Change-making problem

The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. It is a special case of the integer knapsack problem, and has applications wider than just currency.

It is also the most common variation of the coin change problem, a general case of partition in which, given the available denominations of an infinite set of coins, the objective is to find out the number of possible ways of making a change for a specific amount of money, without considering the order of the coins.

It is weakly NP-hard, but may be solved optimally in pseudo-polynomial time by dynamic programming.

Mathematical definition

Coin values can be modeled by a set of n distinct positive integer values (whole numbers), arranged in increasing order as w1 through wn. The problem is: given an amount W, also a positive integer, to find a set of non-negative (positive or zero) integers {x1, x2, ..., xn}, with each xj representing how often the coin with value wj is used, which minimize the total number of coins f(W)

f ( W ) = ∑ j = 1 n x j {\displaystyle f(W)=\sum _{j=1}^{n}x_{j}}

subject to

∑ j = 1 n w j x j = W . {\displaystyle \sum _{j=1}^{n}w_{j}x_{j}=W.}

Non-currency examples

An application of change-making problem can be found in computing the ways one can make a nine dart finish in a game of darts.

Another application is computing the possible atomic (or isotopic) composition of a given mass/charge peak in mass spectrometry.

Methods of solving

Simple dynamic programming

A classic dynamic programming strategy works upward by finding the combinations of all smaller values that would sum to the current threshold. Thus, at each threshold, all previous thresholds are potentially considered to work upward to the goal amount W. For this reason, this dynamic programming approach requires a number of steps that is O(nW), where n is the number of types of coins.

Implementation

The following is a dynamic programming implementation (with Python 3) which uses a matrix to keep track of the optimal solutions to sub-problems, and returns the minimum number of coins, or "Infinity" if there is no way to make change with the coins given. A second matrix may be used to obtain the set of coins for the optimal solution.

Greedy method

For many real-world coin systems, such as those used in the US and many other countries, a greedy algorithm of picking the largest denomination of coin which is not greater than the remaining amount to be made will produce the optimal result. This is not the case for arbitrary coin systems or even some real world systems, though. For instance, if we consider the old (now withdrawn) Indian coin denominations of 5, 10, 20 and 25 paise, then to make 40 paise, the greedy algorithm would choose three coins (25, 10, 5) whereas the optimal solution is two coins (20, 20). Another example is attempting to make 40 US cents without nickels (denomination 25, 10, 1) with similar result — the greedy chooses seven coins (25, 10, and 5 × 1), but the optimal is four (4 × 10). A coin system is called "canonical" if the greedy algorithm always solves its change-making problem optimally. It is possible to test whether a coin system is canonical in polynomial time.