Results 1  10
of
35
Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 57 (2011)
, 2011
"... A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) In other words, Lipschitz functions are not very sensitive to small changes in the input. We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are ɛfar from having the property, that is, differ from every function with the property on at least an ɛ fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f. We consider functions over domains {0, 1} d, {1,..., n} and {1,..., n} d, equipped with ℓ1 distance.
Monotonicity testing and shortestpath routing on the cube
 In Proceedings of the Fourteenth International Workshop on Randomization and Computation (RANDOM
, 2010
"... We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any ℓ disjoint sourcesin ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any ℓ disjoint sourcesink pairs on the directed hypercube can be connected with edgedisjoint paths, then monotonicity of functions f: {0, 1} n → R can be tested with O(n) queries, for any totally ordered range R. More generally, if at least an α(n) fraction of the pairs can always be connected with edgedisjoint paths then the querycomplexity is O(n/α(n)). We construct a family of instances of ℓ = Ω(2n) pairs in ndimensional hypercubes such that no more than roughly a 1 √ fraction of the pairs can be simultaneously connected with n edgedisjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving querycomplexity upper bounds from routing properties cannot yield, by itself, querycomplexity bounds better than ≈ n3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture on routing in the hypercube. In particular, we show that for any δ> 0, the ndimensional hypercube is not n 1 2 −δrealizable with shortest paths, while previously it was only known that hypercubes are not 1realizable with shortest paths. We also prove a lower bound of Ω(n/ɛ) queries for onesided nonadaptive testing of monotonicity over the ndimensional hypercube, as well as additional bounds for specific classes of functions and testers.
Lower bounds for local monotonicity reconstruction from transitiveclosure spanners
, 2010
"... Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. Transitiveclosure spanners are a common abstraction for applications in access contr ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
(Show Context)
Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. Transitiveclosure spanners are a common abstraction for applications in access control, property testing and data structures. We show a connection between 2TCspanners and local monotonicity reconstructors. A local monotonicity reconstructor, introduced by Saks and Seshadhri (SIAM Journal on Computing, 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f: [m] d → R, can quickly evaluate a related function g: [m] d → R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. We show that an efficient local monotonicity reconstructor implies a sparse 2TCspanner of the directed hypergrid (hypercube), providing a new technique for proving lower bounds for local monotonicity reconstructors. Our connection is,
Local Monotonicity Reconstruction
"... We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] d = {1,..., n} d (where d is viewed as a constant). We would like to closel ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
We investigate the problem of monotonicity reconstruction, as defined by Ailon, Chazelle, Comandur and Liu (2004) in a localized setting. We have oracle access to a nonnegative realvalued function f defined on the domain [n] d = {1,..., n} d (where d is viewed as a constant). We would like to closely approximate f by a monotone function g. This should be done by a procedure (a filter) that given as input a point x ∈ [n] d outputs the value of g(x), and runs in time that is polylogarithmic in n. The procedure can (indeed must) be randomized, but we require that all of the randomness be specified in advance by a single short random seed. We construct such an implementation where the time and space per query is (log n) O(1) and the size of the seed is polynomial in log n and d. Furthermore, with high probability, the ratio of the (Hamming) distance between g and f to the minimum possible Hamming distance between a monotone function and f is bounded above by a function of d (independent of n). This allows for a local implementation: one can initialize many copies of the filter with the same short random seed, and they can autonomously handle queries, while producing outputs that are consistent with the same approximating function g.
