CS133 Computational Geometry Computational Geometry on Big Data 1 - - PowerPoint PPT Presentation

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CS133 Computational Geometry Computational Geometry on Big Data 1 - - PowerPoint PPT Presentation

Oct 26, 2022 340 likes β€’565 views

CS133 Computational Geometry Computational Geometry on Big Data 1 Big Geometric Data Geotagged Satellite Imagery Check ins Tweets Billions of check ins More than 17 PB Billions of tweets Millions more every day 2 MapReduce Map Reduce

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CS133

Computational Geometry

Computational Geometry on Big Data

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Big Geometric Data

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More than 17 PB

Satellite Imagery Geotagged Tweets

Billions of tweets Billions of check ins Millions more every day

Check ins

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MapReduce

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Input Big Data Shuffle Map Map Map Map Reduce Reduce Reduce Reduce Output Split Input

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CG Algorithms on big data

Utilize divide and conquer algorithms

  • 1. Partition the input across machines
  • 2. (Optional) prune partitions that do not

contribute to answer

  • 3. Apply the algorithm locally in each partition
  • 4. Combine the partial answers to compute

the final result

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Examples

Convex hull algorithm Closest pair Farthest pair Voronoi diagram/Delaunay triangulation

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Data Partitioning

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Spatial Partitioning

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Skyline (Maximal Vectors)

Select all non-dominated points

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Input Output

𝑦1, 𝑧1 ≻ 𝑦2, 𝑧2 ⟺ 𝑦1 β‰₯ 𝑦2 ∧ 𝑧1 β‰₯ 𝑧2

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Skyline in MapReduce

Non-spatial partitioning Spatial partitioning ο‚„Global skyline Local skyline ο‚‚Pruning Partition

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Skyline Pruning

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π‘ž2 π‘ž1 π‘ž1 π‘ž2 π‘ž3

π‘ž1. π‘¦π‘›π‘—π‘œ, π‘ž1. π‘§π‘›π‘—π‘œ ≻ π‘ž2. 𝑦𝑛𝑏𝑦, π‘ž2. 𝑧𝑛𝑏𝑦 π‘ž1. π‘¦π‘›π‘—π‘œ, π‘ž1. 𝑧𝑛𝑏𝑦 ≻ π‘ž3. 𝑦𝑛𝑏𝑦, π‘ž3. 𝑧𝑛𝑏𝑦 π‘ž1. 𝑦𝑛𝑏𝑦, π‘ž1. π‘§π‘›π‘—π‘œ ≻ π‘ž2. 𝑦𝑛𝑏𝑦, π‘ž2. 𝑧𝑛𝑏𝑦

Partition domination rules

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Convex Hull

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Convex Hull in MapReduce

Non-spatial partitioning Spatial partitioning Partition Local hull ο‚„Global hull ο‚‚Pruning

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Pruning

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The intersection of the four skyline pruning rules with all directions

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Closest Pair

Find the pair of points that have the shortest Euclidean distance

Input Output

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Closest Pair in MapReduce

Non-spatial partitioning Spatial partitioning  Global closest pair ο‚‚ Local closest pair Partition

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Farthest Pair

Find the pair of points that have the largest Euclidean distance

Input Output

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Farthest Pair in MapReduce

Non-spatial partitioning Spatial partitioning Partition Local farthest pair ο‚„Global farthest pair ο‚‚Pruning

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Voronoi Diagram

Partitioning Local VD Pruning Vertical Merge Pruning Horizontal Merge Final output

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Voronoi Diagram Pruning

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Conclusion

Computational geometry algorithms can be parallelized Both non-spatial and spatial partitioning can be used Spatial partitioning enables some pruning techniques This method applies to several computational geometry algorithms