Lecture 16
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Lecture 16

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Lecture 16 Lecture 16 Presentation Transcript

  • Randomized Algorithms CS648 Lecture 16 Randomized Incremental Construction (Backward analysis) 1
  • PROBLEM 1 FIND-MIN PROBLEM 2
  • Find-Min algorithm A 1 2 … ?? 3
  • A 1 2 … 4
  • 5
  • A 1 2 … 6
  • Fact: A is permuted randomly uniformly 7
  • A 1 2 … 8
  • PROBLEM 2 CLOSEST PAIR OF POINTS 9
  • Closest Pair of Points 10
  • 11
  • O(1) O(1) O(1) 12
  • 13
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  • RANDOMIZED INCREMENTAL CONSTRUCTION 19
  • Randomized Incremental Construction 20
  • 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
  • PROBLEM 3 CONVEX HULL OF POINTS 22
  • Convex hull of Points 23
  • Convex hull of Points 24
  • Randomized algorithm for convex hull 25
  • 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
  • Conflict graph : a powerful data structure cones 27
  • Before entering the for loop 28
  • Before entering the for loop cones 29
  • 30
  • 31
  • cones 32
  • cones 33
  • cones 34
  • cones 35
  • cones 36
  • cones 37
  • cones 38
  • cones 39
  • cones 40
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  • Running time of the algorithm 46
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