Lecture 16
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The document discusses randomized incremental construction, a technique for building geometric structures by incrementally adding elements in random order. It provides examples like computing the convex hull of a set of points. The key ideas are using a conflict graph to track inconsistencies and resolving them in random order, resulting in an expected linear time algorithm. The technique can be applied to problems like computing the smallest enclosing sphere of a point set.
Lecture 16
- 1. Randomized Algorithms CS648 Lecture 16 Randomized Incremental Construction (Backward analysis) 1
- 4. A 1 2 … 4
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- 6. A 1 2 … 6
- 7. Fact: A is permuted randomly uniformly 7
- 8. A 1 2 … 8
- 9. PROBLEM 2 CLOSEST PAIR OF POINTS 9
- 10. Closest Pair of Points 10
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- 21. Randomized Incremental Construction • Convex Hull of a set of points • Trapezoidal decomposition of a set of segments. • Convex polytope of a set of half-planes • Smallest sphere enclosing a set of points. • Linear programming in finite dimensions. 21
- 22. PROBLEM 3 CONVEX HULL OF POINTS 22
- 23. Convex hull of Points 23
- 24. Convex hull of Points 24
- 25. Randomized algorithm for convex hull 25
- 26. A simple exercise from geometry Exercise: Given a line L and two points p and q, determine whether the points lie on the same/different sides of L. L p q q 26
- 27. Conflict graph : a powerful data structure cones 27
- 28. Before entering the for loop 28
- 29. Before entering the for loop cones 29
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- 32. cones 32
- 33. cones 33
- 34. cones 34
- 35. cones 35
- 36. cones 36
- 37. cones 37
- 38. cones 38
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- 40. cones 40
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- 46. Running time of the algorithm 46
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