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Shamir's scheme for sharing secrets is closely related to Reed-Solomon coding schemes. Decoding algorithms for Reed-Solomon codes provide extensions and generalizations of Shamir's method.

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motivated in part by practical considerations. Among the many areas treated in the recent literature are sorting [2,3,7,12,15], the evaluation of arithmetic expressions, linear recurrences and polynomials [4,8,10], matrix algorithms [5,6,13], and graph theory [9,14,15]. In this paper we present a parallel algorithm CONNECT which determines the connected components of an undirected graph with n vertices in time O(log2n) using n 2 processors. Next, we modify the algorithm to demonstrate an observation due to F.P. Preparata and R.L. Probert, viz., n[n/lg n] processors 1 suffice to achieve a time bound of O(log2n). Finally, we show that CONNECT can be modified to compute the transitive closure of an nxn symmetric Boolean matrix in time O(log2n) using n[n/lg n] processors. We use the single instruction stream-multiple data stream (SIMD) model of parallel processors. It is assumed that the processors have access to a common memory, and that simultaneous access to the same location is permitted for fetch instructions but not for store instructions. The Algorithm ConnectWe present a parallel algorithm which uses n 2 processors to find the connected components of an undirected graph with n vertices in time O(log2n). An O0og2n) time bound also can be achieved using only n[n/[log2n]] processors. The algorithm can he used to find the transitive closure of a symmetric Boolean matrix. We assume that the processors have access to a common memory. Simultaneous access to the same location is permitted for fetch instructions but not for store instructions.

A complete digital audio amplifier has been developed, implemented and tested. The process is entirely computational, and the output load and filter are the only analog components in the system. The process makes use of digital signal processing and a switching power stage to provide both high fidelity and high efficiency, beginning with a digital audio data stream. The advantages of naturally-sampled pulse-width modulation (PWM) are discussed in depth, including spectral analysis and comparisons to uniformly-sampled PWM. It is shown that natural PWM does not introduce audible distortion at switching frequencies consistent with power electronics practice. Interpolation methods for sample data conversion to natural PWM are discussed, and error analysis is presented based on Lagrange's Expansion Theorem. Noise-shaping processes are used to support high fidelity with practical values of time resolution. A counter conversion process enforces switching dead time in the inverter gate signals. The experimental full-bridge inverter implementation demonstrates that miniaturization is possible. A complete test system delivered more than 50 W into an 8 load with an efficiency of 80% and total harmonic distortion plus noise of 0.02%. Index Terms-Class D amplifier, digital audio amplifier, noiseshaping, pulse-width modulation, uniform and natural sampling. NOMENCLATURE Differences between actual crossing times and noise-shaper output. Duty ratio. Generalized frequency. Carrier waveform (switching) frequency. Low-pass filter cutoff frequency. Signal bandlimit frequency.. Integer cycle number; frequency index. Smallest positive integer that gives .