Locality-Sensitive Orderings ANN -Quadtree Walecki Theorem - PowerPoint PPT Presentation

Locality-Sensitive Orderings Main Result Quadtree Locality-Sensitive Orderings ANN ǫ -Quadtree Walecki Theorem Local-Sensitivity Anil Maheshwari Theorem Applications anil@scs.carleton.ca References School of Computer Science Carleton University Canada

What we want to do? Locality-Sensitive Orderings Main Result Local Ordering Theorem (CHJ2020) Quadtree Consider a unit cube in d -dimensions. For ǫ > 0 , there is ANN ǫ )) orderings of [0 , 1) d such that for a family of O ( 1 ǫ d log( 1 ǫ -Quadtree any p, q ∈ [0 , 1) d , there is an ordering in the family where Walecki Theorem all the points between p and q are within a distance of at Local-Sensitivity Theorem most ǫ || p − q || 2 from p or q . Applications References ǫ || p − q || p ǫ || p − q || q p q

Application: Closest Pair Locality-Sensitive Orderings Input: Set of n points in ℜ d Main Result Output: Closest pair Quadtree ANN Algorithm: ǫ -Quadtree For every ordering, find the pair of consecutive points 1 Walecki Theorem that have minimum distance. Local-Sensitivity Theorem Report the pair that has the least distance among all 2 Applications the orderings. References Time: O ( n × # of orderings ) ≈ O ( n/ǫ d ) Dynamic: Insert/Delete points and maintain closest pair

Tools & Techniques Locality-Sensitive Orderings Main Result Old & New Concepts Quadtree Quadtree. 1 ANN ǫ -Quadtree Linear orderings of points in a Quadtree. 2 Walecki Theorem Shifted Quadtrees and ANN. 3 Local-Sensitivity Theorem Quadtree as union of ǫ -Quadtrees. 4 Applications (Wonderful) Walecki Construction from 19th Century. 5 References Locality-Sensitive Orderings. 6 Applications in ANN, Bi-chromatic ANN, Spanners, ... 7

Quadtree of a point set Locality-Sensitive Orderings Main Result Quadtree ANN ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem Applications References

Quadtree of a point set Locality-Sensitive Orderings Main Result Quadtree ANN ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem Applications References

Quadtree of a point set Locality-Sensitive Orderings Main Result Quadtree ANN ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem Applications References

Quadtree of a point set Locality-Sensitive Orderings Main Result Quadtree ANN A B C D ǫ -Quadtree C D Walecki Theorem Local-Sensitivity Theorem c d a b c d Applications A References B a b

Linear order Locality-Sensitive Orderings Main Result DFS traversal of Quadtree Quadtree Obtain a linear order of points by performing the DFS ANN traversal of the Quadtree. ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem b d c g Applications References i f a h f a i g b j k j d c l e l e k h h f a k e l j i g b d c

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Quadtree Cells & DFS order Locality-Sensitive Orderings Main Result Quadtree b d c ANN g ǫ -Quadtree Walecki Theorem i Local-Sensitivity Theorem f a h f a i g b Applications j k j d c References l e l e k h

Approximate NN from Linear Order Locality-Sensitive Orderings Main Result Approximate NN Quadtree Let q be nearest-neighbor of p . Assume that there is a cell ANN containing p and q in Quadtree with diameter ≈ || p − q || . ǫ -Quadtree Walecki Theorem Local-Sensitivity p x q Theorem Applications References p q = NN ( p ) q diam ≈ || p − q || || p − x || ≈ || p − q || x

Assumption? Locality-Sensitive Orderings How to ensure that the following is true? Main Result Quadtree ANN ǫ -Quadtree There is a cell containing p and q in the Quadtree with Walecki Theorem diameter ≈ || p − q || Local-Sensitivity Theorem Applications References

Quadtrees of Shifted Point Sets Locality-Sensitive Orderings Assume all points in P ⊂ [0 , 1) d . Main Result Construct D = 2 ⌈ d 2 ⌉ + 1 copies of P . Quadtree ANN Shifted Point Sets ǫ -Quadtree Walecki Theorem For i = 0 , . . . , D , define shifted point sets i i i Local-Sensitivity P i = { p j + ( D +1 , D +1 , . . . , D +1 ) |∀ p j ∈ P } Theorem Applications Let Quadtrees of P 0 , P 1 , . . . , P D be T 0 , T 1 , . . . , T D . References Chan (DCG98) For any pair of points p, q ∈ P , there exists a Quadtree T ∈ { T 0 , T 1 , . . . , T D } such that the cell containing p, q in T has diameter c || p − q || (for some constant c ≥ 1 ).

