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- W2071166712 abstract "We introduce the k-payment problem: given a total budget of N units, the problem is to represent this budget as a set of coins, so that any k exact payments of total value at most N can be made using k disjoint subsets of the coins. The goal is to minimize the number of coins for any given N and k, while allowing the actual payments to be made on-line, namely without the need to know all payment requests in advance. The problem is motivated by the electronic cash model, where each coin is a long bit sequence, and typical electronic wallets have only limited storage capacity. The k-payment problem has additional applications in other resource-sharing scenarios.Our results include a complete characterization of the k-payment problem as follows. First, we prove a necessary and sufficient condition for a given set of coins to solve the problem. Using this characterization, we prove that the number of coins in any solution to the k-payment problem is at least k HN/k, where Hn denotes the nth element in the harmonic series. This condition can also be used to efficiently determine k (the maximal number of exact payments) which a given set of coins allows in the worst case. Secondly, we give an algorithm which produces, for any N and k, a solution with a minimal number of coins. In the case that all denominations are available, the algorithm finds a coin allocation with at most (k+1)HN/(k+1) coins. (Both upper and lower bounds are the best possible.) Finally, we show how to generalize the algorithm to the case where some of the denominations are not available." @default.
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- W2071166712 date "2000-01-01" @default.
- W2071166712 modified "2023-09-25" @default.
- W2071166712 title "Exact Analysis of Exact Change: Thek-Payment Problem" @default.
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- W2071166712 doi "https://doi.org/10.1137/s0895480197318118" @default.
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