Partial match queries: a limit process Nicolas Broutin Ralph - PowerPoint PPT Presentation

Partial match queries: a limit process Nicolas Broutin Ralph Neininger Henning Sulzbach Partial match queries: a limit process 1 / 19

Background/Introduction Data structures/Algorithms ◮ Analysis of costs/running times in natural conditions ◮ expected cost ◮ performance guarantee provided by concentration Methodology ◮ complex “objects” that decompose recursively (tree like, or related) ◮ general approach for convergence using contractions Partial match queries: a limit process 2 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Searching geometric data and quadtrees 2 4 1 3 Partial match queries: a limit process 3 / 19

Model and Previous results Point set = { ( U i , V i ) , i ≥ 1 } iid uniform in [ 0 , 1 ] 2 C n ( s ) the number of lines intersecting { x = s } in a quadtree of size n Theorem (Flajolet, Gonnet, Puech and Robson (1993)) For ξ uniform independent of { ( U i , V i ) , i ≥ 1 } √ κ = Γ( 2 β + 2 ) 17 − 3 E [ C n ( ξ )] ∼ κ n β where 2 Γ( β + 1 ) 2 , β = 2 Theorem (Chern and Hwang (2003)) Let φ ( z ) = ( z + 1 )( z + 2 ) − 4 and β > β ′ the roots of φ . For ξ uniform independent of { ( U i , V i ) , i ≥ 1 } , one has the exact expression ( − 1 ) k + 1 2 ( 1 − β ) k − 1 ( 1 − β ′ ) k − 1 � n � � E [ C n ( ξ )] = k k !( k + 1 )! 1 ≤ k ≤ n Corollary (Chern and Hwang (2003)) For ξ uniform independent of { ( U i , V i ) , i ≥ 1 } E [ C n ( ξ )] = κ n β − 1 + O ( n β − 1 ) Partial match queries: a limit process 4 / 19

Idea of the method / heuristic for the constants Recursive decomposition ξ d We have Y = max { U 1 , U 2 } and ( I , J ) = Mult ( Bin ( n − 1 , Y ); V , ( 1 − V )) then ( U , V ) C n ( ξ ) d = 1 + C I ( ξ ′ ) + C J ( ξ ′ ) E [ C n ( ξ )] ≈ 2 E [ C nYV ( ξ ′ )] ⇒ Y Plugging E [ C n ( ξ )] = κ n β yields √ 4 17 − 3 1 = 2 E [ Y β V β ] = 2 E [ Y β ] · E [ V β ] = ⇒ β = ( β + 2 )( β + 1 ) 2 About the variance Var ( C n ( ξ )) Even when conditioning on the first point, the two terms are still dependent on the query line Partial match queries: a limit process 5 / 19

The cost at a fixed query line Idea: ◮ if the query line is fixed at s ∈ ( 0 , 1 ) , then we do have independence ◮ however, its relative position changes in the subproblems ◮ ⇒ consider the entire process ( C n ( s ) , s ∈ ( 0 , 1 )) Theorem (Flajolet, Labelle, Laforest and Salvy 1995) √ 2 − 1 ) = o ( n β ) E [ C n ( 0 )] = Θ( n Note : in particular, E [ C n ( U 1 )] = o ( n β ) , and C n ( s ) is not concentrated. Theorem (Curien and Joseph (2011)) For every fixed s ∈ ( 0 , 1 ) , one has Γ( 2 β + 2 )Γ( β + 2 ) E [ C n ( s )] ∼ K 1 ( s ( 1 − s )) β/ 2 n β , K 1 = 2 Γ( β + 1 ) 3 Γ( β/ 2 + 1 ) 2 . Partial match queries: a limit process 6 / 19

Main result Theorem There exists a random continuous function Z such that, as n → ∞ , � C n ( s ) � d K 1 n β , s ∈ [ 0 , 1 ] → ( Z ( s ) , s ∈ [ 0 , 1 ]) . (1) This convergence in distribution holds in the Banach space ( D [ 0 , 1 ] , � · � ) of right-continuous functions with left limits (c` adl` ag) equipped with the supremum norm. Proposition The distribution of the random function Z in (1) is a fixed point of the following equation � s � s � � �� Z ( s ) d ( UV ) β Z ( 1 ) + ( U ( 1 − V )) β Z ( 2 ) = 1 { s < U } U U � s − U � s − U � � �� (( 1 − U ) V ) β Z ( 3 ) + (( 1 − U )( 1 − V )) β Z ( 4 ) + 1 { s ≥ U } , 1 − U 1 − U where U and V are independent [ 0 , 1 ] -uniform random variables and Z ( i ) , i = 1 , . . . , 4 are independent copies of the process Z, which are also independent of U and V. Furthermore, Z in (1) is the only solution such that E [ Z ( s )] = ( s ( 1 − s )) β/ 2 for all s ∈ [ 0 , 1 ] and E [ � Z � 2 ] < ∞ . Partial match queries: a limit process 7 / 19

