Abstract
We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0,p 1,p 2,… in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k 2) additional storage.
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Communicated by Pankaj K. Agarwal.
M.A.A. was supported by the Netherlands Organization for Scientific Research (NWO) under project no. 612.065.307 and MADALGO, A Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation. M.dB. was supported by the Netherlands Organization for Scientific Research (NWO) under project no. 639.023.301. P.H. was supported by the Netherlands Organization for Scientific Research (NWO) under project no. 639.023.301. A.Z. was supported by Sharif University and a grant from IPM school of CS (no. CS1386-2-01).
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Abam, M.A., de Berg, M., Hachenberger, P. et al. Streaming Algorithms for Line Simplification. Discrete Comput Geom 43, 497–515 (2010). https://doi.org/10.1007/s00454-008-9132-4
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DOI: https://doi.org/10.1007/s00454-008-9132-4