0/1knapsack背包问题:完整思维演推导过程+python

文章目录

• the python code:
• the executed result:
• 01背包问题的推演过程
• • 关于自底向上
  • tips:我们考虑从子问题的角度求解:
• 算法导论习题中伪代码及其解释

the python code:

class Item():
    def __init__(self,weight,value):
        self.value=value
        self.weight=weight
# test data:
items_list=[Item(2,3),Item(3,4),Item(1,5),Item(5,6)]
# items_list=[Item(2,3),Item(3,4),Item(1,5),Item(6,26)]
 
def knapsack01(n, w,items_list):
    """[summary]
 
    Args:
        n (int): the number of the items to be choose （originally)
        w (int): the max weight the knapsack could maintain
        items_list (list): list of items
 
    Returns:
        list: two dimesion list (the last element is the max value of the knapsack)
    """    
    k = [[0 for j in range(w+1)] for i in range(n+1)]
    
    for i in range(1,n+1):
        for j in range(1,w+1):
            item=items_list[i-1]
            if j<item.weight:
                k[i][j]=k[i-1][j]
            else:
                k[i][j]=max(k[i-1][j],k[i-1][j-item.weight]+item.value)
    return k
                
k=knapsack01(4,6,items_list)
for line in k:
    print(line)
print("the max value:",k[-1][-1])

the executed result:

01背包问题的推演过程

关于自底向上

tips:我们考虑从子问题的角度求解:

(往往需要理想化一些条件来使用(比如假设我们有现成的所有子问题规模的下的最优解),基于这样的条件来推导状态方程:

算法导论习题中伪代码及其解释