What does it look like I n = 1000 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Partial match queries: a limit process 8 / 19

What does it look like II 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Partial match queries: a limit process 9 / 19

Moments and supremum Theorem We have for all s ∈ ( 0 , 1 ) , as n → ∞ , � � 2B ( β + 1 , β + 1 ) 2 β + 1 ( s ( 1 − s )) β n 2 β . Var ( C n ( s )) ∼ 3 ( 1 − β ) − 1 � 1 0 x a − 1 ( 1 − x ) b − 1 dx denotes the Eulerian beta integral (a , b > 0 ). Here, B ( a , b ) := Theorem Let S n = sup s ∈ [ 0 , 1 ] C n ( s ) . Then, as n → ∞ , d n − β S n E [ S n ] ∼ n β E [ S ] , Var ( S n ) ∼ n 2 β Var ( S ) . → S = sup Z ( s ) and s ∈ [ 0 , 1 ] Partial match queries: a limit process 10 / 19

Convergence in distribution by contraction I. 2 4 Cost of the construction of the quadtree / path length n � P n = D i with D i the depth of the i -th inserted point i = 1 ◮ I r n the number of points inside the r -th child cell ◮ Q r the volume or the r -th child cell 1 3 We have 4 X n = P n − α n log n � d P n = P I r n + n − 1 and write n r = 1 n ) d ( I 1 n , . . . , I 4 = Mult ( n − 1 ; UV , U ( 1 − V ) , ( 1 − U )( 1 − V ) , ( 1 − U ) V ) . Shifting and rescaling we obtain: � I r � I r � I r 4 � n − α I r n log I r 4 � � P I r P n − α n log n + n − 1 − α log n � � n n n n = + α log I r n n n n n n n r = 1 r = 1 � �� � � �� � � �� � X n A r n b n Partial match queries: a limit process 11 / 19

Convergence in distribution by contraction II. General problem: = � 4 d r = 1 A r n · X r A recursive family of equations X n n + b n with I r ◮ ( A 1 n , . . . , A 4 n , I 1 n , . . . , I 4 n , b n ) independent of (( X 1 ) , . . . , ( X 4 )) ◮ ( X r n , n ≥ 1 ) iid copies of ( X ) The equation ”converges” to a limit equation: n = I r A r n n → Leb ( Q r ) � I r � I r 4 � � 4 b n = n − 1 − α log n � � n n + α log → 1 + α Leb ( Q r ) log Leb ( Q r ) n n n n r = 1 r = 1 4 4 � � Leb ( Q r ) · X r + 1 + α X d = Leb ( Q r ) log Leb ( Q r ) (2) r = 1 r = 1 Formalization: (2) a transfer map on a space of probability measures on R . d 2 ( φ, ϕ ) = inf {� X − Y � 2 : L ( X ) = φ, L ( Y ) = ϕ } � ◮ on M 2 = { probability measures µ : x 2 d µ < ∞} no contraction (can shift!) � ◮ on M 0 2 = { µ ∈ M 2 : xd µ = 0 } contraction Partial match queries: a limit process 12 / 19

Convergence for partial match processes s 2 4 n ) d ( I ( 1 ) n , . . . , I ( 4 ) = Mult ( n − 1 ; UV , U ( 1 − V ) , ( U , V ) ( 1 − U )( 1 − V ) , ( 1 − U ) V ) � s � s � � �� C n ( s ) d C ( 1 ) + C ( 2 ) = 1 + 1 { s < U } I ( 1 ) I ( 2 ) U U n n � 1 − s � 1 − s � � �� C ( 3 ) + C ( 4 ) 1 3 + 1 { s ≥ U } I ( 3 ) I ( 4 ) 1 − U 1 − U n n Partial match queries: a limit process 13 / 19

Convergence for partial match processes s 2 4 n ) d ( I ( 1 ) n , . . . , I ( 4 ) = Mult ( n − 1 ; UV , U ( 1 − V ) , ( U , V ) ( 1 − U )( 1 − V ) , ( 1 − U ) V ) � s � s � � �� C n ( s ) d C ( 1 ) + C ( 2 ) = 1 + 1 { s < U } I ( 1 ) I ( 2 ) U U n n � 1 − s � 1 − s � � �� C ( 3 ) + C ( 4 ) 1 3 + 1 { s ≥ U } I ( 3 ) I ( 4 ) 1 − U 1 − U n n Heuristic : If n − β C n ( · ) converges, we should have n − β C n ( · ) → Z ( · ) satisfying � s � s � � �� Z ( s ) d ( UV ) β Z ( 1 ) + ( U ( 1 − V )) β Z ( 2 ) = 1 { s < U } U U � s − U � s − U � � �� (( 1 − U ) V ) β Z ( 3 ) + (( 1 − U )( 1 − V )) β Z ( 4 ) + 1 { s ≥ U } 1 − U 1 − U Partial match queries: a limit process 13 / 19