Exact analysis of exact change: The k-payment problem

Boaz Patt-Shamir*, Yiannis Tsiounis, Yair Frankel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 kHN/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.

Original languageEnglish
Pages (from-to)436-453
Number of pages18
JournalSIAM Journal on Discrete Mathematics
Issue number4
StatePublished - 2000


  • Change-making
  • Coin allocation
  • Electronic cash
  • Exact change
  • k-payment problem


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