.L. Drysdale, A Comparison of Sequential Delaunay Triangulation Algorithm, Computational Geometry 7, 361-385, 1997 https://doi.org/10.1016/S0925-7721(96)00025-9
  13. K.J. Supowit, The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees, J.ACM 30(3), 428-448, 1983 https://doi.org/10.1145/2402.322386
  14. Y.C. Wee and S. Chaiken and S. S. Ravi, Rectilinear Steiner Tree Heuristics and Minimum Spanning Tree Algorithms Using Geographic Nearest Neighbors, Algorithmica 12, 421-435, 1994 https://doi.org/10.1007/BF01188713
  15. D.E. Willard and Y. C. Wee, Quasi-valid Range Querying and its Implications for Nearest Neighbor Problems, Proceedings of the Fourth Annual Symp. on Computational Geometry, 34-43, 1988 https://doi.org/10.1145/73393.73398
  16. A.C. Yao, On Constructing Minimum Spanning Trees in k-dimensional Spaces and Related Problems, SIAM J. Comput. 11(4), 721-736, 1982 https://doi.org/10.1137/0211059