Lecture 9 November 23, 2015

Transcription

1 CSC244: Discrepancy Theory in Coputer Science Fall 25 Aleksandar Nikolov Lecture 9 Noveber 23, 25 Scribe: Nick Spooner Properties of γ 2 Recall that γ 2 (A) is defined for A R n as follows: γ 2 (A) = in{r(u) c(v ) : UV = A, U R k, V R k n, k N}, where r(u) is the axiu row nor of U, and c(v ) is the axiu colun nor of V We showed in the previous lecture that there exists a constant C such that We give soe other useful properties of γ 2 γ 2 (A) C log rk A herdisc(a) C log γ 2 (A) () Monotonicity γ 2 (A S,T ) γ 2 (A), where A S,T is the subatrix of A whose rows are indexed by S [] and whose coluns are indexed by T [n] 2 Transpose γ 2 (A) = γ 2 (A T ), where A T is the atrix transpose of A (( )) A 3 Diagonal block atrices γ 2 = ax(γ B 2 (A), γ 2 (B)) 4 Triangle inequality γ 2 (A + B) γ 2 (A) + γ 2 (B) 5 Union γ 2 (( A B )) γ2 (A) 2 + γ 2 (B) 2 Most of these properties follow straightforwardly fro the definitions We give a detailed proof of Property 4 Proof of triangle inequality Let U A, V A be such that U A V A = A, and r(u A ) = c(v A ) = γ 2 (A) This can always be achieved siply by scaling the atrices appropriately We take U B, V B siilarly Let U := ( ( ) ) VA U A U B, and V := Then clearly UV = A + B Moreover V B r(u) 2 = ax U i 2 2 = ( ax (UA ) i (U B ) i 2 2) i= i= ax (U A) i ax (U B) i 2 2 = r(u A ) 2 + r(u B ) 2 = γ 2 (A) + γ 2 (B) i= i= The sae inequality holds for c(v ) 2, so γ 2 (A + B) r(a) 2 c(v ) 2 γ 2 (A) + γ 2 (B) Reark Using the bounds in (), we can obtain approxiate versions of the above properties for herdisc

2 Kronecker products For atrices A R p q, B R r s, the Kronecker (tensor) product A B R pr qs is given by the block atrix A B A 2 B A B = A 2 B A 22 B Lea 2 (Property 6) γ 2 (A B) = γ 2 (A)γ 2 (B) Reark 3 This property does not hold for the cobinatorial discrepancy disc(a) Proof We ake use of a basic property of the tensor product: (A B)(C D) = (AC) (BD) Applying this property to the singular value decopositions of A and B, we see that if A has singular values σ,, σ p and B has singular values τ,, τ r then A B has singular values σ τ,, σ τ r,, σ p τ,, σ p τ r First we prove that γ 2 (A B) γ 2 (A)γ 2 (B) We take U A, V A such that A = U A V A and r(u A )c(v A ) = γ 2 (A); siilarly we have U B, V B Let U := U A U B and V := V A V B Then UV = A B, by the basic property of tensor products Moreover we have that r(u) = r(u A )r(u B ), since the rows of U have the for u A u B where u A is a row of A and u B is a row of B, and for any two vectors u, u 2, u u 2 2 = u 2 u 2 2 The sae property holds of the coluns of V and hence of c(v ), and so γ 2 (A B) r(u)c(v ) = (r(u A )c(v A )) (r(u B )c(v B )) = γ 2 (A)γ 2 (B) It reains to show that γ 2 (A B) γ 2 (A)γ 2 (B) For this we ake use of the dual of the seidefinite progra for coputing γ 2 By strong duality, it holds that (cf last lecture) γ 2 (A) = ax{ P AQ tr : P, Q nonnegative diagonal atrices st tr(p 2 ) = tr(q 2 ) = } Let P A, Q A be such that P A AQ A tr = γ 2 (A), and P B, Q B likewise Let P := P A P B and Q = Q A Q B Note that tr((p A P B ) 2 ) = tr(pa 2 P B 2 ) = tr(p A 2) tr(p B 2 ) = by easy properties of the Kronecker product, and the sae holds for Q, hence P, Q is a feasible solution Finally fro the properties of the singular values of Kronecker products, we get P (A B)Q tr = (P A AQ A ) (P B BQ B ) tr = σ i τ j = ( σ i )( τ j ) i j i j = P A AQ A tr P B BQ B tr = γ 2 (A) γ 2 (B) 2 Discrepancy of corners Recall fro Lecture that for y R d we define the corner C(y) := {x R d : x i y i, i =,, d} For d N the set C d := {C(y) : y [, ] d } Let P be a finite subset of [, ] d ; then C d P is the set of subsets of P of the for C(y) P for soe y [, ] d We define the cobinatorial discrepancy 2

