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### Advancements in Homotopic Polygonal Line Simplification for Massive Data Analysis ###

This research delves into homotopic polygonal line simplification to effectively manage and simplify massive datasets, specifically contour lines. Our work introduces both restricted and unrestricted simplification methods while ensuring preservation of homotopy and maintaining topological integrity. We address historical challenges in effective contour representation and propose new algorithms that significantly reduce complexity while simplifying vast amounts of polygon data. Our findings are evidenced in multiple published papers and experimental results, showcasing enhancements in computational efficiency and precision. ###

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### Advancements in Homotopic Polygonal Line Simplification for Massive Data Analysis ###

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  1. Homotopic Polygonal Line SimplificationLasse DeleuranPhD student

  2. Content • Motivation • Our Results • Restricted simplification • Unrestricted simplification • Simplifying massive data

  3. Motivation – Contour Lines

  4. Motivation – Contour Lines

  5. Motivation – Contour Lines

  6. Motivation – Contour Lines

  7. Our Results • “Improving Homotopic Shortest Paths Using Homotopic X-Shortest Paths”, M. Abam and L. Deleuran, TBS • “Computing Homotopic Line Simplification in a Plane”, M. Abam, S. Daneshpajouh, L. Deleuran, S. Ehsani and M. Ghodsi, EuroCG 2011, Submitted to CGTA 2012 • “Simplifying Massive Contour Maps”, L. Arge, L. Deleuran, T. Mølhave, M. Revsbæk, and J. Truelsen, ESA, 2012

  8. Definitions • Polygonal line:P=p1,p2, … ,pn |P| = n-1 • Homotopic poly. lines:

  9. Homotopic X-Shortest Paths • n polygonal paths of combined size m • Endpoints are the obstacles • Compute x-shortest path while maintaining homotopy

  10. Shortest Paths – Previous Work • Efrat et al. ’06: expected time O(nlogε+1n+mlogn) • Bespamyathnikh ’03: O(nlogε+1n+mlogn) • 2-part approach: 1) Compute homotopic x-shortest paths2) Compute homotopic shortest paths • Use their 2. part to achieve O(nlogε+1n+m)

  11. Restricted - Problem Definition • Given a path of size n, compute the paths with fewest points while • Only using original points • Maintaining some error constraint • Maintaining homotopy to m obstacle points • Strong vs weak homotopy:

  12. Restricted - Previous Results • Previous Results • Imai & Iri ‘88: Framework for the problem • Hausdorff: Chan & Chin ’92 O(n2) • Frechét Distance: Alt & Goday ’95 O(n3) • L1 and Uniform metric: Agarwal & Varadarajan ’00 O(n4/3+ ε ) • Our Results • Compute strongly homotopic ”links” • X-monotone path in O(mlog(nm) + nlogn log(nm) + k) • Any path in O(n(m + n)log(nm)) • Compute homotopic shortest path in O(n6m2)

  13. Simplifying a Massive Ammount of Polygons • Previous Results • Problems • Our Algorithm • Experimental Results

  14. Previous Results: Terrain vs Polygons • Simplifying terrain • Agarwal, et. al. ’98 (I/O efficient contour generation) • Agarwal, et. al. ’08 (I/O efficient map generation) • Carr, et. al. ’10 (DEM Simplification) • Garland & Heckbert ’97 (Surface Simplification) • Agarwal, et. al. ’06 (I/O efficient conditioning) • Simplifying polygons • See surveys by Mitchell ‘97, ‘98

  15. Challenges when Simplifying • Too many details • Massive data • Maintain precision • Maintain topology • Prevent intersections

  16. Challenges– Too Many Details

  17. Challenges- Massive Data • Denmark: 26B LIDAR points • 12.4B grid cells

  18. M • Challenges- Massive Data: I/O Model • RAM/internal memory size M • Unbounded disk/external memory • Transfer in blocks of size B • CPU only works on internal memory RAM CPU Disk B

  19. Massive Data - Practical Assumptions • Any polygon fits in memory (smaller than M) • Segments intersecting any vertical line is < M

  20. Challenges- Maintain Precision • Simplification algorithms typically only consider movement in the plane (x and y)

  21. Challenges- Maintain Topology • Topology: Parent / child relationships • Maintain topology through homotopy

  22. Problems – Prevent Intersections • Intersections with other polygons / self intersections

  23. Algorithm Overview • 1: Collect polygons(I/O-efficient) • 2:I/O efficient polygon visiting (I/O-efficient) • 3: Simplify polygons (internal)

  24. 1/3 - Collect Polygons

  25. 2/3 - I/O Efficient Polygon Visiting • Polygons are neighbors if no other poly. divides them • A polygon must be considered together w. neighbors

  26. 3/3 Simplify Polygons • Basic Algorithm: Douglas Peucker

  27. Problems with Douglass Peucker • Running time O(n2), but O(nlogn) in practice • No z-constraint / not constrained by other polygons • Introduces self intersections, no homotopy

  28. Simplifying – Adding Boundaries • Construct Trapezoidal decompositionO(nlogn) • Continue DP until inside of decomposition O(n2logn) • Add contraint for z

  29. Simplifying – Removing Intersections • Sweep to find intersections O(nlogn) • Continue DP on intersecting segments • Repeat => O(n2logn)

  30. Simplifying – Maintain Homotopy • Trapezoidal sequence: ABCDCFCDEDE • Contract XYX -> X • Canonical Sequence:ABCDE

  31. Simplifying – Maintain Homotopy • Check segment for homotopy: O(n) • =>O(n2logn)

  32. Practical Optimizations

  33. Optimization 1 - Conditioning the Terrain • Fill up all holes with depth of less than 0.5m • Do so for small hills too.

  34. Optimization 2 - Exploit Bounding Boxes

  35. Optimization 3 - Minimal Decompositions • Too much time will be spent constructing decompositions • Only use edges that intersect bounding box

  36. Setup • Code in C++ using TPIE and TerraSTREAM • Machine: • 8-core Intel Xenon CPU @ 3.2GHz • 12GB of RAM • disk speed: 400MB/s

  37. Results • Results for Denmark dataset (12B points): • 49 hours to simplify 4B segments on 7M contours • Z-diff: 0.5m (Border contours: 0.2m) • DP-error: 5m • 600.000 self intersections • 8.2% of the points remained after simplifying Thank You

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