arXiv subjects

Rafail Ostrovsky

Publications and source records attributed to Rafail Ostrovsky.

At least 37 records · Page 2Linked to original sources

arXiv2015

Local Correctability of Expander Codes

In this work, we present the first local-decoding algorithm for expander codes. This yields a new family of constant-rate codes that can recover from a constant fraction of errors in the codeword symbols, and where any symbol of the codeword can be recovered with high probability by reading $N^ε$ symbols from the corrupted codeword, where $N$ is the block-length of the code. Expander codes, introduced by Sipser and Spielman, are formed from an expander graph $G = (V,E)$ of degree $d$, and an inner code of block-length $d$ over an alphabet $Σ$. Each edge of the expander graph is associated with a symbol in $Σ$. A string in $Σ^{E}$ will be a codeword if for each vertex in $V$, the symbols on the adjacent edges form a codeword in the inner code. We show that if the inner code has a smooth reconstruction algorithm in the noiseless setting, then the corresponding expander code has an efficient local-correction algorithm in the noisy setting. Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one. This provides an alternative to the multiplicity codes of Kopparty, Saraf and Yekhanin (STOC '11) and the lifted codes of Guo, Kopparty and Sudan (ITCS '13).

cs.IT↗

arXiv2014

It's Not Easy Being Three: The Approximability of Three-Dimensional Stable Matching Problems

In 1976, Knuth asked if the stable marriage problem (SMP) can be generalized to marriages consisting of 3 genders. In 1988, Alkan showed that the natural generalization of SMP to 3 genders ($3$GSM) need not admit a stable marriage. Three years later, Ng and Hirschberg proved that it is NP-complete to determine if given preferences admit a stable marriage. They further prove an analogous result for the $3$ person stable assignment ($3$PSA) problem. In light of Ng and Hirschberg's NP-hardness result for $3$GSM and $3$PSA, we initiate the study of approximate versions of these problems. In particular, we describe two optimization variants of $3$GSM and $3$PSA: maximally stable marriage/matching (MSM) and maximum stable submarriage/submatching (MSS). We show that both variants are NP-hard to approximate within some fixed constant factor. Conversely, we describe a simple polynomial time algorithm which computes constant factor approximations for the maximally stable marriage and matching problems. Thus both variants of MSM are APX-complete.

cs.CC↗

arXiv2014

On The Communication Complexity of Finding an (Approximate) Stable Marriage

In this paper, we consider the communication complexity of protocols that compute stable matchings. We work within the context of Gale and Shapley's original stable marriage problem\cite{GS62}: $n$ men and $n$ women each privately hold a total and strict ordering on all of the members of the opposite gender. They wish to collaborate in order to find a stable matching---a pairing of the men and women such that no unmatched pair mutually prefer each other to their assigned partners in the matching. We show that any communication protocol (deterministic, nondeterministic, or randomized) that correctly ouputs a stable matching requires $Ω(n^2)$ bits of communication. Thus, the original algorithm of Gale and Shapley is communication-optimal up to a logarithmic factor. We then introduce a "divorce metric" on the set of all matchings, which allows us to consider approximately stable matchings. We describe an efficient algorithm to compute the "distance to stability" of a given matching. We then show that even under the relaxed requirement that a protocol only yield an approximate stable matching, the $Ω(n^2)$ communication lower bound still holds.

cs.CC↗

arXiv2014

Universal Streaming

Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? In other words, is it possible to collect some data in small space during the stream, such that any other statistic that can be computed with comparable space can be computed after the fact? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model.

cs.DS↗

arXiv2014

Improved Approximation Algorithms for Earth-Mover Distance in Data Streams

For two multisets $S$ and $T$ of points in $[Δ]^2$, such that $|S| = |T|= n$, the earth-mover distance (EMD) between $S$ and $T$ is the minimum cost of a perfect bipartite matching with edges between points in $S$ and $T$, i.e., $EMD(S,T) = \min_{π:S\rightarrow T}\sum_{a\in S}||a-π(a)||_1$, where $π$ ranges over all one-to-one mappings. The sketching complexity of approximating earth-mover distance in the two-dimensional grid is mentioned as one of the open problems in the literature. We give two algorithms for computing EMD between two multi-sets when the number of distinct points in one set is a small value $k=\log^{O(1)}(Δn)$. Our first algorithm gives a $(1+ε)$-approximation using $O(kε^{-2}\log^{4}n)$ space and works only in the insertion-only model. The second algorithm gives a $O(\min(k^3,\logΔ))$-approximation using $O(\log^{3}Δ\cdot\log\logΔ\cdot\log n)$-space in the turnstile model.

