I am trying to write an algorithm for the Change-making problem.
I am assuming the coin system is canonical, which means the greedy algorithm of always choosing the largest coin that is less than the remaining amount is an optimal algorithm.
coinChange is a function to algorithm. The inputs are the amount of money and the denominations for the given coin system. The function outputs the minimum number of coins that are required for each denomination.
For example:
Input : coinChange 34 [1, 5, 10, 25, 50, 100]
Output : [4,1,0,1,0,0]
The type declaration for coinChange is:
coinChange :: Integer -> [Integer] -> [Integer]
I found this solution and I understand why it works:
coinChange 0 ds = take (length ds) (repeat 0)
coinChange c ds = (coinChange (c `rem` d) (init ds) ) ++ [c `quot` d]
where d = last ds
However, I got several error messages when I tried to make my own solution.
At first, I tried:
coinChange c [] = []
coinChange c ds = reverse ( (c `quot` x): coinChange (c `rem` x) xs )
where (x:xs) = reverse ds
I tried reversing the order of the coin denominations, so I start with the largest first. I thought this would be more convenient than using last and init.
When I run this I get:
coinChange 64 [1,2,5,10,20,50,100]
[64,0,0,0,0,0,0]
It seems strange that I get 64 here.
If I am using a recursive function, I think it is confusing to use reverse on the entire list, as parts of the list will be reversed multiple times. So I tried this:
coinChange c [] = []
coinChange c ds =reverse( (c `quot` x):(coinChange' (c `rem` x) xs ))
where (x:xs) = (reverse ds)
coinChange' c [] = []
coinChange' c ds = (c`quot`x):(coinChange' (c `rem` x) xs )
where (x:xs) = reverse ds
If I try:
coinChange 64 [1,2,5,10,20,50,100]
I thought I would get [0,2,0,1,0,1,0] as my final output:
(64 `quot` 100):(coinChange' (64 `rem` 100) [1,2,5,10,20,50])
0:((64`quot`50):coinChange' (64`rem`50) [1,2,5,10,20])
0:(1:coinChange' 14 [1,2,5,10,20])
0:(1:((14`quot` 20):coinChange' (14 `rem` 20) [1,2,5,10]))
0:(1:(0:coinChange 14 [1,2,5,10]))
0:(1:(0:((14`quot` 10) : coinChange' (14 `rem` 10) [1,2,5])))
0:(1:(0:( 1 : coinChange' (14 `rem` 10) [1,2,5])))
0:(1:(0:( 1 : coinChange' 4 [1,2,5])))
0:(1:(0:( 1 : ((4 `quot` 5):coinChange' (4`rem` 5) [1,2]))))
0:(1:(0:( 1 : (0:coinChange' 4 [1,2]))))
0:(1:(0:( 1 : (0:((4`quot`2):coinChange (4`rem`2) [1])))))
0:(1:(0:( 1 : (0:(2:coinChange 0 [1])))))
0:(1:(0:( 1 : (0:(2:((0`quot`1):coinChange' (0`rem` 1) []))))))
0:(1:(0:( 1 : (0:(2:(0:coinChange' 0 []))))))
0:(1:(0:( 1 : (0:(2:(0:[]))))))
reverse [0,1,0,1,0,2,0]
final output: [0,2,0,1,0,1,0]
However, when I run this code in the terminal, I get:
coinChange 64 [1,2,5,10,20,50,100]
[0,0,0,0,0,64,0]
Why does my approach not work?
I then tried:
coinChange 0 ds = take (length ds) (repeat 0)
coinChange c ds = reverse( (c `quot` x): coinChange' (c `rem` x) xs )
where (x:xs) = reverse ds
coinChange' 0 ds = take (length ds) (repeat 0)
coinChange' c ds = (c `quot` x): coinChange' (c `rem` x) xs
where (x:xs) = reverse ds
I took the line coinChange 0 ds = take (length ds) (repeat 0) from the already-working solution.
However, when I run this code in the terminal, I get the same error:
coinChange 64 [1,2,5,10,20,50,100]
[0,0,0,0,0,64,0]
I think the main problem is with where (x:xs) = reverse ds. Why does this part of the code not work in the way that I expected it to work?
I still get the same error when I use let instead of where.
coinChange 0 ds = take (length ds) (repeat 0)
coinChange c ds = let (x:xs) = reverse ds
in reverse( (c quot x): coinChange' (c rem x) xs )
coinChange' 0 ds = take (length ds) (repeat 0)
coinChange' c ds = let (x:xs) = reverse ds
in (c quot x): coinChange' (c rem x) xs
I thought the problem could be that I am using (x:xs).
So, I tried:
coinChange 0 ds = take (length ds) (repeat 0)
coinChange c ds = let xs = reverse ds
in reverse( (c `quot` (head xs)): coinChange' (c `rem` (head xs)) (tail xs) )
coinChange' 0 ds = take (length ds) (repeat 0)
coinChange' c ds = let xs = reverse ds
in (c `quot` (head xs)): coinChange' (c `rem` (head xs)) (tail xs)
This still gave the same error.
I tried to the concatenation operator (++), instead of the prepend operator :, to see if that would help:
coinChange 0 ds = take (length ds) (repeat 0)
coinChange c ds = let xs = reverse ds
in coinChange (c `rem` (head xs)) (tail xs) ++ [c `quot` (head xs)]
I still got the same error with this code as well.
It appears that the problem must be with my approach of reversing the list of coin denominations.
Why have all my attempts failed? Why do I keep getting this error with 64, when the quotient should be 1?
In this answer I will focus on this attempt of yours:
coinChange c [] = []
coinChange c ds =reverse( (c `quot` x):(coinChange' (c `rem` x) xs ))
where (x:xs) = (reverse ds)
coinChange' c [] = []
coinChange' c ds = (c`quot`x):(coinChange' (c `rem` x) xs )
where (x:xs) = reverse ds
I think the problem is indeed in the where (x:xs) = reverse ds part in the coinChange' function. Each recursive call reverses the list of denominations. So the xs you get there from the pattern match in that where clause is the reversed list of denominations without the largest denomination. But that list is still reversed, so you shouldn't pass that as an argument to the coinChange or coinChange' functions again. You can reverse this list again before passing it to the recursive call, but that is not very efficient. I would suggest only reversing the denominations in the coinChange function.
Also, the coinChange function can be simplified, because it basically only needs to reverse the denominations and the resulting list.
Here is how that would look:
coinChange c ds = reverse (coinChange' c (reverse ds))
coinChange' c [] = []
coinChange' c (x:xs) = (c `quot` x) : coinChange' (c `rem` x) xs