看下面的代码，我用了两种方法来解决这个问题(简单递归和DP)。为什么DP方式比较慢？
你有什么建议？
#!/usr/local/bin/python2.7
# encoding: utf-8
问题:存在一个正整数数组。给定一个正整数 S,\ 找出数字之和为 S 的组合总数。
方法一:
def find_sum_recursive(number_list, sum_to_find):
count = 0
for i in range(len(number_list)):
sub_sum = sum_to_find - number_list[i]
if sub_sum < 0:
continue
elif sub_sum == 0:
count += 1
continue
else:
sub_list = number_list[i + 1:]
count += find_sum_recursive(sub_list, sub_sum)
return count
方法二:
def find_sum_DP(number_list, sum_to_find):
count = 0
if(0 == sum_to_find):
count = 1
elif([] != number_list and sum_to_find > 0):
count = find_sum_DP(number_list[:-1], sum_to_find) + find_sum_DP(number_list[:-1], sum_to_find - number_list[:].pop())
return count
运行它:
def main(argv=None): # IGNORE:C0111
number_list = [5, 5, 10, 3, 2, 9, 8]
sum_to_find = 15
input_setup = ';number_list = [5, 5, 10, 3, 2, 9, 8, 7, 6, 4, 3, 2, 9, 5, 4, 7, 2, 8, 3];sum_to_find = 15'
print 'Calculating...'
print 'recursive starting'
count = find_sum_recursive(number_list, sum_to_find)
print timeit.timeit('count = find_sum_recursive(number_list, sum_to_find)', setup='from __main__ import find_sum_recursive' + input_setup, number=10)
cProfile.run('find_sum_recursive(' + str(number_list) + ',' + str(sum_to_find) + ')')
print 'recursive ended:', count
print 'DP starting'
count_DP = find_sum_DP(number_list, sum_to_find)
print timeit.timeit('count_DP = find_sum_DP(number_list, sum_to_find)', setup='from __main__ import find_sum_DP' + input_setup, number=10)
cProfile.run('find_sum_DP(' + str(number_list) + ',' + str(sum_to_find) + ')')
print 'DP ended:', count_DP
print 'Finished.'
if __name__ == '__main__':
sys.exit(main())
我重新编码了方法二，现在是:
def find_sum_DP(number_list, sum_to_find):
count = [[0 for i in xrange(0, sum_to_find + 1)] for j in xrange(0, len(number_list) + 1)]
for i in range(len(number_list) + 1):
for j in range(sum_to_find + 1):
if (0 == i and 0 == j):
count[i][j] = 1
elif (i > 0 and j > 0):
if (j > number_list[i - 1]):
count[i][j] = count[i - 1][j] + count[i - 1][j - number_list[i - 1]]
elif(j < number_list[i - 1]):
count[i][j] = count[i - 1][j]
else:
count[i][j] = count[i - 1][j] + 1
else:
count[i][j] = 0
return count[len(number_list)][sum_to_find]
比较方法一和方法二:
Calculating...
recursive starting
0.00998711585999
92 function calls (63 primitive calls) in 0.000 seconds
Ordered by: standard name
ncalls tottime percall cumtime percall filename:lineno(function)
1 0.000 0.000 0.000 0.000 <string>:1(<module>)
30/1 0.000 0.000 0.000 0.000 FindSum.py:18(find_sum_recursive)
30 0.000 0.000 0.000 0.000 {len}
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
30 0.000 0.000 0.000 0.000 {range}
recursive ended: 6
DP starting
0.00171685218811
15 function calls in 0.000 seconds
Ordered by: standard name
ncalls tottime percall cumtime percall filename:lineno(function)
1 0.000 0.000 0.000 0.000 <string>:1(<module>)
1 0.000 0.000 0.000 0.000 FindSum.py:33(find_sum_DP)
3 0.000 0.000 0.000 0.000 {len}
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
9 0.000 0.000 0.000 0.000 {range}
DP ended: 6
Finished.
最佳答案
如果您使用的是 iPython，%prun 是您的 friend 。
看一下递归版本的输出:
2444 function calls (1631 primitive calls) in 0.002 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
814/1 0.002 0.000 0.002 0.002 <ipython-input-1-7488a6455e38>:1(find_sum_recursive)
814 0.000 0.000 0.000 0.000 {range}
814 0.000 0.000 0.000 0.000 {len}
1 0.000 0.000 0.002 0.002 <string>:1(<module>)
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
现在，对于 DP 版本:
10608 function calls (3538 primitive calls) in 0.007 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
7071/1 0.007 0.000 0.007 0.007 <ipython-input-15-3535e3ab26eb>:1(find_sum_DP)
3535 0.001 0.000 0.001 0.000 {method 'pop' of 'list' objects}
1 0.000 0.000 0.007 0.007 <string>:1(<module>)
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
7071比814高了不少!
