Notes on Bit-reversal Broadcast Scheduling

This report contains revision and extension of some results about RBO from [14]. RBO is a simple and efficient broadcast scheduling of n = 2 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 [κ′, κ′′]. RBO receives at most 2k + 1 other frames’ keys before receiving the first key from [κ′, κ′′] or noticing that there are no such keys in the broadcast cycle. As a simple corollary, 4k + 2 is upper bound the number of keys outside [κ′, κ′′] that will ever be received. In unreliable network the expected number of efforts to receive such frames is bounded by (8k + 4)/p + 2(1 − p)/p, where p is probability of successful reception, and the reception rate of the frames with the keys in [κ′, κ′′] is p – the highest possible. The receiver’s protocol state consists of the values k, κ′ and κ′′, 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.