# 动态规划思想 dp方程式如下
# dp[0] = 0
# dp[i] = min{dp[i - coins[j]] + 1}, 且 其中 i >= coins[j], 0 <= j < coins.length
# 回溯法，输出可找的硬币方案
# path[i] 表示经过本次兑换后所剩下的面值，即 i - path[i] 可得到本次兑换的硬币值。
 
 
def changeCoins(coins, n):
  if n < 0: return None
  dp, path = [0] * (n+1), [0] * (n+1) # 初始化
  for i in range(1, n+1):
    minNum = i # 初始化当前硬币最优值
    for c in coins: # 扫描一遍硬币列表，选择一个最优值
      if i >= c and minNum > dp[i-c]+1:
        minNum, path[i] = dp[i-c]+1, i - c
    dp[i] = minNum # 更新当前硬币最优值
 
  print('最少硬币数:', dp[-1])
  print('可找的硬币', end=': ')
  while path[n] != 0:
    print(n-path[n], end=' ')
    n = path[n]
  print(n, end=' ')
 
 
if __name__ == '__main__':
  coins, n = [1, 4, 5], 22 # 输入可换的硬币种类，总金额n
  changeCoins(coins, n)