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What This Country Needs is an 18¢ Piece
, 2002
"... We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18¢ piece. ..."
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We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18¢ piece.
Exact Analysis of Exact Change
, 1997
"... We consider the kpayment 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, w ..."
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We consider the kpayment 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 online, 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 kpayment problem has additional applications in other resourcesharing scenarios. Our results include a complete characterization of the kpayment 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 kpayment problem is at least kH N=k , where H n denotes the ...
Exact analysis of exact change: the kpayment problem
 SIAM Journal on Discrete Mathematics
"... Abstract. We introduce the kpayment 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 ..."
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Abstract. We introduce the kpayment 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 online, 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 kpayment problem has additional applications in other resourcesharing scenarios. Our results include a complete characterization of the kpayment 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 kpayment problem is at least kH N/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 minimal number of coins. In the case that all denominations are available, the algorithm finds a coin allocation with at most (k+1)H N/(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.
COMBINATORICS OF THE CHANGEMAKING PROBLEM
, 801
"... Abstract. We investigate the structure of the currencies (systems of coins) for which the greedy changemaking algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such s ..."
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Abstract. We investigate the structure of the currencies (systems of coins) for which the greedy changemaking algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such systems. We also uncover some relations between such currencies and their subcurrencies. 1.
Table of contents Table of figures.................................................................................................................................. III
"... 2. The market basket database used............................................................................................. 9 3. A formal representation of the change making problem...................................................... 11 4. A practical example of payment analysis............... ..."
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2. The market basket database used............................................................................................. 9 3. A formal representation of the change making problem...................................................... 11 4. A practical example of payment analysis................................................................................ 13 5. Managerial implications........................................................................................................... 22
WHAT’S IN YOUR WALLET?!
"... Abstract. We use Markov chains and numerical linear algebra — and several CPU hours — to determine the most likely set of coins in your wallet under reasonable spending assumptions. We also compute a number of additional statistics. In particular, the expected number of coins carried by a person in ..."
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Abstract. We use Markov chains and numerical linear algebra — and several CPU hours — to determine the most likely set of coins in your wallet under reasonable spending assumptions. We also compute a number of additional statistics. In particular, the expected number of coins carried by a person in the United States in our model is 10. 1.
Research and development of fringe projectionbased methods in 3D shape reconstruction
, 2006
"... ..."
Two Topics in Dominance Relations for the Unbounded Knapsack Problem
, 2008
"... On the unbounded knapsack problem, dominance relations play a crucial role to reduce items to be considered in a given instance. This article picks up two topics in dominance relations. One is a connection between dominance relations and polynomially solvable special cases, and the other is on unus ..."
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On the unbounded knapsack problem, dominance relations play a crucial role to reduce items to be considered in a given instance. This article picks up two topics in dominance relations. One is a connection between dominance relations and polynomially solvable special cases, and the other is on unusual dominance relations.