cs.DS↗

arXiv2013

Secure End-to-End Communication with Optimal Throughput in Unreliable Networks

We demonstrate the feasibility of end-to-end communication in highly unreliable networks. Modeling a network as a graph with vertices representing nodes and edges representing the links between them, we consider two forms of unreliability: unpredictable edge-failures, and deliberate deviation from protocol specifications by corrupt nodes. We present a robust routing protocol for end-to-end communication that is simultaneously resilient to both forms of unreliability. In particular, we prove rigorously that our protocol is SECURE against the actions of the corrupt nodes, achieves correctness (Receiver gets ALL of the messages from Sender, in order and without modification), and enjoys provably optimal throughput performance, as measured using competitive analysis. Furthermore, our protocol does not incur any asymptotic memory overhead as compared to other protocols that are unable to handle malicious interference of corrupt nodes. In particular, our protocol requires O(n^2) memory per processor, where n is the size of the network. This represents an O(n^2) improvement over all existing protocols that have been designed for this network model.

cs.NI↗

arXiv2013

How to Catch L_2-Heavy-Hitters on Sliding Windows

Finding heavy-elements (heavy-hitters) in streaming data is one of the central, and well-understood tasks. Despite the importance of this problem, when considering the sliding windows model of streaming (where elements eventually expire) the problem of finding L_2-heavy elements has remained completely open despite multiple papers and considerable success in finding L_1-heavy elements. In this paper, we develop the first poly-logarithmic-memory algorithm for finding L_2-heavy elements in sliding window model. Since L_2 heavy elements play a central role for many fundamental streaming problems (such as frequency moments), we believe our method would be extremely useful for many sliding-windows algorithms and applications. For example, our technique allows us not only to find L_2-heavy elements, but also heavy elements with respect to any L_p for 0 2 this task is impossible. Our method may have other applications as well. We demonstrate a broader applicability of our novel yet simple method on two additional examples: we show how to obtain a sliding window approximation of other properties such as the similarity of two streams, or the fraction of elements that appear exactly a specified number of times within the window (the rarity problem). In these two illustrative examples of our method, we replace the current expected memory bounds with worst case bounds.

cs.DS↗

arXiv2013

How Hard is Counting Triangles in the Streaming Model

The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of $Ω(m)$ for graphs $G$ with $m$ edges on $n$ vertices. If a constant number of passes is allowed, we show a lower bound of $Ω(m/T)$, $T$ the number of triangles. We match, in some sense, this lower bound with a 2-pass $O(m/T^{1/3})$-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least $T$ triangles. We present a new graph parameter $ρ(G)$ -- the triangle density, and conjecture that the space complexity of the triangles problem is $Ω(m/ρ(G))$. We match this by a second algorithm that solves the distinguishing problem using $O(m/ρ(G))$-memory.

cs.DS↗

arXiv2012

Approximating Large Frequency Moments with Pick-and-Drop Sampling

Given data stream $D = \{p_1,p_2,...,p_m\}$ of size $m$ of numbers from $\{1,..., n\}$, the frequency of $i$ is defined as $f_i = |\{j: p_j = i\}|$. The $k$-th \emph{frequency moment} of $D$ is defined as $F_k = \sum_{i=1}^n f_i^k$. We consider the problem of approximating frequency moments in insertion-only streams for $k\ge 3$. For any constant $c$ we show an $O(n^{1-2/k}\log(n)\log^{(c)}(n))$ upper bound on the space complexity of the problem. Here $\log^{(c)}(n)$ is the iterative $\log$ function. To simplify the presentation, we make the following assumptions: $n$ and $m$ are polynomially far; approximation error $ε$ and parameter $k$ are constants. We observe a natural bijection between streams and special matrices. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a \emph{pick-and-drop sampling}; it samples a heavy element (i.e., element $i$ with frequency $Ω(F_k)$) with probability $Ω(1/n^{1-2/k})$ and gives approximation $\tilde{f_i} \ge (1-ε)f_i$. In addition, the estimations never exceed the real values, that is $ \tilde{f_j} \le f_j$ for all $j$. As a result, we reduce the space complexity of finding a heavy element to $O(n^{1-2/k}\log(n))$ bits. We apply our method of recursive sketches and resolve the problem with $O(n^{1-2/k}\log(n)\log^{(c)}(n))$ bits.

