On the On-line Number of Snacks Problem |
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Authors: | Weimin Ma Jane You Yinfeng Xu James Liu Kanliang Wang |
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Affiliation: | (1) School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China, 710049;(2) Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
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Abstract: | In the number of snacks problem (NSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as : CA()/COPT(), where denotes a sequence of numbers of customers, COPT() is the cost of satisfying by an optimal off-line algorithm, and CA() is the cost of satisfying by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm(ENHA), with competitive ratio 1+p(M-m)/(M+m), where M and m are two extreme numbers of customers over the total period of the game, and p is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP. |
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Keywords: | On-line number of snacks problem Competitive algorithms Competitive ratio |
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