Dynamic ANN Locality-Sensitive Orderings Chan’s ANN Algorithm: Main Result Quadtree Construct linear (dfs) order for each of the Quadtrees 1 ANN T 0 , T 1 , . . . , T D . ǫ -Quadtree For each point p , find its neighbor in each of the 2 Walecki Theorem linear orders that minimizes the distance. Local-Sensitivity Theorem Let q be the neighbor of p with the minimum distance. 3 Applications Report q as the ANN of p . 4 References Chan (1998, 2006) For fixed dimension d , in O ( n log n ) preprocessing time and O ( n ) space, we can find a c -approximate nearest neighbor of any point in P in O (log n ) time ( c = f ( d ) ).

ǫ -Quadtree Locality-Sensitive Orderings Main Result ǫ -Quadtree Quadtree For a constant ǫ > 0 , recursively partition a cube [0 , 1) d ANN evenly into 1 ǫ d sub-cubes ( ǫ = 1 / 2 = ⇒ Standard ǫ -Quadtree Quadtree). Walecki Theorem Local-Sensitivity Theorem l × l × . . . × l Applications ǫl × ǫl × . . . × ǫl References ǫ 2 l × ǫ 2 l × . . . × ǫ 2 l l ǫ 3 l × ǫ 3 l × . . . × ǫ 3 l l

Quadtree as union of ǫ -Quadtrees Locality-Sensitive Orderings Main Result Partitioning a Quadtree T into log 1 ǫ ǫ -Quadtrees Quadtree Let ǫ = 2 − 3 . T = T B ǫ ∪ T R ǫ ∪ T U ǫ . ANN ǫ -Quadtree Walecki Theorem T B ǫ Local-Sensitivity Theorem T R Applications ǫ T References T U ǫ

Walecki’s Result Locality-Sensitive Orderings Main Result Permuting cells of a node of an ǫ -Quadtree Quadtree Let ǫ = 2 − 2 . Any two cells are neighbors in at least one of ANN the 8 permutations. ǫ -Quadtree Walecki Theorem ABPCODNEMFLGKHJI Local-Sensitivity Theorem A B C D BCADPEOFNGMHLIKJ Applications CDBEAFPGOHNIMJLK References E F G H DECFBGAHPIOJNKML EFDGCHBIAJPKOLNM I J K L FGEHDICJBKALPMON GHFIEJDKCLBMANPO M N O P HIGJFKELDMCNBOAP

(Wonderful) Walecki Result Locality-Sensitive Orderings Main Result Walecki Theorem Quadtree For n elements { 0 , 1 , 2 , . . . , n − 1 } , there is a set of ⌈ n 2 ⌉ ANN permutations of the elements, such that, for all ǫ -Quadtree i, j ∈ { 1 , 2 , . . . , n − 1 } , there is a permutation in which i Walecki Theorem and j are adjacent. Local-Sensitivity Theorem Applications References

Proof (by Figure) Locality-Sensitive Orderings Partition K 8 in 4 Hamiltonian Paths Main Result Quadtree ANN ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem Applications References

Linear orders of points of P ⊂ [0 , 1) d Locality-Sensitive Orderings Main Result DFS Traversal of an ǫ -Quadtree T ǫ Quadtree ANN # children of any node of T ǫ = O (1 /ǫ d ) . 1 ǫ -Quadtree Walecki Theorem Construct O (1 /ǫ d ) permutations of cells using 2 Local-Sensitivity Walecki’s construction. Theorem Generate O (1 /ǫ d ) linear orders of points in P by Applications 3 References performing DFS traversal of T ǫ with respect to each permutation.

Structure of Cells Locality-Sensitive Orderings Main Result Quadtree A B C D ANN ǫ -Quadtree E F G H Walecki Theorem Local-Sensitivity Theorem I J K L Applications References M N O P A B P C O D N E M F L G K H J I

What have we learnt so far? Locality-Sensitive Orderings Point set P ⊂ [0 , 1) d . Main Result 1 Quadtree Shifted points sets P 0 , P 1 , . . . , P D and their 2 ANN Quadtrees T 0 , T 1 , . . . , T D . ǫ -Quadtree Each Quadtree T i partitioned into log 1 ǫ ǫ -Quadtrees. 3 Walecki Theorem Local-Sensitivity Permutations of cells of a node in an ǫ -Quadtree 4 Theorem (Walecki’s result). Applications Linear orders of points in P from DFS (for each References 5 permutation) of ǫ -Quadtrees. Total # Linear Orders 6 = O ( D × log 1 ǫ × 1 ǫ d ) = O ( 1 ǫ d log 1 ǫ ) . These linear orders satisfy the “locality” condition. 7