cs.DS↗

arXiv2012

Near-Optimal Radio Use For Wireless Network Synchronization

We consider the model of communication where wireless devices can either switch their radios off to save energy, or switch their radios on and engage in communication. We distill a clean theoretical formulation of this problem of minimizing radio use and present near-optimal solutions. Our base model ignores issues of communication interference, although we also extend the model to handle this requirement. We assume that nodes intend to communicate periodically, or according to some time-based schedule. Clearly, perfectly synchronized devices could switch their radios on for exactly the minimum periods required by their joint schedules. The main challenge in the deployment of wireless networks is to synchronize the devices' schedules, given that their initial schedules may be offset relative to one another (even if their clocks run at the same speed). We significantly improve previous results, and show optimal use of the radio for two processors and near-optimal use of the radio for synchronization of an arbitrary number of processors. In particular, for two processors we prove deterministically matching $Θ(\sqrt{n})$ upper and lower bounds on the number of times the radio has to be on, where $n$ is the discretized uncertainty period of the clock shift between the two processors. (In contrast, all previous results for two processors are randomized.) For $m=n^β$ processors (for any $β< 1$) we prove $Ω(n^{(1-β)/2})$ is the lower bound on the number of times the radio has to be switched on (per processor), and show a nearly matching (in terms of the radio use) $Õ(n^{(1-β)/2})$ randomized upper bound per processor, with failure probability exponentially close to 0. For $β\geq 1$ our algorithm runs with at most $poly-log(n)$ radio invocations per processor. Our bounds also hold in a radio-broadcast model where interference must be taken into account.

cs.DS↗

arXiv2011

Position-Based Quantum Cryptography: Impossibility and Constructions

In this work, we study position-based cryptography in the quantum setting. The aim is to use the geographical position of a party as its only credential. On the negative side, we show that if adversaries are allowed to share an arbitrarily large entangled quantum state, no secure position-verification is possible at all. We show a distributed protocol for computing any unitary operation on a state shared between the different users, using local operations and one round of classical communication. Using this surprising result, we break any position-verification scheme of a very general form. On the positive side, we show that if adversaries do not share any entangled quantum state but can compute arbitrary quantum operations, secure position-verification is achievable. Jointly, these results suggest the interesting question whether secure position-verification is possible in case of a bounded amount of entanglement. Our positive result can be interpreted as resolving this question in the simplest case, where the bound is set to zero. In models where secure positioning is achievable, it has a number of interesting applications. For example, it enables secure communication over an insecure channel without having any pre-shared key, with the guarantee that only a party at a specific location can learn the content of the conversation. More generally, we show that in settings where secure position-verification is achievable, other position-based cryptographic schemes are possible as well, such as secure position-based authentication and position-based key agreement.

quant-ph↗

arXiv2010

Recursive Sketching For Frequency Moments

In a ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to compute $F_k$ (for $k>2$) in space complexity $O(\mbox{\em poly-log}(n,m)\cdot n^{1-\frac2k})$, which is optimal up to (large) poly-logarithmic factors in $n$ and $m$, where $m$ is the length of the stream and $n$ is the upper bound on the number of distinct elements in a stream. The best known lower bound for large moments is $Ω(\log(n)n^{1-\frac2k})$. A follow-up work of Bhuvanagiri, Ganguly, Kesh and Saha (SODA 2006) reduced the poly-logarithmic factors of Indyk and Woodruff to $O(\log^2(m)\cdot (\log n+ \log m)\cdot n^{1-{2\over k}})$. Further reduction of poly-log factors has been an elusive goal since 2006, when Indyk and Woodruff method seemed to hit a natural "barrier." Using our simple recursive sketch, we provide a different yet simple approach to obtain a $O(\log(m)\log(nm)\cdot (\log\log n)^4\cdot n^{1-{2\over k}})$ algorithm for constant $ε$ (our bound is, in fact, somewhat stronger, where the $(\log\log n)$ term can be replaced by any constant number of $\log $ iterations instead of just two or three, thus approaching $log^*n$. Our bound also works for non-constant $ε$ (for details see the body of the paper). Further, our algorithm requires only $4$-wise independence, in contrast to existing methods that use pseudo-random generators for computing large frequency moments.

cs.DS↗

arXiv2010

Rademacher Chaos, Random Eulerian Graphs and The Sparse Johnson-Lindenstrauss Transform

The celebrated dimension reduction lemma of Johnson and Lindenstrauss has numerous computational and other applications. Due to its application in practice, speeding up the computation of a Johnson-Lindenstrauss style dimension reduction is an important question. Recently, Dasgupta, Kumar, and Sarlos (STOC 2010) constructed such a transform that uses a sparse matrix. This is motivated by the desire to speed up the computation when applied to sparse input vectors, a scenario that comes up in applications. The sparsity of their construction was further improved by Kane and Nelson (ArXiv 2010). We improve the previous bound on the number of non-zero entries per column of Kane and Nelson from $O(1/ε\log(1/δ)\log(k/δ))$ (where the target dimension is $k$, the distortion is $1\pm ε$, and the failure probability is $δ$) to $$ O\left({1\overε} \left({\log(1/δ)\log\log\log(1/δ) \over \log\log(1/δ)}\right)^2\right). $$ We also improve the amount of randomness needed to generate the matrix. Our results are obtained by connecting the moments of an order 2 Rademacher chaos to the combinatorial properties of random Eulerian multigraphs. Estimating the chance that a random multigraph is composed of a given number of node-disjoint Eulerian components leads to a new tail bound on the chaos. Our estimates may be of independent interest, and as this part of the argument is decoupled from the analysis of the coefficients of the chaos, we believe that our methods can be useful in the analysis of other chaoses.

