An Optimization Problem Related to VoD Broadcasting

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).