Notes on Bit-reversal Broadcast Scheduling

TL;DR

This paper extends results on RBO, an efficient broadcast scheduling method for battery-powered radios, analyzing its performance, bounds, and protocol complexity for key-based data retrieval in unreliable networks.

Contribution

It provides revised bounds and extended analysis of RBO's efficiency, including performance in unreliable networks and computational complexity of its protocol.

Findings

RBO transmits at most 2k+1 frames outside the key interval before success

Expected effort to receive desired frames is bounded by a function of p and k

Protocol's wake-up time computation is O(k) operations

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.