Coin changing algorithm

Question

Suppose I have a set of coins having denominations a1, a2, ... ak.

One of them is known to be equal to 1.

I want to make change for all integers 1 to n using

Answer 1

Here's a sample implementation of a dynamic programming algorithm in Python. It is simpler than the algorithm that Jason describes, because it only calculates 1 row of the 2D table he describes.

Please note that using this code to cheat on homework will make Zombie Dijkstra cry.

import sys
def get_best_coins(coins, target):
    costs = [0]
    coins_used = [None]
    for i in range(1,target + 1):
        if i % 1000 == 0:
            print '...', 
        bestCost = sys.maxint
        bestCoin = -1
        for coin in coins:
            if coin <= i:
                cost = 1 + costs[i - coin]
                if cost < bestCost:
                    bestCost = cost
                    bestCoin = coin
        costs.append(bestCost)
        coins_used.append(bestCoin)
    ret = []    
    while target > 0:
        ret.append(coins_used[target])
        target -= coins_used[target]
    return ret

coins = [1,10,25]
target = 100033
print get_best_coins(coins, target)