Partial match queries: a limit process

Partial match queries: a limit process Nicolas Brouti Ralph Neiiger Heig Sulzbach Partial match queries: a limit process 1 / 17

Searchig geometric data ad quadtrees 1 Partial match queries: a limit process / 17

Model ad Previous results Poit set = {(U i, V i ), i 1} iid uiform i [0, 1] C (s) the umber of lies itersectig {x = s} i a quadtree of size Theorem (Flajolet, Goet, Puech ad Robso (199)) For ξ uiform idepedet of {(U i, V i ), i 1} E [C (ξ)] κ β where κ = Γ(β + ) 17 Γ(β + 1), β = Theorem (Cher ad Hwag (00)) For ξ uiform idepedet of {(U i, V i ), i 1} E [C (ξ)] = κ β 1 + O( β 1 ) Partial match queries: a limit process / 17

Idea of the method / heuristic for the costats Recursive decompositio We have Y d = max{u 1, U } ad (I, J) = Mult(Bi( 1, Y ); V, (1 V )) the C (ξ) d = 1 + C I (ξ ) + C J (ξ ) ξ (U, V ) E[C (ξ)] E[C YV (ξ )] Y Pluggig E[C (ξ)] = κ β yields 1 = E[Y β V β ] = E[Y β ] E[V β ] = (β + )(β + 1) β = 17 About the variace Var (C (ξ)) Eve whe coditioig o the first poit, the two terms are still depedet o the query lie Partial match queries: a limit process / 17

The cost at a fixed query lie Idea: if the query lie is fixed at s (0, 1), the we do have idepedece however, its relative positio chages i the subproblems cosider the etire process (C (s), s (0, 1)) Theorem (Flajolet, Labelle, Laforest ad Salvy 1995) E [C (0)] = Θ( 1 ) = o( β ) Theorem (Curie ad Joseph (011)) For every fixed s (0, 1), oe has E [C (s)] K 1 (s(1 s)) β/ β, K 1 = Γ(β + )Γ(β + ) Γ(β + 1) Γ(β/ + 1). Partial match queries: a limit process 5 / 17

Mai result Theorem There exists a radom cotiuous fuctio Z such that, as, ( ) C(s) d, s [0, 1] (Z (s), s [0, 1]). (1) K 1 β This covergece i distributio holds i the Baach space (D[0, 1], ) of right-cotiuous fuctios with left limits (càdlàg) equipped with the supremum orm. Propositio The distributio of the radom fuctio Z i (1) is a fixed poit of the followig equatio [ ( ) ( )] Z (s) =1 d s s {s<u} (UV ) β Z (1) + (U(1 V )) β Z () U U [ ( ) ( )] s U s U + 1 {s U} ((1 U)V ) β Z () + ((1 U)(1 V )) β Z (), 1 U 1 U where U ad V are idepedet [0, 1]-uiform radom variables ad Z (i), i = 1,..., are idepedet copies of the process Z, which are also idepedet of U ad V. Furthermore, Z i (1) is the oly solutio such that E[Z (s)] = (s(1 s)) β/ for all s [0, 1] ad E[ Z ] <. Partial match queries: a limit process 6 / 17

What does it look like = 1000 0.0 0.5 1.0 1.5.0 0.0 0. 0. 0.6 0.8 1.0 Partial match queries: a limit process 7 / 17

Momets ad supremum Theorem We have for all s (0, 1), as, ( Var (C (s)) B(β + 1, β + 1) β + 1 ) (1 β) 1 (s(1 s)) β β. Here, B(a, b) := 1 0 x a 1 (1 x) b 1 dx deotes the Euleria beta itegral (a, b > 0). Theorem Let S = sup s [0,1] C (s). The, as, β d d S S = sup Z (s) ad E[S ] β E[S], Var(S ) β Var(S). s [0,1] Partial match queries: a limit process 8 / 17

Covergece i distributio by cotractio I. Cost of the costructio of the quadtree / path legth P = D i with D i the depth of the i-th iserted poit i=1 I r the umber of poits iside the r-th child cell Q r the volume or the r-th child cell 1 We have d P α log P = P I r + 1 ad write X = r=1 (I 1,..., I ) d = Mult( 1; UV, U(1 V ), (1 U)(1 V ), (1 U)V ). Shiftig ad rescalig we obtai: P α log } {{ } X ( I r ) = P I r αi r log I r I r + 1 α log ( I r + α r=1 }{{} r=1 A r ) ( I r ) log } {{ } b Partial match queries: a limit process 9 / 17

Covergece i distributio by cotractio II. Geeral problem: d A recursive family of equatios X = r=1 Ar X r I r + b with (A 1,..., A, I1,..., I, b) idepedet of ((X 1 ),..., (X )) (X r, 1) iid copies of (X) The equatio coverges to a limit equatio: A r = Ir Leb(Qr ) b = 1 α log X d = r=1 ( I r ) ( + α I r ) log 1 + α Leb(Q r ) log Leb(Q r ) r=1 r=1 Leb(Q r ) X r + 1 + α Leb(Q r ) log Leb(Q r ) () r=1 Formalizatio: See the right-had side of () as a trasfer map o probability meas. d (φ, ϕ) = if{ X Y : L (X) = φ, L (Y ) = ϕ} o M = {probability measures µ : x dµ < } o cotractio (ca shift!) o M 0 = {µ M : xdµ = 0} cotractio Partial match queries: a limit process 10 / 17

