Results 1  10
of
13
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.
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
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.
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.
Improved Approximation for the Directed Spanner Problem
, 2011
"... We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonne ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d: E → R ≥0 and a stretch k ≥ 1, a subgraph H = (V, EH) is a kspanner of G if for every edge (s, t) ∈ E, the graph H contains a path from s to t of length at most k · d(s, t). The previous best approximation ratio was Õ(n 2/3), due to Dinitz and Krauthgamer (STOC ’11). We also improve the approximation ratio for the important special case of directed 3spanners with unit edge lengths from Õ ( √ n) to O(n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer’s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly
Finding Sparser Directed Spanners
"... A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, a subgraph H = (V, EH) is a kspanner of a graph G = (V, E) if for every pair of vertices u, v ∈ V, the shortest path distance distH(u, v) from u to v in H is at most k · distG(u, ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, a subgraph H = (V, EH) is a kspanner of a graph G = (V, E) if for every pair of vertices u, v ∈ V, the shortest path distance distH(u, v) from u to v in H is at most k · distG(u, v). We focus on spanners of directed graphs and a related notion of transitiveclosure spanners. The latter captures the idea that a spanner should have a small diameter but preserve the connectivity of the original graph. We study the computational problem of finding the sparsest kspanner (resp., kTCspanner) of a given directed graph, which we refer to as Directed kSpanner (resp., kTCSpanner). We improve all known approximation algorithms for Directed kSpanner and kTCSpanner for k ≥ 3. (For k = 2, the current ratios are tight, assuming P̸=NP.) Along the way, we prove several structural results about the size of the sparsest spanners of directed graphs.
Steiner TransitiveClosure Spanners Of Lowdimensional Posets
"... Given a directed graph G = (V, E) and an integer k ≥ 1, a Steiner ktransitiveclosurespanner (Steiner kTCspanner) of G is a directed graph H = (VH, EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V, the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwis ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Given a directed graph G = (V, E) and an integer k ≥ 1, a Steiner ktransitiveclosurespanner (Steiner kTCspanner) of G is a directed graph H = (VH, EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V, the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TCspanners of directed acyclic graphs or, equivalently, partially ordered sets. We study the relationship between the dimension of a poset and the size, denoted Sk, of its sparsest Steiner kTCspanner. We present a nearly tight lower bound on S2 for ddimensional directed hypergrids. Our bound is derived from an explicit dual solution to a linear programming relaxation of the 2TCspanner problem. We also give an efficient construction of Steiner 2TCspanners, of size matching the lower bound, for all lowdimensional posets. Finally, we present a nearly tight lower bound on Sk for ddimensional posets.
Approximation Algorithms for Spanner Problems and Directed Steiner Forest
, 2013
"... We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonne ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We present an O ( √ n log n)approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d: E → R ≥0 and a stretch k ≥ 1, a subgraph H = (V, EH) is a kspanner of G if for every edge (s, t) ∈ E, the graph H contains a path from s to t of length at most k · d(s, t). The previous best approximation ratio was Õ(n 2/3), due to Dinitz and Krauthgamer (STOC ’11). We also improve the approximation ratio for the important special case of directed 3spanners with unit edge lengths from Õ ( √ n) to O(n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer’s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly