Algorithmic inference: Difference between revisions

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The '''change-making problem''' addresses the following question: how can a given amount of money be made with the least number of coins of given denominations? It is a [[knapsack problem| knapsack type problem]], and has applications wider than just currency.
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==Mathematical definition==
Given a set of integer coin values {''w''<sub>1</sub>, ''w''<sub>2</sub>, ..., ''w''<sub>''n''</sub>} where ''w''<sub>1</sub> = 1 and ''w''<sub>''j''</sub> < ''w''<sub>''j''+1</sub> for 1 &le; ''j'' &le; ''n''&nbsp;&minus;&nbsp;1, and a positive integer ''W'', find a set of non-negative integers {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>} which minimize
 
: <math>\sum_{j=1}^n x_j</math>
 
subject to
 
: <math>\sum_{j=1}^n w_j x_j = W.</math>
 
==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.
 
==Methods of solving==
 
===Dynamic programming===
Making a list of ways to make each amount from one upwards (an example of a [[dynamic programming]] method)
 
===Linear programming===
[[Linear_programming#Integer_unknowns | Integer Linear Programming]] is often a quick way to solve this kind of problem, but the time it will take to resolve the problem is not certain, and may be slow in some cases
 
===Greedy method===
In the US (and most other) coin systems, a [[greedy algorithm]] of picking the largest denomination of coin which is not greater than the remaining amount to be made will always produce the optimal result. This is not automatically the case, though: if the coin denominations were 1, 3 and 4, then to make 6, the greedy algorithm would choose three coins (4,1,1) whereas the optimal solution is two coins (3,3).
 
== Related problems ==
 
The "optimal [[denomination (currency) | denomination]] problem"<ref>
{{cite journal |author=J. Shallit |title=What this country needs is an 18c piece |journal=[[Mathematical Intelligencer]] |volume=25 |issue=2 |pages=20–23 |doi=10.1007/BF02984830 |url=http://www.cs.uwaterloo.ca/~shallit/Papers/change2.pdf}}
</ref>
is a problem for people who design entirely new currencies:
What denominations should be chosen for the coins
in order to minimize the average cost of making change --
i.e., the average number of coins needed to make change?
The version of this problem assumed that the people making change will use the minimum number of coins
(from the denominations available).
One variation of this problem assumes that the people making change will use the "greedy algorithm" for making change, even when that requires more than the minimum number of coins.
Most current currencies use a [[1-2-5 series]],
but some other set of denominations would require fewer denominations of coins or a smaller average number of coins to make change or both.
 
==See also==
* [[List of knapsack problems]]
* [[Coin problem]]
* [[package-merge algorithm#The coin collector's problem | The coin collector's problem]]
 
==References==
{{reflist}}
* {{cite journal |author=X. Cai |title=Canonical Coin Systems for Change-Making Problems |journal=Proceedings of the Ninth International Conference on Hybrid Intelligent Systems |year=2009 |pages=499–504 |doi=10.1109/HIS.2009.103 |arxiv=0809.0400}}
* {{cite journal |author=M. Adamaszek, A. Niewiarowska |title=Combinatorics of the change-making problem |journal=European Journal of Combinatorics |volume=31 |issue=1 |year=2010 |pages=47–63 |doi=10.1016/j.ejc.2009.05.002 |arxiv=0801.0120}}
 
[[Category:Number theory]]
[[Category:Recreational mathematics]]
[[Category:Combinatorial optimization]]

Latest revision as of 16:11, 29 May 2014

The littler boat is a lot easier to manipulate as well as weigh and coast commercial enterprise. Kayakers have to keep in mind that lower flatwater kayaks need short tracking functions and don't search very fast at the same time. Having said that, these are typically suitable for people please do not prefer to project gold watches considerably (beyond 3 kilometers). All kayaks are really easy to play in comparison with bigger flatwater kayaks, the main factor as part of coast sportfishing.

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