Hard Properties with (Very) Short PCPPs and Their Applications

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ?, we construct a property P^(?)? {0,1}^n satisfying the following: Any testing algorithm for P^(?) requires ?(n) many queries, and yet P^(?) has a constant query PCPP whose proof size is O(n?log^(?)n), where log^(?) denotes the ? times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n?polylog(n)). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ?, we construct a property that has a constant-query tester, but requires ?(n/log^(?)(n)) queries for every tolerant or erasure-resilient tester

Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time \tildeO(\sqrt{N}), where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity \tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter \e. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter \e. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree Graphs, One-Sided vs Two-Sided Error Probability Updated versio

External Sampling

36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part IWe initiate the study of sublinear-time algorithms in the external memory model [1]. In this model, the data is stored in blocks of a certain size B, and the algorithm is charged a unit cost for each block access. This model is well-studied, since it reflects the computational issues occurring when the (massive) input is stored on a disk. Since each block access operates on B data elements in parallel, many problems have external memory algorithms whose number of block accesses is only a small fraction (e.g. 1/B) of their main memory complexity. However, to the best of our knowledge, no such reduction in complexity is known for any sublinear-time algorithm. One plausible explanation is that the vast majority of sublinear-time algorithms use random sampling and thus exhibit no locality of reference. This state of affairs is quite unfortunate, since both sublinear-time algorithms and the external memory model are important approaches to dealing with massive data sets, and ideally they should be combined to achieve best performance. In this paper we show that such combination is indeed possible. In particular, we consider three well-studied problems: testing of distinctness, uniformity and identity of an empirical distribution induced by data. For these problems we show random-sampling-based algorithms whose number of block accesses is up to a factor of 1/√B smaller than the main memory complexity of those problems. We also show that this improvement is optimal for those problems. Since these problems are natural primitives for a number of sampling-based algorithms for other problems, our tools improve the external memory complexity of other problems as well.David & Lucile Packard Foundation (Fellowship)Center for Massive Data Algorithmics (MADALGO)Marie Curie (International Reintegration Grant 231077)National Science Foundation (U.S.) (Grant 0514771)National Science Foundation (U.S.) (Grant 0728645)National Science Foundation (U.S.) (Grant 0732334)Symantec Research Labs (Research Fellowship

Testing formula satisfaction

We study the query complexity of testing for properties defined by read once formulae, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in \epsilon and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from satisfying the property. For formulae only involving And/Or gates, we provide a more efficient test whose query complexity is only quasi-polynomial in \epsilon. On the other hand we show that such testability results do not hold in general for formulae over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over alphabets of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size

Synchronization of chaotic nonlinear optical ring oscillators

Abstract We find, by a numerical study, that coupled chaotic nonlinear optical ring cavities, described by delay-differential equations, can show synchronization. This can be useful for encryption in optical communication. The study of chaos has been attracting and fascinating scientists in the last three decades. Although from practical and applicative points of view, chaos with its complicated dynamics, seems mostly useless or at least a state which any system should be avoided from, a number of recent works have been done, in which dynamical systems were designed to take advantage of the chaotic state. Ott, Grebogi, and Yorke [l] showed that by time-dependent perturbations of a parameter in the system, it can be controlled to stay in one of the periodic orbits that compose the chaotic attractor. This method can be used as a stabilization mechanism for unstable chaotic systems; in optics, specifically, a control of laser chaos was reported in Refs. [2] and ]3]. The fact that a system in a chaotic regime is unpredictable, and at the same time deterministic, can also be utilized. Pecora and Carroll [4] showed that in certain circumstances, two chaotic systems can be synchronized to produce the same chaotic signal. This synchronization can be used for encryption of information in private communication channels. In their scheme the synchronization is achieved in the following way: A nonlinear chaotic system is divided into two parts. One part is replicated to produce a non-autonomous second system driven by the corresponding signals in the parent (Master) system. It was shown [4] that the two subsystems become asymptotically synchronized if all of the conditional Lyapunov exponents of the replicated subsystem are negative. Such a scheme can be used for encryption by using the signals of the slave subsystem in the transmission area as a keying code, while sending the driving signals of the master system to the receiving part. When synchronization is achieved, the signals in the receiving area are used for the decoding process and recovering the original information. The synchronization scheme and its use for private communication can be shown i

Can You Solve Closest String Faster than Exhaustive Search?

We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X \subseteq \Sigma^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $\bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $\Sigma^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-\epsilon) d} poly(nd))$, for any $\epsilon > 0$, unless the Strong Exponential Time Hypothesis fails. $\bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 \pm o(1)}$ whenever the dimension is $\omega(\log n) < d < n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$

Customer satisfaction survey pilot project : Charles Towne Landing State Historic Site

This details the South Carolina Department of Parks, Recreation and Tourism's program that proactively seeks to establish an internal process to measure and report external customer satisfaction and loyalty on an on-going basis, in compliance with Malcolm Baldrige criteria at Charles Towne Landing State Historic Site

Large core-area erbium-doped fiber laser

Abstract Ž 2 . A large core-area 804 mm erbium-doped fiber laser is demonstrated. The laser power showed oscillatory or spiky behavior at a stable frequency, which, we believe, results from relaxation oscillation, seeded by noises in the high erbium concentration gain medium. q 1998 Elsevier Science B.V

Global value chains and regional systems of innovation: Towards a critical juncture?

Over recent years, the world has witnessed unexpected challenges - including the COVID-19 pandemic and significant geopolitical tensions. These events have had substantial impacts on both Global Value Chains and Regional Innovation Systems – two complementary analytical scopes that compose the complex geography of innovation. This has led governments to take drastic measures on different fronts and scholars to argue about the surging of a phase of de-globalization in which Global Value Chains are being transformed and restructured, potentially altering the geography of economic activity that has been forged over the last decades. It is uncertain how countries, regions, firms and individuals will respond to multifaceted crises and productive rearrangements, which ones will be more resilient and better capable of doing so than others. In this introduction to the Special Issue “Global Value Chains and Regional Systems of Innovation: Towards a Critical Juncture?” we discuss the local-global dynamics of innovation and propose a critical appraisal on how key contextual parameters have changed, on the one hand, and the potential outcomes of these shifts, on the other. We outline pressing issues for debate among scholars, policymakers and practitioners as well as offer elements to begin a discussion on the critical junctures that lay ahead. We also present the insightful articles that compose this Special Issue