Source Broadcasting to the Masses: Separation has a Bounded Loss

This work discusses the source broadcasting problem, i.e. transmitting a source to many receivers via a broadcast channel. The optimal rate-distortion region for this problem is unknown. The separation approach divides the problem into two complementary problems: source successive refinement and broadcast channel transmission. We provide bounds on the loss incorporated by applying time-sharing and separation in source broadcasting. If the broadcast channel is degraded, it turns out that separation-based time-sharing achieves at least a factor of the joint source-channel optimal rate, and this factor has a positive limit even if the number of receivers increases to infinity. For the AWGN broadcast channel a better bound is introduced, implying that all achievable joint source-channel schemes have a rate within one bit of the separation-based achievable rate region for two receivers, or within $\log_2 T$ bits for $T$ receivers

Computing π(N)\pi(N): An elementary approach in O~(N)\tilde{O}(\sqrt{N}) time

We present an efficient and elementary algorithm for computing the number of primes up to $N$ in $\tilde{O}(\sqrt N)$ time, improving upon the existing combinatorial methods that require $\tilde{O}(N ^ {2/3})$ time. Our method has a similar time complexity to the analytical approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. While the most time-efficient version of our algorithm requires $\tilde{O}(\sqrt N)$ space, we present a continuous space-time trade-off, showing, e.g., how to reduce the space complexity to $\tilde{O}(\sqrt[3]{N})$ while slightly increasing the time complexity to $\tilde{O}(N^{8/15})$. We apply our techniques to improve the state-of-the-art complexity of elementary algorithms for computing other number-theoretic functions, such as the the Mertens function (in $\tilde{O}(\sqrt N)$ time compared to the known $\tilde{O}(N^{0.6})$), summing Euler's totient function, counting square-free numbers and summing primes. Implementation code is provided