Notes on Bit-reversal Broadcast Scheduling

Abstract: This report contains revision and extension of some results about RBO [arXiv:1108.5095]. RBO is a simple and efficient broadcast scheduling of $n = 2^k$ uniform frames for battery powered radio receivers. Each frame contains a key from some arbitrary linearly ordered universe. The broadcast cycle -- a sequence of frames sorted by the keys and permuted by $k$-bit reversal -- is transmitted in a round robin fashion by the broadcaster. At arbitrary time during the transmission, the receiver may start a simple protocol that reports to him all the frames with the keys that are contained in a specified interval of the key values $[K', K"]$. RBO receives at most $2 k + 1$ other frames' keys before receiving the first key from $[K', K"]$ or noticing that there are no such keys in the broadcast cycle. As a simple corollary, $4 k + 2$ is upper bound the number of keys outside $[K', K"]$ that will ever be received. In unreliable network the expected number of efforts to receive such frames is bounded by $(8 k + 4) / p + 2 (1 - p) / p^2$, where $p$ is probability of successful reception, and the reception rate of the requested frames is $p$ -- the highest possible. The receiver's protocol state consists of the values $k$, $K'$ and $K"$, one wake-up timer and two other $k$-bit variables. Its only nontrivial computation -- the computation of the next wake-up time slot -- can be performed in $O (k)$ simple operations, such as arithmetic/bit-wise operations on $k$-bit numbers, using only constant number of $k$-bit variables.