Optimal bounds for monotonicity and Lipschitz testing over the hypercube and hypergrids
, 2012
"... The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f: [k]n 7 → R (for some ordered range R). The hypergrid/cube has a natural partial order given by coordinatewise ordering, denoted by ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f: [k]n 7 → R (for some ordered range R). The hypergrid/cube has a natural partial order given by coordinatewise ordering, denoted by ≺. A function is monotone if for all pairs x ≺ y, f(x) ≤ f(y). The distance to monotonicity, εf, is the minimum fraction of values of f that need to be changed to make f monotone. For k = 2 (the boolean hypercube), the usual tester is the edge tester, which checks monotonicity on adjacent pairs of domain points. It is known that the edge tester using O(ε−1n log R) samples can distinguish a monotone function from one where εf> ε. On the other hand, the best lower bound for monotonicity testing over general R is Ω(n). We resolve this long standing open problem and prove that O(n/ε) samples suffice for the edge tester. For hypergrids, known testers require O(ε−1n log k log R) samples, while the best known (nonadaptive) lower bound is Ω(ε−1n log k). We give a (nonadaptive) monotonicity tester for hypergrids running in O(ε−1n log k) time. Our techniques lead to optimal property testers (with the same running time) for the natural Lipschitz property on hypercubes and hypergrids. (A cLipschitz function is one where f(x) − f(y)  ≤ c‖x − y‖1.) In fact, we give a general unified proof for O(ε−1n log k)query testers for a class of “boundedderivative ” properties, a class containing both monotonicity and Lipschitz. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complex
Extensions and Limits to Vertex Sparsification
"... Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, EH) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, EH) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(logk/loglogk)factor where K  = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, EH) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in [22]. We also prove an Ω(loglogk) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. Our proof crucially relies on a geometric phenomenon pertaining to the unit congestion polytope of an expander graph, which we exploit in order to prove suboptimal but superconstant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flowtype problems designed by a black box reduction to a "uniform " case on k nodes (see [22] for examples) must incur a superconstant cost in the approximation ratio.
Balancing degree, diameter and weight in Euclidean spanners
 In Proc. of 18th ESA
, 2010
"... Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
(Show Context)
Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) edges and diameter α(n), where α stands for the inverseAckermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(log n)andweightO(log n)w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+ɛ)spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n log n)w(MST(S)), and O(n) edges.Fork = O(1) this gives rise to maximum degree O(1), diameter O(log n) andweightO(log 2 n)w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n 1/α(n) (log n)α(n))w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log 2 n, but the diameter is allowed to grow beyond log n.
Directed Spanners via FlowBased Linear Programs
, 2011
"... We examine directed spanners through flowbased linear programming relaxations. We design an Õ(n2/3)approximation algorithm for the directed kspanner problem that works for all k ≥ 1, which is the first sublinear approximation for arbitrary edgelengths. Even in the more restricted setting of unit ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
We examine directed spanners through flowbased linear programming relaxations. We design an Õ(n2/3)approximation algorithm for the directed kspanner problem that works for all k ≥ 1, which is the first sublinear approximation for arbitrary edgelengths. Even in the more restricted setting of unit edgelengths, our algorithm improves over the previous Õ(n1−1/k) approximation [BGJ + 09] when k ≥ 4. For the special case of k = 3 we design a different algorithm achieving an Õ(√n)approximation, improving the previous Õ(n 2/3) [EP05, BGJ + 09] (independently of our work, an Õ(n 1−1/⌈k/2 ⌉ ) was recently devised [BRR10]). Both of our algorithms easily extend to the faulttolerant setting, which has recently attracted attention but not from an approximation viewpoint. We also prove a nearly matching integrality gap of ˜ Ω(n 1/3−ɛ) for every constant ɛ> 0. A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flowbased relaxation, and show how to approximately solve it even when its size is not polynomial. The main challenge is to design a rounding scheme that “coordinates ” the choices of flowpaths between the many demand pairs while using few edges overall. We achieve this, roughly speaking, by randomization at the level of vertices.
TransitiveClosure Spanners: A Survey
"... We survey results on transitiveclosure spanners and their applications. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. These spanner ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We survey results on transitiveclosure spanners and their applications. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. These spanners were studied implicitly in different areas of computer science, and properties of these spanners have been rediscovered over the span of 20 years. The common task implicitly tackled in these diverse applications can be abstracted as the problem of constructing sparse TCspanners. In this article, we survey combinatorial bounds on the size of sparsest TCspanners, and algorithms and inapproximability results for the problem of computing the sparsest TCspanner of a given directed graph. We also describe multiple applications of TCspanners, including property testing, property reconstruction, key management in access control hierarchies and data structures.