3 of C d, disc(n, C d ) := sup P disc(c d P ), where the supreu is taken over subsets P [, ] d of size n Recall also that the continuous discrepancy D(n, C d ) O() disc(n, C d ) Open proble We know that (approxiately) log (d )/2 n D(n, C d ) log d n Can we get a tighter bound? The following theore suggests that better bounds for D(n, C d ) are unlikely to coe fro better bounds for disc(n, C d ) Theore 4 For d N, it holds that Ω(log d n) disc(n, C d ) O(log d+/2 n) Proof sketch Let Q := [n] d [, n] d be the set of d-vectors whose coordinates are positive integers at ost n (Note: we can scale this set into [, ] d so that it fits the definitions above) Let S := C d Q Then S := {[y ] [y d ] : y,, y d [n]}, and the incidence atrix A of S is, the d-wise Kronecker product of the n n lower triangular atrix with s below (and on) the diagonal T d n Proposition 5 γ 2 (T n ) = Θ(log n) Given the proposition, it is not too difficult to show that the upper and lower bounds hold Lower bound By Lea 2, γ 2 (A) = Θ(log d n) Then inequality () gives that herdisc(a) Ω(γ 2 (A)/ log rk A) = Ω(log d n) By the definition of hereditary discrepancy, there exists a subset P Q, P = n, such that the discrepancy disc(s P ) = Ω(log d n) Since S = C d Q, and P Q, clearly S P = C d P, so disc(n, C d ) disc(s P ) = Ω(log d n) Upper bound We ay assue P [n] d, since for any P [, n] d we can transfor it into P [n] d such that P = P and disc(c d P ) disc(c d P ) Then disc(n, C d ) herdisc(s) The γ 2 upper bound gives herdisc(s) = O( log n d )γ 2 (A) = O(log d+/2 n), which concludes the proof Proof of Proposition 5 We show the upper and lower bounds separately γ 2 (T n ) = O(log n) Notice that we can decopose T n as follows: = + Or, written as a su of block atrices, ( ) ( ) Tn/2 T n = + T n/2 (( )) (( )) Tn/2 Note that γ 2 = Fro Property 3 of γ 2, γ 2 = γ T 2 (T n/2 ) n/2 Thus γ 2 (T n ) + γ 2 (T n/2 ) by the triangle inequality, and solving the recurrence gives the upper bound 3

4 2 γ 2 (T n ) = Ω(log n) Here we once again ake use of the dual By the norality conditions on P and Q, for any atrix A R k k we have that P = Q = k I k is a feasible solution, and so ( γ 2 (A) P AQ tr = k A Tn Tn tr Let B := T ) It is not difficult to see that this is a T T n circulant atrix, ie each colun is the previous colun rotated by one row The eigenvalues of such a atrix are the DFT coefficients of its first colun, and so the i-th singular value of B is approxiately n/i, so B tr = Θ(n log n) Then finally which proves the clai T n γ 2 (T n ) 4 γ 2(B) 8n B tr = Ω(log n), 3 Data structure lower bounds 3 Range counting Let d N, point set P R d, and weight function w : P Z We are interested in designing a data structure which supports two operations: Update Given a pair (p, x) P Z, set w(p) := w(p) + x Query Given z R d, return p C(z) P w(p) 32 The oblivious group odel We define a restricted odel of a data structure In this odel, the data structure retains s values y := (y,, y s ) where each y i R is a linear cobination of the w(p) for p P Let U, V be such that UV = A, where A is the incidence atrix of C d P V encodes the linear cobinations of group eleents which are used to copute y: y = V w U encodes the linear cobinations of y i which are used to answer queries; the condition UV = A is necessary for correctness Then our operations are constrained to be of the following for: Update Given a pair (p, x), y := y + xv p, where V p is the p-th colun of V Query Given z, return U i, y, where the i-th row of A is the indicator vector of the set C(z) P The tie coplexity of updates, t u := ax p nnz(v p ), where for a vector u, nnz(u) is defined as the nuber of non-zero entries in u Siilarly the tie coplexity of queries t q := ax z nnz(u z ) The one-diensional case is well-understood Theore 6 (Fredan 82 []) For d =, t u + t q = Ω(log n) For higher diensions, we require an additional paraeter, a bound on the absolute values of entries in U, V Theore 7 (Larsen [2]) For d N, t u t q d 2 Ω(log 2 n)/ 4 4

5 Proof sketch It is not hard to see that r(u) t q, and c(v ) t u Then γ 2 (A) 2 t u t q, so t u t q Ω(log nd ) 2 For any natural data structures, = Indeed for any d N there exists a data structure with = which atches the lower bound In the case d =, the Fredan bound shows that the dependence on is not required, but for d > it reains open whether larger values of allow for ore efficient data structures References [] Michael L Fredan 982 The Coplexity of Maintaining an Array and Coputing Its Partial Sus J ACM 29, (January 982), [2] Larsen, KG 2 On Range Searching in the Group Model and Cobinatorial Discrepancy IEEE 52nd Annual Syposiu on the Foundations of Coputer Science (FOCS), pp