Understanding change-making algorithm

Question

I was looking for a good solution to the Change-making problem and I found this code(Python):

target = 200
coins = [1,2,5,10,20,50,100,200]
ways = [1]+[0]*ta         


        

Answer 1

The main idea behind the code is the following: "On each step there are ways ways to make change of i amount of money given coins [1,...coin]".

So on the first iteration you have only a coin with denomination of 1. I believe it is evident to see that there is only one way to give a change having only these coins for any target. On this step ways array will look like [1,...1] (only one way for all targets from 0 to target).

On the next step we add a coin with denomination of 2 to the previous set of coins. Now we can calculate how many change combinations there are for each target from 0 to target. Obviously, the number of combination will increase only for targets >= 2 (or new coin added, in general case). So for a target equals 2 the number of combinations will be ways[2](old) + ways[0](new). Generally, every time when i equals a new coin introduced we need to add 1 to previous number of combinations, where a new combination will consist only from one coin.

For target > 2, the answer will consist of sum of "all combinations of target amount having all coins less than coin" and "all combinations of coin amount having all coins less than coin itself".

Here I described two basic steps, but I hope it is easy to generalise it.

Let me show you a computational tree for target = 4 and coins=[1,2]:

ways[4] given coins=[1,2] =

ways[4] given coins=[1] + ways[2] given coins=[1,2] =

1 + ways[2] given coins=[1] + ways[0] given coins=[1,2] =

1 + 1 + 1 = 3

So there are three ways to give a change: [1,1,1,1], [1,1,2], [2,2].

The code given above is completely equivalent to the recursive solution. If you understand the recursive solution, I bet you understand the solution given above.