0/1背包问题求解算法性能比较：蛮力法，回溯法和分支限界法

import random
import time
import matplotlib.pyplot as plt
import numpy as np

# 生成随机的物品重量和价值
def generate_items(n):
    weights = [random.randint(1, 10) for _ in range(n)]
    values = [random.randint(10, 50) for _ in range(n)]
    return weights, values

# 蛮力法求解0/1背包问题
def brute_force_knapsack(weights, values, capacity):
    n = len(weights)
    max_value = 0
    for i in range(2 ** n):           # 遍历所有可能的物品组合
        current_weight = 0
        current_value = 0
        for j in range(n):
            if (i >> j) & 1:               # 判断第j个物品是否在当前组合中
                current_weight += weights[j]
                current_value += values[j]
        if current_weight <= capacity and current_value > max_value:
            max_value = current_value
    return max_value

# 回溯法求解0/1背包问题
def backtrack_knapsack(weights, values, capacity):
    def backtrack(i, current_weight, current_value):
        nonlocal max_value
        if current_weight > capacity:
            return
        if current_value > max_value:
            max_value = current_value
        if i == n:
            return
        backtrack(i + 1, current_weight, current_value)
        backtrack(i + 1, current_weight + weights[i], current_value + values[i])

    n = len(weights)
    max_value = 0
    backtrack(0, 0, 0)
    return max_value

# 分支限界法求解0/1背包问题
def branch_bound_knapsack(weights, values, capacity):
    class Node:
        def __init__(self, level, weight, value, bound):
            self.level = level
            self.weight = weight
            self.value = value
            self.bound = bound

    def bound(node):
        if node.weight >= capacity:
            return 0
        bound = node.value
        j = node.level + 1
        total_weight = node.weight
        while j < n and total_weight + weights[j] <= capacity:
            total_weight += weights[j]
            bound += values[j]
            j += 1
        if j < n:
            bound += (capacity - total_weight) * values[j] / weights[j]
        return bound

    n = len(weights)
    max_value = 0
    Q = []
    root = Node(-1, 0, 0, 0)
    Q.append(root)
    while Q:
        node = Q.pop(0)
        if node.level == n - 1:
            continue
        left = Node(node.level + 1, node.weight, node.value, 0)
        left.bound = bound(left)
        if left.bound > max_value:
            Q.append(left)
        right = Node(node.level + 1, node.weight + weights[node.level + 1], node.value + values[node.level + 1], 0)
        right.bound = bound(right)
        if right.weight <= capacity and right.value > max_value:
            max_value = right.value
        if right.bound > max_value:
            Q.append(right)
    return max_value

# 测试程序
N = [4, 8, 16, 24]
times_brute_force = []
times_backtrack = []
times_branch_bound = []
for n in N:
    weights, values = generate_items(n)
    capacity = sum(weights) // 2

    start_time = time.time()
    brute_force_knapsack(weights, values, capacity)
    end_time = time.time()
    times_brute_force.append(end_time - start_time)

    start_time = time.time()
    backtrack_knapsack(weights, values, capacity)
    end_time = time.time()
    times_backtrack.append(end_time - start_time)

    start_time = time.time()
    branch_bound_knapsack(weights, values, capacity)
    end_time = time.time()
    times_branch_bound.append(end_time - start_time)

# Fit a polynomial curve to the data points
curve_brute_force = np.polyfit(N, times_brute_force, 10)
curve_backtrack = np.polyfit(N, times_backtrack, 10)
curve_branch_bound = np.polyfit(N, times_branch_bound, 10)

# Generate a smooth curve using the fitted polynomial coefficients
smooth_N = np.linspace(min(N), max(N), 100)
smooth_times_brute_force = np.polyval(curve_brute_force, smooth_N)
smooth_times_backtrack = np.polyval(curve_backtrack, smooth_N)
smooth_times_branch_bound = np.polyval(curve_branch_bound, smooth_N)

# Plot the curves
plt.plot(smooth_N, smooth_times_brute_force, label='Brute Force (Curve)')
plt.plot(smooth_N, smooth_times_backtrack, label='Backtrack (Curve)')
plt.plot(smooth_N, smooth_times_branch_bound, label='Branch and Bound (Curve)')

plt.xlabel('N')
plt.ylabel('Time (s)')
plt.legend()
plt.show()