cs.DS↗

arXiv2010

Deterministic and Energy-Optimal Wireless Synchronization

We consider the problem of clock synchronization in a wireless setting where processors must power-down their radios in order to save energy. Energy efficiency is a central goal in wireless networks, especially if energy resources are severely limited. In the current setting, the problem is to synchronize clocks of $m$ processors that wake up in arbitrary time points, such that the maximum difference between wake up times is bounded by a positive integer $n$, where time intervals are appropriately discretized. Currently, the best-known results for synchronization for single-hop networks of $m$ processors is a randomized algorithm due to \cite{BKO09} of O(\sqrt {n /m} \cdot poly-log(n)) awake times per processor and a lower bound of Omega(\sqrt{n/m}) of the number of awake times needed per processor \cite{BKO09}. The main open question left in their work is to close the poly-log gap between the upper and the lower bound and to de-randomize their probabilistic construction and eliminate error probability. This is exactly what we do in this paper. That is, we show a {deterministic} algorithm with radio use of Theta(\sqrt {n /m}) that never fails. We stress that our upper bound exactly matches the lower bound proven in \cite{BKO09}, up to a small multiplicative constant. Therefore, our algorithm is {optimal} in terms of energy efficiency and completely resolves a long sequence of works in this area. In order to achieve these results we devise a novel {adaptive} technique that determines the times when devices power their radios on and off. In addition, we prove several lower bounds on the energy efficiency of algorithms for {multi-hop networks}. Specifically, we show that any algorithm for multi-hop networks must have radio use of Omega(\sqrt n) per processor.

cs.DC↗

arXiv2010

Position-Based Quantum Cryptography

This paper is replaced by arXiv:1009.2490. The new paper includes a general impossibility result and restricted possibility results, and it has two additional authors.

quant-ph↗

arXiv2010

AMS Without 4-Wise Independence on Product Domains

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm ε)$ approximation (with probability $1-δ$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$.

cs.DS↗

arXiv2009

Throughput in Asynchronous Networks

We introduce a new, "worst-case" model for an asynchronous communication network and investigate the simplest (yet central) task in this model, namely the feasibility of end-to-end routing. Motivated by the question of how successful a protocol can hope to perform in a network whose reliability is guaranteed by as few assumptions as possible, we combine the main "unreliability" features encountered in network models in the literature, allowing our model to exhibit all of these characteristics simultaneously. In particular, our model captures networks that exhibit the following properties: 1) On-line; 2) Dynamic Topology; 3)Distributed/Local Control 4) Asynchronous Communication; 5) (Polynomially) Bounded Memory; 6) No Minimal Connectivity Assumptions. In the confines of this network, we evaluate throughput performance and prove matching upper and lower bounds. In particular, using competitive analysis (perhaps somewhat surprisingly) we prove that the optimal competitive ratio of any on-line protocol is 1/n (where n is the number of nodes in the network), and then we describe a specific protocol and prove that it is n-competitive. The model we describe in the paper and for which we achieve the above matching upper and lower bounds for throughput represents the "worst-case" network, in that it makes no reliability assumptions. In many practical applications, the optimal competitive ratio of 1/n may be unacceptable, and consequently stronger assumptions must be imposed on the network to improve performance. However, we believe that a fundamental starting point to understanding which assumptions are necessary to impose on a network model, given some desired throughput performance, is to understand what is achievable in the worst case for the simplest task (namely end-to-end routing).

cs.NI↗

arXiv2009

Measuring Independence of Datasets

A data stream model represents setting where approximating pairwise, or $k$-wise, independence with sublinear memory is of considerable importance. In the streaming model the joint distribution is given by a stream of $k$-tuples, with the goal of testing correlations among the components measured over the entire stream. In the streaming model, Indyk and McGregor (SODA 08) recently gave exciting new results for measuring pairwise independence. The Indyk and McGregor methods provide $\log{n}$-approximation under statistical distance between the joint and product distributions in the streaming model. Indyk and McGregor leave, as their main open question, the problem of improving their $\log n$-approximation for the statistical distance metric. In this paper we solve the main open problem posed by of Indyk and McGregor for the statistical distance for pairwise independence and extend this result to any constant $k$. In particular, we present an algorithm that computes an $(ε, δ)$-approximation of the statistical distance between the joint and product distributions defined by a stream of $k$-tuples. Our algorithm requires $O(({1\over ε}\log({nm\over δ}))^{(30+k)^k})$ memory and a single pass over the data stream.

cs.DS↗