Abstract
Consider a tree T of depth 2 whose root has s child nodes and the k th child node from left has n k child leaves. Considered as a round-robin tree, T represents a schedule in which each page assigned to a leaf under node k (1≤ k ≤ s) appears with period sn k . By varying s, we want to maximize the total number n = \(\sum_{k=1}^{s}\) n k of pages assigned to the leaves with the following constraints: for 1≤ k≤ s, \(n_{k} = \lfloor(m + \sum_{j=1}^{ k-1}n_{j} )/s\rfloor\), where m is a given integer parameter. This problem arises in the optimization of a video-on-demand scheme, called Fixed-Delay Pagoda Broadcasting.
Due to the floor functions involved, the only known algorithm for finding the optimal s is essentially exhaustive, testing m/2 different potential optimal values of size O(m) for s. Since computing n for a given value of s incurs time O(s), the time complexity of finding the optimal s is O(m 2). This paper analyzes this combinatorial optimization problem in detail and limits the search space for the optimal s down to \(\kappa \sqrt{m}\) different values of size \(O \sqrt{m}\) each, where κ ≈ 0.9, thus improving the time complexity down to O(m).
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© 2005 Springer-Verlag Berlin Heidelberg
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Kameda, T., Sun, Y., Goddyn, L. (2005). An Optimization Problem Related to VoD Broadcasting. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_13
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DOI: https://doi.org/10.1007/11602613_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
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