On Constructing Spanners from Random Gaussian Projections

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work

Orthogonal Vectors Indexing

In the recent years, intensive research work has been dedicated to prove conditional lower bounds in order to reveal the inner structure of the class P. These conditional lower bounds are based on many popular conjectures on well-studied problems. One of the most heavily used conjectures is the celebrated Strong Exponential Time Hypothesis (SETH). It turns out that conditional hardness proved based on SETH goes, in many cases, through an intermediate problem - the Orthogonal Vectors (OV) problem. Almost all research work regarding conditional lower bound was concentrated on time complexity. Very little attention was directed toward space complexity. In a recent work, Goldstein et al.[WADS \u2717] set the stage for proving conditional lower bounds regarding space and its interplay with time. In this spirit, it is tempting to investigate the space complexity of a data structure variant of OV which is called OV indexing. In this problem n boolean vectors of size clogn are given for preprocessing. As a query, a vector v is given and we are required to verify if there is an input vector that is orthogonal to it or not. This OV indexing problem is interesting in its own, but it also likely to have strong implications on problems known to be conditionally hard, in terms of time complexity, based on OV. Having this in mind, we study OV indexing in this paper from many aspects. We give some space-efficient algorithms for the problem, show a tradeoff between space and query time, describe how to solve its reporting variant, shed light on an interesting connection between this problem and the well-studied SetDisjointness problem and demonstrate how it can be solved more efficiently on random input

1st doctoral symposium of the international conference on software language engineering (SLE) : collected research abstracts, October 11, 2010, Eindhoven, The Netherlands

The first Doctoral Symposium to be organised by the series of International Conferences on Software Language Engineering (SLE) will be held on October 11, 2010 in Eindhoven, as part of the 3rd instance of SLE. This conference series aims to integrate the different sub-communities of the software-language engineering community to foster cross-fertilisation and strengthen research overall. The Doctoral Symposium at SLE 2010 aims to contribute towards these goals by providing a forum for both early and late-stage Ph.D. students to present their research and get detailed feedback and advice from researchers both in and out of their particular research area. Consequently, the main objectives of this event are: – to give Ph.D. students an opportunity to write about and present their research; – to provide Ph.D. students with constructive feedback from their peers and from established researchers in their own and in different SLE sub-communities; – to build bridges for potential research collaboration; and – to foster integrated thinking about SLE challenges across sub-communities. All Ph.D. students participating in the Doctoral Symposium submitted an extended abstract describing their doctoral research. Based on a good set of submisssions we were able to accept 13 submissions for participation in the Doctoral Symposium. These proceedings present final revised versions of these accepted research abstracts. We are particularly happy to note that submissions to the Doctoral Symposium covered a wide range of SLE topics drawn from all SLE sub-communities. In selecting submissions for the Doctoral Symposium, we were supported by the members of the Doctoral-Symposium Selection Committee (SC), representing senior researchers from all areas of the SLE community.We would like to thank them for their substantial effort, without which this Doctoral Symposium would not have been possible. Throughout, they have provided reviews that go beyond the normal format of a review being extra careful in pointing out potential areas of improvement of the research or its presentation. Hopefully, these reviews themselves will already contribute substantially towards the goals of the symposium and help students improve and advance their work. Furthermore, all submitting students were also asked to provide two reviews for other submissions. The members of the SC went out of their way to comment on the quality of these reviews helping students improve their reviewing skills

1st doctoral symposium of the international conference on software language engineering (SLE) : collected research abstracts, October 11, 2010, Eindhoven, The Netherlands

The first Doctoral Symposium to be organised by the series of International Conferences on Software Language Engineering (SLE) will be held on October 11, 2010 in Eindhoven, as part of the 3rd instance of SLE. This conference series aims to integrate the different sub-communities of the software-language engineering community to foster cross-fertilisation and strengthen research overall. The Doctoral Symposium at SLE 2010 aims to contribute towards these goals by providing a forum for both early and late-stage Ph.D. students to present their research and get detailed feedback and advice from researchers both in and out of their particular research area. Consequently, the main objectives of this event are: – to give Ph.D. students an opportunity to write about and present their research; – to provide Ph.D. students with constructive feedback from their peers and from established researchers in their own and in different SLE sub-communities; – to build bridges for potential research collaboration; and – to foster integrated thinking about SLE challenges across sub-communities. All Ph.D. students participating in the Doctoral Symposium submitted an extended abstract describing their doctoral research. Based on a good set of submisssions we were able to accept 13 submissions for participation in the Doctoral Symposium. These proceedings present final revised versions of these accepted research abstracts. We are particularly happy to note that submissions to the Doctoral Symposium covered a wide range of SLE topics drawn from all SLE sub-communities. In selecting submissions for the Doctoral Symposium, we were supported by the members of the Doctoral-Symposium Selection Committee (SC), representing senior researchers from all areas of the SLE community.We would like to thank them for their substantial effort, without which this Doctoral Symposium would not have been possible. Throughout, they have provided reviews that go beyond the normal format of a review being extra careful in pointing out potential areas of improvement of the research or its presentation. Hopefully, these reviews themselves will already contribute substantially towards the goals of the symposium and help students improve and advance their work. Furthermore, all submitting students were also asked to provide two reviews for other submissions. The members of the SC went out of their way to comment on the quality of these reviews helping students improve their reviewing skills

Cryptography from Information Loss

© Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod. Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former.1 The subject of this work is “lossy” reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and powerful consequences for lifting hardness into “useful” hardness, namely cryptography. Our first, conceptual, contribution is a definition of lossy reductions in the language of mutual information. Roughly speaking, our definition says that a reduction C is t-lossy if, for any distribution X over its inputs, the mutual information I(X; C(X)) ≤ t. Our treatment generalizes a variety of seemingly related but distinct notions such as worst-case to average-case reductions, randomized encodings (Ishai and Kushilevitz, FOCS 2000), homomorphic computations (Gentry, STOC 2009), and instance compression (Harnik and Naor, FOCS 2006). We then proceed to show several consequences of lossy reductions: 1. We say that a language L has an f-reduction to a language L0 for a Boolean function f if there is a (randomized) polynomial-time algorithm C that takes an m-tuple of strings X = (x1, . . ., xm), with each xi ∈ {0, 1}n, and outputs a string z such that with high probability, L0(z) = f(L(x1), L(x2), . . ., L(xm)) Suppose a language L has an f-reduction C to L0 that is t-lossy. Our first result is that one-way functions exist if L is worst-case hard and one of the following conditions holds: f is the OR function, t ≤ m/100, and L0 is the same as L f is the Majority function, and t ≤ m/100 f is the OR function, t ≤ O(m log n), and the reduction has no error This improves on the implications that follow from combining (Drucker, FOCS 2012) with (Ostrovsky and Wigderson, ISTCS 1993) that result in auxiliary-input one-way functions. 2. Our second result is about the stronger notion of t-compressing f-reductions – reductions that only output t bits. We show that if there is an average-case hard language L that has a t-compressing Majority reduction to some language for t = m/100, then there exist collision-resistant hash functions. This improves on the result of (Harnik and Naor, STOC 2006), whose starting point is a cryptographic primitive (namely, one-way functions) rather than average-case hardness, and whose assumption is a compressing OR-reduction of SAT (which is now known to be false unless the polynomial hierarchy collapses). Along the way, we define a non-standard one-sided notion of average-case hardness, which is the notion of hardness used in the second result above, that may be of independent interest