如何处理以下贪婪/动态问题?

Question

我必须检查是否可以从给定的数字中获取总和>严格大于给定的总和

例如：

如果给定的总和为 9，给定的数字为 [3,5,7] 那么正确答案是 1*3 +1*7=10>9，所以输出应该是 [1,0,1]

例2：

如果给定的总和为 13，数字为 [1,3,5] 那么正确答案是 0*1+3*3+1*5=14>13，但 9*1+0*3+5=14 是>错误的，所以正确的输出是 [0,3,1] 如何处理这个问题

Answer 1

可以使用无界背包算法的版本获得解决方案。

在这里，我们修改来自 Python Unbound Knapsack 的代码

修改有两个目标：

1. 最小化值的计数 从阵列中使用

2. 创建小于限制的最大总和(即 给定总和 + 1(

def knapsack_unbounded_dp(arr, C):
C += 1  # actual limit
# Form index value pairs (to keep track of the original indexes after sorting)
items = [(i, v) for i, v in enumerate(arr)]
# Sort values in descending order
items = sorted(items, key=lambda item: item[1], reverse=True)
# Sack keeps track of:
#  max value so i.e. sack[0] and
# count of how many each item are in the sack i.e. sack[1]
sack = [(0, [0 for i in items]) for i in range(0, C+1)]   # value, [item counts]
for i,item in enumerate(items):
# For current item we check if a previous entry could have done better
# by adding this item to the sack
index, value = item
for c in range(value, C+1):   # check all previous values
sackwithout = sack[c-value]  # previous max sack to try adding this item to
trial = sackwithout[0] + value  # adding value to max without using it
used = sackwithout[1][i]        # count of i-them item
if sack[c][0] < trial:
# old max sack with this added item is better
sack[c] = (trial, sackwithout[1][:])
sack[c][1][i] +=1   # use one more
value, bagged = sack[C]
# index and count of each array value
new_bagged = [(i, v) for (i, _), v in zip(items, bagged)]
# Re-sort based upon original order of array indexes
new_bagged.sort(key=lambda t: t[0])
# counts based upon original array order
cnts = [v for i, v in new_bagged]
return sum(cnts), value, cnts

测试代码

for t in [([1, 3, 5], 8), ([3, 5, 7], 9), ([1, 3, 5], 13)]:
result = knapsack_unbounded_dp(t[0], t[1])
print(f'Test: {t[0]}, Given Sum {t[1]}')
print(f'Result counts {result[2]}, with max sum {result[1]}, Total Count {result[0]}n')

输出

Test: [1, 3, 5], Given Sum 8
Result counts [0, 3, 0], with max sum 9, Total Count 3
Test: [3, 5, 7], Given Sum 9
Result counts [0, 2, 0], with max sum 10, Total Count 2
Test: [1, 3, 5], Given Sum 13
Result counts [0, 3, 1], with max sum 14, Total Count 4