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The Robinson-Foulds (RF) metric is the measure most widely used in comparing phylogenetic trees; it can be computed in linear time using Day's algorithm. When faced with the need to compare large numbers of large trees, however, even linear time becomes prohibitive. We present a randomized approximation scheme that provides, in sublinear time and with high probability, a (1 + epsilon) approximation of the true RF metric. Our approach is to use a sublinear-space embedding of the trees, combined with an application of the Johnson-Lindenstrauss lemma to approximate vector norms very rapidly. We complement our algorithm by presenting an efficient embedding procedure, thereby resolving an open issue from the preliminary version of this paper. We have also improved the performance of Day's (exact) algorithm in practice by using techniques discovered while implementing our approximation scheme. Indeed, we give a unified framework for edge-based tree algorithms in which implementation tradeoffs are clear. Finally, we present detailed experimental results illustrating the precision and running-time tradeoffs as well as demonstrating the speed of our approach. Our new implementation, FastRF, is available as an open-source tool for phylogenetic analysis.

High-frequency needle ultrasound transducers with an aperture size of 0.4 mm were fabricated using lead magnesium niobate-lead titanate (PMN-33% PT) as the active piezoelectric material. The active element was bonded to a conductive silver particle matching layer and a conductive epoxy backing through direct contact curing. An outer matching layer of parylene was formed by vapor deposition. The active element was housed within a polyimide tube and a 20-gauge needle housing. The magnitude and phase of the electrical impedance of the transducer were 47 omega and -38 degrees, respectively. The measured center frequency and -6 dB fractional bandwidth of the PMN-PT needle transducer were 44 MHz and 45%, respectively. The two-way insertion loss was approximately 15 dB. In vivo high-frequency, pulsed-wave Doppler patterns of blood flow in the posterior portion and in vitro ultrasonic backscatter microscope (UBM) images of the rabbit eye were obtained with the 44-MHz needle transducer.

The development of a high frequency (> 50 MHz) annular array ultrasonic transducer is presented. The array was constructed by bonding a 9 microm P(VDF-TrFE) film to a two-sided polyimide flexible circuit with annuli electrodes on the top layer. Each annulus was separated by a 30 microm kerf and had several electroplated microvias that connected to electrode traces on the bottom side of the flex circuit. In order to improve device sensitivity, each element was electrically matched to an impedance magnitude of 50 omega and 0 degrees phase at resonance using a serial inductor and high impedance coaxial cable. The array's performance was evaluated by measuring the electrical impedance, pulse echo response, and cross talk between elements. The average round trip insertion loss was -33.5 dB after compensating for diffractive and attenuative losses. The measured average center frequency and bandwidth for an element was 55 MHz and 47%, respectively. The measured cross talk between adjacent elements remained below -29 dB at the center frequency in water. A vertical wire phantom was imaged using a single focus transmit beamformer and dynamic focusing receive beamformer. This image showed a significant improvement in lateral resolution over a range of 9 mm after the dynamic focusing receive algorithm was applied. These results correlated well with predictions from a Field II simulation. After beamforming, the minimum lateral resolution achieved by the array (-6 dB) was 108 microm at the focus.