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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 ..."
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Cited by 19 (4 self)
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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.
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 ..."
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Cited by 13 (7 self)
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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,
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 ..."
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Cited by 6 (0 self)
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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
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 ..."
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Cited by 3 (2 self)
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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 ..."
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Cited by 3 (0 self)
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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
Õ ( √ n) Approximation for Directed Spanners
, 2010
"... We study the computational problem of finding the sparsest kspanner of a given directed graph G, that is, a kspanner with the smallest number of edges. 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) w ..."
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We study the computational problem of finding the sparsest kspanner of a given directed graph G, that is, a kspanner with the smallest number of edges. 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 → N and a stretch k ≥ 1, a subgraph H = (V, EH) is a kspanner of G if for all edges (u, v) ∈ E, the graph H contains a path from u to v of length at most k · d(u, v). We describe an approximation algorithm with O ( √ n log n) ratio. This improves all known approximation algorithms for k> 4 by ˜ Ω(n1/6). Our approach builds on the recent work of Bhattacharyya et al. [3] and Dinitz and Krauthgamer [6]. Namely, we also find a linear programming formulation, round its fractional solution and combine it with random sampling. Moreover, as in [6], our linear program is flowbased. However, our linear program, as well as our rounding scheme, are quite different, allowing us to use a simpler separation oracle and to obtain a better approximation ratio. 1
Limitations of Local Filters of Lipschitz and Monotone Functions
"... We study local filters for two properties of functions f: {0, 1} d → R: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given blackbox access to a function f and a query point x in the domain of f. Its output is a value F (x), such t ..."
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We study local filters for two properties of functions f: {0, 1} d → R: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given blackbox access to a function f and a query point x in the domain of f. Its output is a value F (x), such that (i) the reconstructed function F (x) satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function f satisfies the property, then for every point x in the domain (with high constant probability) the reconstructed value F (x) differs from f(x) by at most a. Local filters were introduced by Saks and Seshadhri (SICOMP 2010) and the relaxed definition we study is due to Bhattacharyya et al. (RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy. We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, i.e., with a = 0) applied only to more restrictive class of filters, e.g., nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a cconnector. Then we present a lower bound on the maximum outdegree of a cconnector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.