Abstract
An online algorithm is typically unaware of the length of the input request sequence that it is called upon. Consequently, it cannot determine whether it has already processed most of its input or whether the bulk of work is still ahead.
In this paper, we are interested in whether some sort of orientation within the request sequence is nevertheless possible. Our objective is to preemptively guess the center of a request sequence of unknown length n: While processing the input, the online algorithm maintains a guess for the central position 
and is only allowed to update its guess to the position of the current element under investigation. We show that there is a randomized algorithm that in expectation places the guess at a distance of 0.172n from the central position 
, and we prove that this is best possible. We also give upper and lower bounds for a natural extension to weighted sequences.
This problem has an application to preemptively partitioning integer sequences and is connected to the online bidding problem.
C. Konrad is supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1. T. Tonoyan is supported by grants no. 152679-05 and 174484-05 from the Icelandic Research Fund.
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Notes
- 1.
An exception are online algorithms with advice, where the online algorithm receives additional input bits prior to the processing of the request sequence. These bits can be used to encode the input length. See [1] for a recent survey.
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Konrad, C., Tonoyan, T. (2018). Preemptively Guessing the Center. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_24
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