What you are looking for is Dynamic Programming.
You don't actually have to enumerate all the possible combinations for every possible values, because you can build it on top of previous answers.
You algorithm need to take 2 parameters:
- The list of possible coin values, here
[1, 5, 10, 25]
- The range to cover, here
[1, 99]
And the goal is to compute the minimal set of coins required for this range.
The simplest way is to proceed in a bottom-up fashion:
Range Number of coins (in the minimal set)
1 5 10 25
[1,1] 1
[1,2] 2
[1,3] 3
[1,4] 4
[1,5] 5
[1,5]* 4 1 * two solutions here
[1,6] 4 1
[1,9] 4 1
[1,10] 5 1 * experience tells us it's not the most viable one :p
[1,10] 4 2 * not so viable either
[1,10] 4 1 1
[1,11] 4 1 1
[1,19] 4 1 1
[1,20] 5 1 1 * not viable (in the long run)
[1,20] 4 2 1 * not viable (in the long run)
[1,20] 4 1 2
It is somewhat easy, at each step we can proceed by adding at most one coin, we just need to know where. This boils down to the fact that the range [x,y] is included in [x,y+1] thus the minimal set for [x,y+1] should include the minimal set for [x,y].
As you may have noticed though, sometimes there are indecisions, ie multiple sets have the same number of coins. In this case, it can only be decided later on which one should be discarded.
It should be possible to improve its running time, when noticing that adding a coin usually allows you to cover a far greater range that the one you added it for, I think.
For example, note that:
[1,5] 4*1 1*5
[1,9] 4*1 1*5
we add a nickel to cover [1,5] but this gives us up to [1,9] for free!
However, when dealing with outrageous input sets [2,3,5,10,25] to cover [2,99], I am unsure as how to check quickly the range covered by the new set, or it would be actually more efficient.