Question

I’m reviewing some old notes from my algorithms course and the dynamic programming problems are seeming a bit tricky to me. I have a problem where we have an unlimited supply of coins, with some denominations x1, x2, … xn and we want to make change for some value X. We are trying to design a dynamic program to decide whether change for X can be made or not (not minimizing the number of coins, or returning which coins, just true or false).

I’ve done some thinking about this problem, and I can see a recursive method of doing this where it’s something like…

MakeChange(X, x[1..n this is the coins])
    for (int i = 1; i < n; i++)
    {
        if ( (X - x[i] ==0) || MakeChange(X - x[i]) )
            return true;
    }
    return false;

Converting this a dynamic program is not coming so easily to me. How might I approach this?

Answer 1

Your code is a good start. The usual way to convert a recursive solution to a dynamic-programming one is to do it “bottom-up” instead of “top-down”. That is, if your recursive solution calculates something for a particular X using values for smaller x, then instead calculate the same thing starting at smaller x, and put it in a table.

In your case, change your MakeChange recursive function into a canMakeChange table.

canMakeChange[0] = True
for X = 1 to (your max value):
   canMakeChange[X] = False
   for i=1 to n:
     if X>=x[i] and canMakeChange[X-x[i]]==True: 
       canMakeChange[X]=True