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<html>
<head>
<title>
CHANGE_MAKING - Choosing Coins To Match a Given Value
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055" >
<h1 align = "center">
CHANGE_MAKING <br> Choosing Coins To Match a Given Value
</h1>
<hr>
<p>
<b>CHANGE_MAKING</b>
a C++ library which
considers the change making problem,
in which a given sum is to be formed using coins of various denominations.
</p>
<p>
The change making problem is given a target value C and a set of N coin
denominations with value W. The task is to determine the minimum number
of coins needed to form the given value.
</p>
<p>
In the unbounded change making problem, there are a limitless supply of
coins of each denomination. In the bounded change making problem, each
denomination is available only up to some given limit.
</p>
<p>
For some sets of denominations, there will be target values that cannot
be formed. (This relates to Frobenius's problem.)
</p>
<p>
A greedy algorithm for solving the change making problem repeatedly selects
the largest coin denomination available that does not exceed the remainder.
A greedy algorithm is simple, but it is not guaranteed to find a solution
when one exists, and it is not guaranteed to find a minimal solution.
</p>
<p>
For certain sets of coin denominations, such as the US system of 1, 5, 10,
25, 50, 100, there is always a solution, and the greedy algorithm will always
find the minimal solution.
</p>
<p>
For a set of coins such as the old British system, using 1, 2, 6, 12, 24, 48
and 60, there is always a solution, but the greedy algorithm will not always
find the minimal solution.
</p>
<p>
For artificial systems such as 20, 25, 40, it is easy to see that there are many
cases where there is no solution (you can't make ANY value that isn't a multiple
of 5) but that the greedy will fail to find a solution for 50, even though that
is easily formed as 25 + 25.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>CHANGE_MAKING</b> is available in
<a href = "../../c_src/change_making/change_making.html">a C version</a> and
<a href = "../../cpp_src/change_making/change_making.html">a C++ version</a> and
<a href = "../../f_src/change_making/change_making.html">a FORTRAN90 version</a> and
<a href = "../../f77_src/change_making/change_making.html">a FORTRAN77 version</a> and
<a href = "../../m_src/change_making/change_making.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/change_making/change_making.html">
CHANGE_MAKING</a>,
a dataset directory which
contains test data for the change making problem;
</p>
<p>
<a href = "../../cpp_src/combination_lock/combination_lock.html">
COMBINATION_LOCK</a>,
a C++ program which
simulates the process of determining the secret combination of a lock.
</p>
<p>
<a href = "../../cpp_src/combo/combo.html">
COMBO</a>,
a C++ library which
includes many combinatorial routines.
</p>
<p>
<a href = "../../cpp_src/knapsack_01/knapsack_01.html">
KNAPSACK_01</a>,
a C++ library which
uses brute force to solve small versions of the 0/1 knapsack problem;
</p>
<p>
<a href = "../../cpp_src/partition_problem/partition_problem.html">
PARTITION_PROBLEM</a>,
a C++ library which
seeks solutions of the partition problem, splitting a set of integers into
two subsets with equal sum.
</p>
<p>
<a href = "../../cpp_src/satisfy/satisfy.html">
SATISFY</a>,
a C++ program which
demonstrates, for a particular circuit, an exhaustive search
for solutions of the circuit satisfiability problem.
</p>
<p>
<a href = "../../cpp_src/subset/subset.html">
SUBSET</a>,
a C++ library which
enumerates, generates, ranks and unranks combinatorial objects
including combinations, partitions, subsets, index sets, and trees.
</p>
<p>
<a href = "../../cpp_src/subset_sum/subset_sum.html">
SUBSET_SUM</a>,
a C++ library which
seeks solutions of the subset sum problem, in which it is desired
to find a subset of a set of integers which has a given sum.
</p>
<p>
<a href = "../../cpp_src/tsp_brute/tsp_brute.html">
TSP_BRUTE</a>,
a C++ program which
reads a file of city-to-city distances and solves the
traveling salesperson problem, using brute force.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "change_making.cpp">change_making.cpp</a>, the source code.
</li>
<li>
<a href = "change_making.hpp">change_making.hpp</a>, the includd file.
</li>
<li>
<a href = "change_making.sh">change_making.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "change_making_prb.cpp">change_making_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "change_making_prb.sh">change_making_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "change_making_prb_output.txt">change_making_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHANGE_MAKING_LIST</b> solves the change making problem.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 21 August 2014.
</i>
<!-- John Burkardt -->
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