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

This is a classical example of dynamic programming. It uses caching to avoid the pitfall of counting things like 3+2 = 5 twice (because of another possible solution: 2+3). A recursive algorithm falls into that pitfall. Let's focus on a simple example, where target = 5 and coins = [1,2,3]. The piece of code you posted counts:

1. 3+2
2. 3+1+1
3. 2+2+1
4. 1+2+1+1
5. 1+1+1+1+1

while the recursive version counts:

1. 3+2
2. 2+3
3. 3+1+1
4. 1+3+1
5. 1+1+3
6. 2+1+2
7. 1+1+2
8. 2+2+1
9. 2+1+1+1
10. 1+2+1+1
11. 1+1+2+1
12. 1+1+1+2
13. 1+1+1+1+1

Recursive code:

coinsOptions = [1, 2, 3]
def numberOfWays(target):
    if (target < 0):
        return 0
    elif(target == 0):
        return 1
    else:
        return sum([numberOfWays(target - coin) for coin in coinsOptions])
print numberOfWays(5)

Dynamic programming:

target = 5
coins = [1,2,3]
ways = [1]+[0]*target
for coin in coins:
    for i in range(coin, target+1):
        ways[i]+=ways[i-coin]
print ways[target]