基于python代码，实现500个节点进行随机多项式，计算运行时间和所占内存，然后将节点分区按照四种方法分区之后，再次计算得出时间和内存的对比结果，并绘折线图对比

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首先，我们需要用 Python 生成 500 个随机多项式。这里我们使用 NumPy 库来生成随机数，并使用 SymPy 库来进行多项式运算。代码如下：

import numpy as np
import sympy as sp

np.random.seed(0)

# 生成随机多项式
def generate_polynomials(num_polynomials, max_degree):
    polynomials = []
    for i in range(num_polynomials):
        degree = np.random.randint(1, max_degree + 1)
        coeffs = np.random.randint(-10, 10, degree)
        x = sp.symbols('x')
        poly = sp.Poly(coeffs.tolist()[::-1], x)
        polynomials.append(poly)
    return polynomials

polynomials = generate_polynomials(500, 10)

接下来，我们需要计算程序的运行时间和所占内存。这里我们可以使用 Python 内置的 time 和 memory_profiler 库来进行计时和内存监测。代码如下：

import time
from memory_profiler import memory_usage

# 计算程序运行时间和所占内存
def calculate_time_and_memory_usage(func):
    start_time = time.time()
    memory_usage_before = memory_usage()[0]
    result = func()
    memory_usage_after = memory_usage()[0]
    end_time = time.time()
    time_usage = end_time - start_time
    memory_usage = memory_usage_after - memory_usage_before
    return time_usage, memory_usage, result

# 测试计算多项式的函数
def test_polynomials(polynomials):
    results = []
    for poly in polynomials:
        result = poly.evalf(subs={'x': np.random.rand()})
        results.append(result)
    return results

# 计算未分区时的运行时间和内存占用
time_usage_1, memory_usage_1, _ = calculate_time_and_memory_usage(lambda: test_polynomials(polynomials))
print('Time usage (no partition):', time_usage_1)
print('Memory usage (no partition):', memory_usage_1)

接下来，我们将节点分区。这里我们使用四种常用的分区方法：等分法、负载均衡法、随机分配法和聚类法。代码如下：

import math
from sklearn.cluster import KMeans

# 等分法
def partition_equally(num_partitions, nodes):
    partitions = []
    num_nodes_per_partition = math.ceil(len(nodes) / num_partitions)
    for i in range(num_partitions):
        start_index = i * num_nodes_per_partition
        end_index = min((i + 1) * num_nodes_per_partition, len(nodes))
        partition = nodes[start_index:end_index]
        partitions.append(partition)
    return partitions

# 负载均衡法
def partition_load_balanced(num_partitions, nodes, weights):
    partitions = [[] for i in range(num_partitions)]
    total_weight = sum(weights)
    weight_per_partition = total_weight / num_partitions
    current_partition_index = 0
    current_partition_weight = 0
    for i in range(len(nodes)):
        node = nodes[i]
        weight = weights[i]
        if current_partition_weight + weight > weight_per_partition and current_partition_index < num_partitions - 1:
            current_partition_index += 1
            current_partition_weight = 0
        partitions[current_partition_index].append(node)
        current_partition_weight += weight
    return partitions

# 随机分配法
def partition_randomly(num_partitions, nodes):
    np.random.shuffle(nodes)
    return partition_equally(num_partitions, nodes)

# 聚类法
def partition_by_clustering(num_partitions, nodes, features):
    kmeans = KMeans(n_clusters=num_partitions)
    kmeans.fit(features)
    labels = kmeans.labels_
    partitions = [[] for i in range(num_partitions)]
    for i in range(len(nodes)):
        node = nodes[i]
        partition_index = labels[i]
        partitions[partition_index].append(node)
    return partitions

# 测试分区方法的运行时间和内存占用
num_partitions = 10
weights = np.random.rand(len(polynomials)) * 10
features = np.random.rand(len(polynomials), 2)
nodes = list(range(len(polynomials)))

time_usage_2, memory_usage_2, partitions = calculate_time_and_memory_usage(lambda: partition_equally(num_partitions, nodes))
time_usage_3, memory_usage_3, partitions = calculate_time_and_memory_usage(lambda: partition_load_balanced(num_partitions, nodes, weights))
time_usage_4, memory_usage_4, partitions = calculate_time_and_memory_usage(lambda: partition_randomly(num_partitions, nodes))
time_usage_5, memory_usage_5, partitions = calculate_time_and_memory_usage(lambda: partition_by_clustering(num_partitions, nodes, features))

print('Time usage (equally):', time_usage_2)
print('Memory usage (equally):', memory_usage_2)
print('Time usage (load balanced):', time_usage_3)
print('Memory usage (load balanced):', memory_usage_3)
print('Time usage (randomly):', time_usage_4)
print('Memory usage (randomly):', memory_usage_4)
print('Time usage (clustering):', time_usage_5)
print('Memory usage (clustering):', memory_usage_5)

最后，我们将结果绘制成折线图。这里我们使用 Matplotlib 库来进行绘图。代码如下：

import matplotlib.pyplot as plt

# 绘制时间和内存占用的折线图
times = [time_usage_1, time_usage_2, time_usage_3, time_usage_4, time_usage_5]
memories = [memory_usage_1, memory_usage_2, memory_usage_3, memory_usage_4, memory_usage_5]
labels = ['No partition', 'Equally', 'Load balanced', 'Randomly', 'Clustering']

plt.plot(labels, times, label='Time')
plt.plot(labels, memories, label='Memory')
plt.legend()
plt.show()

这样，我们就完成了基于 Python 代码的 500 个节点随机多项式的计算、分区、运行时间和内存占用的对比，并绘制了折线图。