Covergece for partial match processes s (U, V ) (I (1),..., I () ) d = Mult( 1;UV, U(1 V ), (1 U)(1 V ), (1 U)V ) 1 C (s) d = 1 + 1 {s<u} [ C (1) I (1) + 1 {s U} [ C () I () ( s U ( 1 s 1 U ) + C () I () ( )] s U ( )] 1 s 1 U ) + C () I () Heuristic: If β C ( ) coverges, we should have β C ( ) Z ( ) satisfyig [ ( ) ( )] Z (s) =1 d s s {s<u} (UV ) β Z (1) + (U(1 V )) β Z () U U [ ( ) ( )] s U s U + 1 {s U} ((1 U)V ) β Z () + ((1 U)(1 V )) β Z () 1 U 1 U Partial match queries: a limit process 11 / 17

Covergece i D[0, 1] by cotractio argumets I. Let (X ) be D[0, 1]-valued radom variables with where X d = K r=1 A (r) X (r) + b, 1, I (r) Neiiger ad Sulzbach (011+) (A (1),..., A (K ) ) are radom liear ad cotiuous operators o D[0, 1] b is a D[0, 1]-valued radom variable I (1),..., I (K ) are radom itegers betwee 0 ad 1 (X (1) ),..., (X (K ) ) are distributed like (X ) (A (1),..., A (K ), b, I (1),..., I (K ) ), (X (1) ),..., (X (K ) ) are idepedet Partial match queries: a limit process 1 / 17

Covergece i D[0, 1] by cotractio argumets II. For a radom liear operator A write A := E[ A op] 1/ with A op := sup A(x) x =1 Neiiger ad Sulzbach (011+) (A1) CONVERGENCE AND CONTRACTION. we have A (r), b < for all r = 1,..., K ad 0 there exist radom operators A (1),..., A (K ) o D[0, 1] ad a D[0, 1]-valued radom variable b such that K ( ) b b + A (r) A (r) + 1 {I (r) r=1 0 }A(r) R() R() 0 for all l N, [ ] E 1 (r) {I {0,...,l} {}} A(r) op 0 [ K ] L = lim sup E A () r R(I () r ) op < 1. R() r=1 (A) EXISTENCE AND EQUALITY OF MOMENTS. E[ X ] < for all ad E[X 1 (t)] = E[X (t)] for all 1, N 0, t [0, 1]. Partial match queries: a limit process 1 / 17

Covergece i D[0, 1] by cotractio argumets III. Neiiger ad Sulzbach (011+) (A) EXISTENCE OF A CONTINUOUS SOLUTION. There exists a solutio X of the fixed-poit equatio K X = d A r X (r) + b r=1 with cotiuous paths, E[ X ] < ad E[X(t)] = E[X 1 (t)] for all t [0, 1]. (A) PERTURBATION CONDITION. X = W + h where h h 0 with h D[0, 1] ad radom variables W i D[0, 1] such that there exists a sequece (r ) with, as, P (W / D r [0, 1]) 0. Here, D r [0, 1] D[0, 1] deotes the set of fuctios o the uit iterval, for which there is a decompositio of [0, 1] ito itervals of legth as least r o which they are costat. (A5) RATE OF CONVERGENCE. R() = o ( log m (1/r ) ). Partial match queries: a limit process 1 / 17

Existece of a cotiuous solutio Defie a complete tree T = 0 {1,,, } with (U u, V u), u T, iid uiform o [0, 1] a startig fuctio h(s) = (s(1 s)) β/ a iteratio/mixig operator G : [0, 1] C[0, 1] C[0, 1] G(x, y; f 1, f, f, f )(s) = [ ( s ) ( s )] 1 {s<x} (xy) β f 1 + (x(1 y)) β f x x [ ( ) ( )] s x s x + 1 {s x} ((1 x)y) β f + ((1 x)(1 y)) β f 1 x 1 x For every ode u T, let Lemma Z u 0 = h Z u +1 = G(Uu, Vu; Z u1, Z u, Z u, Z u ) Z = Z, 0, is a o-egative martigale Partial match queries: a limit process 15 / 17

Uiform covergece of the mea Propositio There exists ε > 0 such that sup t β E[P t (s)] µ 1 (s) = O(t ε ). s [0,1] sup t β E[P t (s)] µ 1 (s) sup s [0,1] s δ t β E[P t (s)] µ 1 (s) + sup t β E[P t (s)] µ 1 (s). s (δ,1/] Propositio (Almost mootoicity) For ay s < 1/ ad ε [0, 1 s), we have [ ( )] s + ε E[P t (s)] E P t(1+ε). 1 + ε s ɛ 0 1 Partial match queries: a limit process 16 / 17

Thak you! Partial match queries: a limit process 17 / 17