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Question: There are n libraries L1, L2, . . . , Ln. We want to store copies of a book in some of the libraries. Storing a copy at Li incurs an integer purchase cost ci > 0. A copy of the book is always stored at Ln. If a user requests the book from Li and Li does not have it, then Li+1, Li+2, . . . are searched sequentially until a copy of the book is found at Lj , for some j > i. This results in a user delay of j - i. (Note that, in this case, no library Lk with an index k smaller than i is searched; also, if the user finds the book at Li , then the user delay is 0.) We define the total cost as the sum of all the purchase costs and all the user delays, assuming that there is one user in each of the n libraries. For example, if there are 4 libraries, and copies of the book are stored at L1 and L4, the total cost is c1 + c4 + 2 + 1. Give an efficient algorithm that returns (a) the minimum total cost; and (b) a set of libraries where copies of the books must be placed to achieve the minimum total cost.
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