0-1背包问题算法

0-1背包问题利用Python实现

1，问题描述

背包问题指这样一类问题，题意往往可以抽象成：给定一组物品，每种物品都有自己的重量和价格，在限定的总重量内，我们如何选择，才能使得物品的总价格最高。（来自百度百科）
这次课老师给了我们50组数据，依次按物品编号、物品重量和物品价值顺序如图：

2，算法实现

def dp(weight, count, weights, costs):
    """
    动态规划模板，时间复杂度O(weight * count), 空间复杂度O(weight)
    :param weight: 背包最大能装的重量
    :param count: 物品数量
    :param weights: 每件物品的重量
    :param costs: 每件物品的价值
    :return: 背包装下的最大价值
    """
    pre, cur = [0] * (weight + 1), [0] * (weight + 1)
    for i in range(count):
        for j in range(weight + 1):
            if weights[i] <= j:
                cur[j] = max(pre[j], costs[i] + pre[j - weights[i]])
        pre = cur[:]
    return cur[weight]


# count：物品数量； weight：背包总重量； costs： 每件物品的价值； weight：每件物品的重量
count, weight = 50, 1500
costs = [77,10,34,68,74,55,37,101,52,46,14,63,92,26,105,36,48,22,109,104,21,43,74,21,78,
         54,67,69,51,88,45,52,16,52,58,15,39,60,11,58,33,64,50,86,18,49,33,48,92,51]
weights = [51,44,79,88,72,15,91,71,105,37,101,12,102,31,28,57,81,79,77,45,13,32,83,51,63,
           57,72,47,33,39,26,100,98,50,74,56,100,80,16,103,39,94,66,76,41,54,36,47,28,39]
print(dp(weight, count, weights, costs))

利用Jupyter Notebook（python3）实现得出最优解为1979。

3，后续改进

上述算法只得出了最终的最优解，但是未得到实现最优解的物品编号，需要有后续改进得出最完整的算法程序。

def dp(weight, count, weights, costs):
    """
    动态规划模板，时间复杂度O(weight * count), 空间复杂度O(weight)
    :param weight: 背包最大能装的重量
    :param count: 物品数量
    :param weights: 每件物品的重量
    :param costs: 每件物品的价值
    :return: 背包装下的最大价值
    """
    pre, cur = [0] * (weight + 1), [0] * (weight + 1)
    for i in range(count):
        for j in range(weight + 1):
            if weights[i] <= j:
                cur[j] = max(pre[j], costs[i] + pre[j - weights[i]])
        pre = cur[:]
    return cur[weight]
def show(count,weight,weights,value):
    print('最大价值为:', value[count][weight])
    x = [False for i in range(count)]
    j = weight
    for i in range(count, 0, -1):
        if value[i][j] > value[i - 1][j]:
            x[i - 1] = True
            j -= weights[i - 1]
    print('背包中所装物品为:')
    for i in range(count):
        if x[i]:
            print('第', i+1, '个,', end='')
# count：物品数量； weight：背包总重量； costs： 每件物品的价值； weight：每件物品的重量
count, weight = 50, 1500
costs = [77,10,34,68,74,55,37,101,52,46,14,63,92,26,105,36,48,22,109,104,21,43,74,21,78,
         54,67,69,51,88,45,52,16,52,58,15,39,60,11,58,33,64,50,86,18,49,33,48,92,51]
weights = [51,44,79,88,72,15,91,71,105,37,101,12,102,31,28,57,81,79,77,45,13,32,83,51,63,
           57,72,47,33,39,26,100,98,50,74,56,100,80,16,103,39,94,66,76,41,54,36,47,28,39]
show(count,weight,weights,value)
print(dp(weight, count, weights, costs))