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 Authors: Timothy Chan, Sariel Har-Peled, Mitchell Theorem Jones (ITCS 2019) Applications Presenter: Anil Maheshwari Carleton University Ottawa, Canada

What we want to do? Locality-Sensitive Orderings Main Result Local Ordering Theorem (CHJ2019) 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 Local-Sensitivity all the points between p and q are within a distance of at Theorem most ǫ || p − q || 2 from p or q . Applications ǫ || p − q || ǫ || p − q || p q p q

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

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

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

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

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

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

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

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

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

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

Approximate NN from Linear Order Locality-Sensitive Orderings Approximate NN Main Result 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 p q = NN ( p ) q diam ≈ || p − q || || p − x || ≈ || p − q || x

Quadtrees of Shifted Point Sets Locality-Sensitive Orderings Main Result Quadtree Assume all points in P ∈ [0 , 1) d . ANN Construct D = 2 ⌈ d 2 ⌉ + 1 copies of P . ǫ -Quadtree Walecki Theorem Shifted Point Sets Local-Sensitivity For i = 0 , . . . , D , define shifted point sets Theorem i i i P i = { p j + ( D +1 , D +1 , . . . , D +1 ) |∀ p j ∈ P } Applications Let Quadtrees of P 0 , P 1 , . . . , P D be T 0 , T 1 , . . . , T D . 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 Main Result Quadtree Chan’s ANN Algorithm: ANN Construct linear (dfs) order for each of the Quadtrees 1 ǫ -Quadtree T 0 , T 1 , . . . , T D . Walecki Theorem For each point p , find its neighbor in each of the 2 Local-Sensitivity Theorem linear orders that minimizes the distance. Applications Let q be the neighbor of p with the minimum distance. 3 Report q as the ANN of p . 4 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 ǫ -Quadtree Main Result 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 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 Partitioning a Quadtree T into log 1 ǫ ǫ -Quadtrees Main Result 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 ǫ Applications T T U ǫ

Walecki’s Result Locality-Sensitive Orderings Main Result Ordering cells of a node of an ǫ -Quadtree Quadtree Let ǫ = 2 − 3 . Any two cells are neighbors in at least one of ANN the 8 orders. ǫ -Quadtree Walecki Theorem ABPCODNEMFLGKHJI Local-Sensitivity Theorem A B C D BCADPEOFNGMHLIKJ Applications CDBEAFPGOHNIMJLK E F G H DECFBGAHPIOJNKML EFDGCHBIAJPKOLNM I J K L FGEHDICJBKALPMON GHFIEJDKCLBMANPO M N O P HIGJFKELDMCNBOAP

(Wonderful) Walecki Result Locality-Sensitive Orderings Walecki Theorem Main Result Quadtree A complete graph on n vertices can be partitioned into ⌈ n ANN 2 ⌉ Hamiltonian paths. ǫ -Quadtree Walecki Theorem Local-Sensitivity Theorem Applications

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

Structure of Cells Locality-Sensitive Orderings Main Result Quadtree ANN A B C D ǫ -Quadtree Walecki Theorem Local-Sensitivity E F G H Theorem Applications I J K L 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 Main Result Quadtree Point set P ∈ [0 , 1) d . ANN 1 ǫ -Quadtree Shifted points sets P 0 , P 1 , . . . , P D and their 2 Walecki Theorem Quadtrees T 0 , T 1 , . . . , T D . Local-Sensitivity Theorem Each Quadtree T i partitioned into log 1 ǫ ǫ -Quadtrees. 3 Applications Linear orders of cells of a node in an ǫ -Quadtree. 4 Permutations of points of P obtained from DFS (for 5 each linear order) of ǫ -Quadtrees. Total # Permutations 6 = O ( D × log 1 ǫ × 1 ǫ d ) = O ( 1 ǫ d log 1 ǫ ) . These permutations satisfy “locality” condition. 7

Locality Property Locality-Sensitive Orderings Locality-Sensitive Orderings Main Result Let the Quadtree T i ∈ { T 0 , T 1 , . . . , T D } has a cell Quadtree ANN containing p and q with diameter ≈ || p − q || . ǫ -Quadtree Walecki Theorem Local-Sensitivity T i Theorem Applications

Locality Property Locality-Sensitive Orderings Locality-Sensitive Orderings Main Result Quadtree Let the Quadtree T i ∈ { T 0 , T 1 , . . . , T D } has a cell ANN containing p and q with diameter ≈ || p − q || . ǫ -Quadtree Walecki Theorem Local-Sensitivity T i Theorem Applications c || p − q || ǫc || p − q || c || p − q ||

Locality Property Locality-Sensitive Orderings Locality-Sensitive Orderings Main Result Quadtree Let the Quadtree T i ∈ { T 0 , T 1 , . . . , T D } has a cell ANN containing p and q with diameter ≈ || p − q || . ǫ -Quadtree Walecki Theorem Local-Sensitivity T i Theorem Applications q c || p − q || ǫc || p − q || p c || p − q ||

Locality Property Locality-Sensitive Orderings Locality-Sensitive Orderings Main Result Quadtree Let the Quadtree T i ∈ { T 0 , T 1 , . . . , T D } has a cell ANN containing p and q with diameter ≈ || p − q || . ǫ -Quadtree Walecki Theorem Local-Sensitivity T i Theorem Applications q c || p − q || ǫc || p − q || p c || p − q ||