We study the large-coalition asymptotics of fingerprinting and group testing and derive explicit decoders that provably achieve capacity for many of the considered models. We do this both for simple decoders (which are fast but commonly require larger code lengths) and for joint decoders (which may be slower but achieve the best code lengths). We further make the distinction between informed decoding, where the pirate strategy is exactly known, and uninformed decoding, and we design decoding schemes for both settings. For fingerprinting, we show that if the pirate strategy is known, the Neyman-Pearson-based log-likelihood decoders provably achieve capacity, regardless of the strategy. The decoder built against the interleaving attack is further shown to be a universal decoder, able to deal with arbitrary attacks and achieving the uninformed capacity. This universal decoder is shown to be closely related to the Lagrange-optimized decoder of and the empirical mutual information decoder of Moulin. Joint decoders are also proposed, and we conjecture that these also achieve the corresponding joint capacities. For group testing, the simple decoder for the classical model is shown to be more efficient than the one of and it provably achieves the simple group testing capacity. For generalizations of this model such as noisy group testing, the resulting simple decoders also achieve the corresponding simple capacities.
Work on the fingerprinting game described above started in the late 90s, and lower bounds on the code length were established of the order @math @cite_25 , until in 2003 Tardos @cite_8 proved a lower bound of the order @math and described a scheme with @math , showing this bound is tight. Since the leading constants of the upper and lower bounds did not match, later work on fingerprinting focused on finding the optimal leading constant.
@cite_25 @cite_8
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We construct binary codes for fingerprinting digital documents. Our codes for n users that are e-secure against c pirates have length O(c2log(n e)). This improves the codes proposed by Boneh and Shaw l1998r whose length is approximately the square of this length. The improvement carries over to works using the Boneh--Shaw code as a primitive, for example, to the dynamic traitor tracing scheme of Tassa l2005r. By proving matching lower bounds we establish that the length of our codes is best within a constant factor for reasonable error probabilities. This lower bound generalizes the bound found independently by l2003r that applies to a limited class of codes. Our results also imply that randomized fingerprint codes over a binary alphabet are as powerful as over an arbitrary alphabet and the equal strength of two distinct models for fingerprinting. We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they globally minimize diameter. For the local-average-distance version, we prove an upper bound of @math , a lower bound of 3, and a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the local-diameter version, we prove a lower bound of @math and a tight upper bound of 3 for trees. The same bounds apply, up to constant factors, to the price of anarchy. Our network creation games are closely related to the previously studied unilateral network creation game. The main difference is that our model has no parameter @math for the link creation cost, so our results effectively apply for all values of @math without additional effort; furthe... This work provides tight upper- and lower-bounds for the problem of mean estimation under @math -differential privacy in the local model, when the input is composed of @math i.i.d. drawn samples from a normal distribution with variance @math . Our algorithms result in a @math -confidence interval for the underlying distribution's mean @math of length @math . In addition, our algorithms leverage binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, we prove a matching lower-bound (up to poly-log factors), showing that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length @math . [See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the @math -Approximate Near Neighbor Search problem. For the @math -dimensional Euclidean space and @math -point datasets, we develop a data structure with space @math and query time @math for every @math such that: This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor @math , improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for @math , improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes. For any real number @math , we nearly completely characterize the space complexity of estimating @math for @math matrices @math in which each row and each column has @math non-zero entries and whose entries are presented one at a time in a data stream model. Here the @math are the singular values of @math , and when @math , @math is the @math -th power of the Schatten @math -norm. We show that when @math is not an even integer, to obtain a @math -approximation to @math with constant probability, any @math -pass algorithm requires @math bits of space, where @math as @math and @math is a constant independent of @math . However, when @math is an even integer, we give an upper bound of @math bits of space, which holds even in the turnstile data stream model. The latter is optimal up to @math factors. Our results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices @math : the previous best lower bound was @math for @math , @math for @math and @math for @math . We note for @math , while our lower bound for even integers is the same, for other @math in this range our lower bound is @math , which is considerably stronger than the previous @math for small enough constant @math . We obtain similar near-linear lower bounds for Ky-Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to our work. The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. In an index coding with side-information (ICSI) problem there is a sender with a set of independent messages @math and there are receivers @math , each identified with ( @math ), where @math is the set of messages wanted by receiver @math and @math is the set of messages known a priori to receiver @math . We call a ICSI problem to be single-prior if @math In addition, if @math it is called a single-uniprior problem and has been studied in detail by Ong, Ho and Lim OHL . Error Correcting index codes (ECIC) have been studied by Dau, Skachek and Chee DSC and lower and upper bounds on the optimal length of ECICs have been reported. In this paper we show that for single-prior ICSI problems the lower and upper bounds on the optimal ECICs coincide. This leads to construction of optimal length linear ECICs. This paper considers the problem of variable-length coding over a discrete memoryless channel with noiseless feedback. This paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence, termed extrinsic Jensen–Shannon (EJS) divergence. It is shown that strictly positive lower bounds on EJS divergence provide nonasymptotic upper bounds on the expected code length. This paper presents strictly positive lower bounds on EJS divergence, and hence nonasymptotic upper bounds on the expected code length, for the following two coding schemes: 1) variable-length posterior matching and 2) MaxEJS coding scheme that is based