Dynamic Programming Optimal Coin Change

Question

I\'ve been reviewing some dynamic programming problems, and I have had hard time wrapping my head around some code in regards to finding the smallest number of coins to make

Answer 1

Here is a way to think about the coin changing problem that may be useful, if you are comfortable with graph theory.

Assume you have a graph defined in the following way:

• There is one node for every unit of money (e.g., pennies) from 0 up to the value you are interested in (e.g., 39 cents, or whatever.)
• There is one arc between any two nodes separated by exactly the value of a coin you are allowed to use (e.g., a node between 34 cents and 29 cents if you are allowed to use nickels.)

Now you can think of the coin changing problem as a shortest path problem from your value of interest down to zero, because the number of coins will be exactly the same as the number of arcs in your path.

The algorithm doesn't use a graph theory terminology, but it is doing basically the same thing: The outer loop is ranging over all the "cents" (or nodes, in the graph theory framework) and the inner loop is ranging over all the arcs (the values in coinValueList) from the present arc to the next arc. All together, they are looking for the shortest path from zero up to your value of interest. (Value down to zero, zero up to value, doesn't matter. Traditionally we search downward to zero, though.)

I only really started to understand dynamic programming when I realized many problems could be cast as graph problems. (Be careful, though-- not all of them can. Some are hypergraphs, and some are probably not even that. But it helped me a lot.)