Probability and Delaunay triangulations 1 Randomized algorithms for - PowerPoint PPT Presentation

Walk Complexity O ( n ) Better bounds for random points e r u t c e l y t i l i b a b o r p r e s a e T 15 - 5

Jump and walk 16 - 1

Jump and walk 16 - 2

Jump and walk 16 - 3

Jump and walk Hopefully shorter walk Designed for random points p n ) expected location time O ( 3 16 - 4

Jump and walk (no distribution hypothesis) 17 - 1

Jump and walk (no distribution hypothesis) ] = n E [ ] of in k 17 - 2

Jump and walk (no distribution hypothesis) ] = n E [ ] of in k � n � Walk length = O k p n choose k = 2 17 - 3

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k � n � Walk length = O k p n choose k = 2 17 - 4

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in n k k 1 � n � Walk length = O k p n choose k = 2 17 - 5

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k 1 + k 1 n k k 2 � n � Walk length = O k p n choose k = 2 17 - 6

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k 1 + k 1 n + k 2 k k 3 + . . . k 2 � n � Walk length = O k p n choose k = 2 17 - 7

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k 1 + k 1 n + k 2 k k 3 + . . . k 2 � n � Walk length = O k i choose k i +1 = ↵ k p n choose k = 2 17 - 8

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k 1 + k 1 n + k 2 k k 3 + . . . k 2 � n � Walk length = O k i choose k i +1 = ↵ k p n choose k = 2 point location in O ( ↵ log α n ) 17 - 9

Jump and walk (no distribution hypothesis) Delaunay hierarchy ] = n E [ ] of in k 1 + k 1 n + k 2 k k 3 + . . . k 2 � n � Walk length = O k i choose k i +1 = ↵ k p n choose k = 2 point location in O ( ↵ log α n ) point location in O ( p ↵ log α n ) 17 - 10

Technical detail � n ( ) = O � Walk length = O ] of in k 18 - 1

Technical detail � n ( ) = O � Walk length = O ] of in k random point not a random point 18 - 2

Technical detail � n ( ) = O � Walk length = O ] of in k random point not a random point ] = 1 d � NN ( q ) = 1 X X X E [d � d � v n n q q v = NN ( q ) = 1 d � v  1 X X X 6d � v  36 n n 18 - 3 v v q ; v = NN ( q )

P d � Technical detail � n ( ) = O � Walk length = O ] of in k random point not a random point ] = 1 d � NN ( q ) = 1 X X X E [d � d � v n n q q v = NN ( q ) = 1 d � v  1 X X X 6d � v  36 n n 18 - 4 v v q ; v = NN ( q )

Randomization How many randomness is necessary? If the data are not known in advance shu ffl e locally 19