on a greedy maximization of the EJS divergence. As an asymptotic corollary of the main results, this paper also provides a rate–reliability test. Variable-length coding schemes that satisfy the condition(s) of the test for parameters @math and @math are guaranteed to achieve a rate @math and an error exponent @math . The results are specialized for posterior matching and MaxEJS to obtain deterministic one-phase coding schemes achieving capacity and optimal error exponent. For the special case of symmetric binary-input channels, simpler deterministic schemes of optimal performance are proposed and analyzed. A range counting problem is specified by a set @math of size @math of points in @math , an integer weight @math associated to each point @math , and a range space @math . Given a query range @math , the target output is @math . Range counting for different range spaces is a central problem in Computational Geometry. We study @math -differentially private algorithms for range counting. Our main results are for the range space given by hyperplanes, that is, the halfspace counting problem. We present an @math -differentially private algorithm for halfspace counting in @math dimensions which achieves @math average squared error. This contrasts with the @math lower bound established by the classical result of Dinur and Nissim [PODS 2003] for arbitrary subset counting queries. We also show a matching lower bound on average squared error for any @math -differentially private algorithm for halfspace counting. Both bounds are obtained using discrepancy theory. For the lower bound, we use a modified discrepancy measure and bound approximation of @math -differentially private algorithms for range counting queries in terms of this discrepancy. We also relate the modified discrepancy measure to classical combinatorial discrepancy, which allows us to exploit known discrepancy lower bounds. This approach also yields a lower bound of @math for @math -differentially private orthogonal range counting in @math dimensions, the first known superconstant lower bound for this problem. For the upper bound, we use an approach inspired by partial coloring methods for proving discrepancy upper bounds, and obtain @math -differentially private algorithms for range counting with polynomially bounded shatter function range spaces. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math Q @math $ . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. We prove lower bounds on the complexity of finding @math -stationary points (points @math such that @math ) of smooth, high-dimensional, and potentially non-convex functions @math . We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of @math at a query point @math . We show that for any (potentially randomized) algorithm @math , there exists a function @math with Lipschitz @math th order derivatives such that @math requires at least @math queries to find an @math -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized @math th order regularization are worst-case optimal within their natural function classes. We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs. The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math . We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of @math nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level @math . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of estimating the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a trade-off between approximation and communication, asserting, for example, that protocols with @math bits of communication can obtain only approximation @math , where @math is the length of the input strings. This case of @math communication is of particular importance since it captures constant-size sketches as well as embeddings into spaces like @math and squared- @math , two prevailing algorithmic approaches for dealing with edit distance. Indeed, the known nontrivial communication upper bounds are all derived from embeddings into @math . By excluding low-communication protocols for edit distance, we rule out a strictly richer class of algorithms than previous results. Furthermore, our lower bound holds not only for strings over a binary alphabet but also for strings that are permutations (aka the Ulam metric). For this case, our bound nearly matches an upper bound known via embedding the Ulam metric into @math . Our proof uses a new technique that relies on Fourier analysis in a rather elementary way. We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position, needs code length @math Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies @math This implies that several known PIR schemes are close to optimal. Our results generalize those of [8], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [12], our classical bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs. We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized @math lower bound per operation for several data structural problems on @math elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of @math for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first @math lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update. We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an @math lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [, SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show @math lower bounds for computing @math -approximations of the diameter or the radius, and for computing a @math -approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [, SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of @math -spanners, for @math . A general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices is given. This technique is used to prove (i) A tight bound of @math for the two-dimensional torus; (ii) A tight bound of @math for trees with maximum degree @math ; (iii) Tight bounds of @math for rapidly mixing walks on vertex transitive graphs, where @math denotes the maximum expected hitting time between vertices.In addition to these new results, the technique allows several known lower bounds on cover times to be systematically proved, often in a much simpler way.Finally, a different technique is used to prove an @math lower bound on the cover time, where @math is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [... A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any zero-error @math -query locally correctable code @math that can correct a constant fraction of corrupted symbols must have @math . We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability @math when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error. Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was @math due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield @math -server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a @math -server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error @math -query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet. For our proof of the result, we need a new decomposition lemma for directed graphs that may be of independent interest. Given a dense directed graph @math , our decomposition uses the directed version of Szemeredi regularity lemma due to Alon and Shapira (STOC 2003) to partition almost all of @math into a constant number of subgraphs which are either edge-expanding or empty.