Exact Analysis of Exact ChangePreliminary

We consider the k-payment problem: given a total budget of N units, represent this budget as a set of coins, so that any k exact payment requests 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 The actual payments are made on-line, namely without knowing all payment requests in advance. Our motivation is electronic cash, 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 suucient 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 kH N=k , where H n denotes the nth element in the harmonic series. This condition can also be used to eeciently determine k (the maximal number of exact payments) which a given set of coins allows. Secondly, we give an algorithm which produces, for any N and k, a solution with minimal number of coins. In the case that all denominations are available, the algorithm nds a coin allocation with at most (k + 1)H N=(k+1) coins. (Both upper and lower bounds are the best possible.) The algorithm generalizes naturally to the case where some of the denominations are not available.