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 find-min problem and closest pair of points problem are introduced. The main part describes a randomized algorithm for computing the convex hull of points by maintaining a conflict graph and resolving conflicts as new points are inserted. The algorithm runs in expected linear time.

1. Randomized Algorithms
CS648
Lecture 16
Randomized Incremental Construction
(Backward analysis)
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2. PROBLEM 1
FIND-MIN PROBLEM
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3. Find-Min algorithm
A
1
2
…
??
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4. A
1
2
…
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5. 5

6. A
1
2
…
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7. Fact: A is permuted randomly uniformly
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8. A
1
2
…
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9. PROBLEM 2
CLOSEST PAIR OF POINTS
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10. Closest Pair of Points
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11. 11

12. O(1)
O(1)
O(1)
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19. RANDOMIZED INCREMENTAL
CONSTRUCTION
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20. Randomized Incremental Construction
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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.
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22. PROBLEM 3
CONVEX HULL OF POINTS
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23. Convex hull of Points
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24. Convex hull of Points
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25. Randomized algorithm for convex hull
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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
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27. Conflict graph : a powerful data structure
cones
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28. Before entering the for loop
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29. Before entering the for loop
cones
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31. 31

32. cones
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33. cones
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34. cones
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35. cones
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36. cones
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37. cones
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38. cones
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39. cones
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40. cones
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46. Running time of the algorithm
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