Towards using the history in online computation with advice

Recently, advice complexity has been introduced as a new framework to analyze online algorithms. There, an online algorithm has access to an infinite binary advice tape during the computation. The contents of this tape were prepared beforehand by an omniscient oracle. One is interested in analyzing the number of accessed advice bits necessary and/or sufficient to achieve a certain solution quality. Among others, the bit guessing problem was analyzed in this framework. Here, an algorithm needs to guess a binary string bit by bit, either with or without getting immediate feedback after each bit. The bit guessing problem can be used to obtain lower bounds on the advice complexity of a variety of other online problems. In this paper, we analyze the difference between the two bit guessing variants. More precisely, we show that getting feedback after each request helps save advice bits when we allow errors to be made. This is by no means obvious - for optimality, both problem versions need the same amount of advice, and without advice, knowing the history does not help at all.