Randomized Algorithms CS 648 Lecture 16 Randomized Incremental

Randomized Algorithms CS 648 Lecture 16 Randomized Incremental Construction (Backward analysis) 1

PROBLEM 1 FIND-MIN PROBLEM 2

Find-Min algorithm • • A 1 2 … ? ? 3

• Fact: A is permuted randomly uniformly 7

PROBLEM 2 CLOSEST PAIR OF POINTS 9

Closest Pair of Points • 10

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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

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cones 32

cones 33

cones 34

cones 35

cones 36

cones 37

cones 38

cones 39

cones 40

Running time of the algorithm • 46

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