An Optimal Algorithm for the Distinct Elements Problem

We give the first optimal algorithm for estimating the number of distinct elements in a data stream, closing a long line of theoretical research on this problem begun by Flajolet and Martin in their seminal paper in FOCS 1983. This problem has applications to query optimization, Internet routing, network topology, and data mining. For a stream of indices in {1,...,n}, our algorithm computes a $(1 \pm \epsilon)$-approximation using an optimal $O(1/\epsilon^{-2} + log(n))$ bits of space with 2/3 success probability, where $0<\epsilon<1$ is given. This probability can be amplified by independent repetition. Furthermore, our algorithm processes each stream update in O(1) worst-case time, and can report an estimate at any point midstream in O(1) worst-case time, thus settling both the space and time complexities simultaneously. We also give an algorithm to estimate the Hamming norm of a stream, a generalization of the number of distinct elements, which is useful in data cleaning, packet tracing, and database auditing. Our algorithm uses nearly optimal space, and has optimal O(1) update and reporting times.Engineering and Applied Science

On the Exact Space Complexity of Sketching and Streaming Small Norms

We settle the 1-pass space complexity of $(1 \pm \epsilon)$-approximating the $L_p$ norm, for real p with 1 ≤ p ≤ 2, of a length-n vector updated in a length-m stream with updates to its coordinates. We assume the updates are integers in the range [–M, M]. In particular, we show the space required is $\Theta(\epsilon^{−2} log(mM) + log log(n))$ bits. Our result also holds for 0 < p < 1; although $L_p$ is not a norm in this case, it remains a well-defined function. Our upper bound improves upon previous algorithms of [Indyk, JACM ‘06] and [Li, SODA ‘08]. This improvement comes from showing an improved derandomization of the $L_p$ sketch of Indyk by using k-wise independence for small k, as opposed to using the heavy hammer of a generic pseudorandom generator against space-bounded computation such as Nisan's PRG. Our lower bound improves upon previous work of [Alon-Matias-Szegedy, JCSS ‘99] and [Woodruff, SODA ‘04], and is based on showing a direct sum property for the 1-way communication of the gap-Hamming problem.Engineering and Applied Science

Fast Moment Estimation in Data Streams in Optimal Space

We give a space-optimal streaming algorithm with update time $O(log^2(1/\epsilon)loglog(1/\epsilon))$ for approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream up to a factor of $1 \pm \epsilon$. This provides a nearly exponential improvement over the previous space optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time $\Omega(1/\epsilon^2)$. When combined with the work of [Harvey-Nelson-Onak, FOCS 2008], we also obtain the first algorithm for entropy estimation in turnstile streams which simultaneously achieves near-optimal space and fast update time.Engineering and Applied Science

Detecting a Light Gravitino at Linear Collider to Probe the SUSY Breaking Scale

If supersymmetry is dynamically broken at a low scale ($M_{susy}$), within a few orders of magnitude of the weak scale, then the lightest supersymmetric partner is the gravitino and the next to lightest supersymmetric partner is a neutralino $\chi^0_1$ with mass $m_{\chi^0_1}$, which can decay into a photon ($\gamma$) plus a gravitino ($\widetilde{G}$). We study the detection of $e^{-}e^{+}\rightarrow \chi^0_1 \chi^0_1 \rightarrow \gamma\widetilde{G}\gamma\widetilde{G}$ at the proposed Linear Collider, and find the range of the parameters $M_{susy}$ and $m_{\chi^0_1}$ that can be accessible with a right-hand polarized electron beam at $\sqrt{S}=500$\,GeV, with 50\,${\rm fb}^{-1}$ integrated luminosity. We also discuss briefly the accessible range for current electron and hadron colliders.Comment: 20 pages, LaTeX file and postscript figure

On the Stellar Companion to the Exoplanet Hosting Star 30 Arietis B

A crucial aspect of understanding planet formation is determining the binarity of the host stars. Results from radial velocity (RV) surveys and the follow-up of Kepler exoplanet candidates have demonstrated that stellar binarity certainly does not exclude the presence of planets in stable orbits and the configuration may in fact be relatively common. Here we present new results for the 30 Arietis system which confirms that the B component hosts both planetary and stellar companions. Keck AO imaging provides direct detection of the stellar companion and additional RV data are consistent with an orbiting star. We present a revised orbit of the known planet along with photometry during predicted transit times. Finally, we provide constraints on the properties of the stellar companion based on orbital stability considerations

Adverse events related to ultrasound-guided regional anesthesia performed by Emergency Physicians: Systematic review protocol

The use of ultrasound-guided regional anesthesia for pain management has become increasingly prevalent in Emergency Medicine, with studies noting excellent pain control while sparing opioid use. However, the use of ultrasound-guided regional anesthesia may be hampered by concern about risks for patient harm. This systematic review protocol describes our approach to evaluate the incidence of adverse events from the use of ultrasound-guided regional anesthesia by Emergency Physicians as described in the literature. This project will also seek to document the scope of ultrasound-guided regional anesthesia applications being performed in Emergency Medicine literature, and potentially serve as a framework for future systematic reviews evaluating adverse events in Emergency Medicine

Physical Optimization of Quantum Error Correction Circuits

Quantum error correcting codes have been developed to protect a quantum computer from decoherence due to a noisy environment. In this paper, we present two methods for optimizing the physical implementation of such error correction schemes. First, we discuss an optimal quantum circuit implementation of the smallest error-correcting code (the three bit code). Quantum circuits are physically implemented by serial pulses, i.e. by switching on and off external parameters in the Hamiltonian one after another. In contrast to this, we introduce a new parallel switching method that allows faster gate operation by switching all external parameters simultaneously. These two methods are applied to electron spins in coupled quantum dots subject to a Heisenberg coupling H=J(t) S_1*S_2 which can generate the universal quantum gate `square-root-of-swap'. Using parallel pulses, the encoding for three-bit quantum error correction in a Heisenberg system can be accelerated by a factor of about two. We point out that parallel switching has potential applications for arbitrary quantum computer architectures.Comment: 13 pages, 6 figure