将X中的所有x_i拆分为K组s.t. var（k中的k的和（k（k））被最小化

Question

我有X个正数，索引为x_i。每个x_i需要进入K组之一（其中K是预定的）。设S_j为K_j中所有x_i的总和。我需要分配所有x_i，以便最小化所有S_j的方差。什么算法实现了这个？我确定有一类算法可以解决这样的问题，但我不知道。

由于

Answer 1

那是。鉴于此类大多数问题都是import numpy as np import timeit import matplotlib.pyplot as plt CPUHz = 3.3e9 divpd_cycles = 4.5 L2cachesize = 2*2**20 L3cachesize = 8*2**20 def timeit_command(command, pieces, size): return min(timeit.repeat("for i in xrange(%d): %s" % (pieces, command), "import numpy; A = numpy.random.rand(%d)" % size, number = 6)) def run(): totaliterations = 1e7 commands=["A/=0.5", "A/=0.51", "A/0.5", "A*=2.0", "A*2.0", "A+=2.0"] styles=['-', '-', '--', '-', '--', '-'] def draw_graph(command, style, compute_overhead = False): sizes = [] y = [] for pieces in np.logspace(0, 5, 11): size = int(totaliterations / pieces) sizes.append(size * 8) # 8 bytes per double time = timeit_command(command, pieces, (4 if compute_overhead else size)) # Divide by 2 because SSE instructions process two doubles each cycles = time * CPUHz / (size * pieces / 2) y.append(cycles) if compute_overhead: command = "numpy overhead" plt.semilogx(sizes, y, style, label = command, linewidth = 2, basex = 10) plt.figure() for command, style in zip(commands, styles): print command draw_graph(command, style) # Plot overhead draw_graph("A+=1.0", '-', compute_overhead=True) plt.legend(loc = 'best', prop = {'size':9}, handlelength = 3) plt.xlabel('Array size in bytes') plt.ylabel('CPU cycles per SSE instruction') # Draw vertical and horizontal lines ymin, ymax = plt.ylim() plt.vlines(L2cachesize, ymin, ymax, color = 'orange', linewidth = 2) plt.vlines(L3cachesize, ymin, ymax, color = 'red', linewidth = 2) xmin, xmax = plt.xlim() plt.hlines(divpd_cycles, xmin, xmax, color = 'blue', linewidth = 2) ，您不太可能找到有效的最优算法。

试图最小化最大组的大小的

packing problem有一个简单的4/3 - 1 /（3K）近似算法（来自 Multiprocessor scheduling）：

对数字进行排序，然后将它们分配给目前为止最小的组。