One-side Energy costs of the RBO receiver

Let $n = 2^k$ be the length of the broadcast cycle of the RBO broadcast scheduling protocol (see [arXiv:1108.5095] and [arXiv:1201.3318]). Let $lb$ and $ub$ be the variables of the RBO receiver as defined in [ arXiv:1201.3318 ]. We show that the number of changes of $lb$ (the "left-side energy") is not greater than $k + 1$. We also show that the number of changes of $rb$ (the "right-side energy") is not greater than $k + 2$. Thus the "extra energy" (defined in [arXiv:1201.3318]) is bounded by $2 k + 3$. This updates the previous bound from [arXiv:1201.3318], which was $4 k + 2$.