这里的问题是您的动态规划方法不是动态规划!动态规划的要点是，当您遇到重叠子问题的问题时，就像您在这里所做的那样，您存储每个子问题的结果，然后当您再次需要结果时，您可以从该存储中获取它而不是重新计算。您的代码不会这样做:每次调用 find_sum_DP 时，您都在重新计算，即使已经完成了相同的计算。结果是你的 _DP 方法实际上不仅是递归的，而且递归的函数调用比你的递归方法更多。
(我目前正在写一个DP版来演示)
编辑:
我需要补充一点，虽然我应该了解更多关于动态规划的知识，但我很尴尬地不知道。我也在深夜快速地写这篇文章，有点像对自己的练习。不过，这里是该函数的动态编程实现:
import numpy as np
def find_sum_realDP( number_list, sum_to_find ):
memo = np.zeros( (len(number_list),sum_to_find+1) ,dtype=np.int)-1
# This will store our results. memo[l][n] will give us the result
# for number_list[0:l+1] and a sum_to_find of n. If it hasn't been
# calculated yet, it will give us -1. This is not at all efficient
# storage, but isn't terribly bad.
# Now that we have that, we'll call the real function. Instead of modifying
# the list and making copies or views, we'll keep the same list, and keep
# track of the index we're on (nli).
return find_sum_realDP_do( number_list, len(number_list)-1, sum_to_find, memo ),memo
def find_sum_realDP_do( number_list, nli, sum_to_find, memo ):
# Our count is 0 by default.
ret = 0
# If we aren't at the sum to find yet, do we have any numbers left after this one?
if ((sum_to_find > 0) and nli>0):
# Each of these checks to see if we've already stored the result of the calculation.
# If so, we use that, if not, we calculate it.
if memo[nli-1,sum_to_find]>=0:
ret += memo[nli-1,sum_to_find]
else:
ret += find_sum_realDP_do(number_list, nli-1, sum_to_find, memo)
# This one is a bit tricky, and was a bug when I first wrote it. We don't want to
# have a negative sum_to_find, because that will be very bad; we'll start using results
# from other places in memo because it will wrap around.
if (sum_to_find-number_list[nli]>=0) and memo[nli-1,sum_to_find-number_list[nli]]>=0:
ret += memo[nli-1,sum_to_find-number_list[nli]]
elif (sum_to_find-number_list[nli]>=0):
ret += find_sum_realDP_do(number_list, nli-1, sum_to_find-number_list[nli], memo)
# Do we not actually have any sum to find left?
elif (0 == sum_to_find):
ret = 1
# If we only have one number left, will it get us there?
elif (nli == 0) and (sum_to_find-number_list[nli] == 0 ):
ret = 1
# Store our result.
memo[nli,sum_to_find] = ret
# Return our result.
return ret
请注意，这使用了 numpy .您很可能没有安装它，但我不确定没有它如何在 Python 中编写性能合理的动态编程算法；我认为 Python 列表的性能远不及 Numpy 数组。另请注意，这与您的代码处理零的方式不同，因此与其调试它，我只想说这段代码适用于数字列表中的非零正整数。现在，使用这个算法，分析给我们:
243 function calls (7 primitive calls) in 0.001 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
237/1 0.001 0.000 0.001 0.001 <ipython-input-155-4a624e5a99b7>:9(find_sum_realDP_do)
1 0.000 0.000 0.001 0.001 <ipython-input-155-4a624e5a99b7>:1(find_sum_realDP)
1 0.000 0.000 0.000 0.000 {numpy.core.multiarray.zeros}
1 0.000 0.000 0.001 0.001 <string>:1(<module>)
2 0.000 0.000 0.000 0.000 {len}
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
243 甚至比递归版本要好得多!但是您的示例数据足够小，无法真正显示动态规划算法有多好。
让我们试试 nlist2 = [7, 6, 2, 3, 7, 7, 2, 7, 4, 2, 4, 5, 6, 1, 7, 4, 6, 3, 2, 1 , 1, 1, 4,
2, 3, 5, 2, 4, 4, 2, 4, 5, 4, 2, 1, 7, 6, 6, 1, 5, 4, 5, 3, 2, 3, 7,
1, 7, 6, 6]，具有相同的 sum_to_find=15。这有 50 个值，以及 900206 种方法来获得 15...
使用find_sum_recursive:
3335462 function calls (2223643 primitive calls) in 14.137 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
1111820/1 13.608 0.000 14.137 14.137 <ipython-input-46-7488a6455e38>:1(find_sum_recursive)
1111820 0.422 0.000 0.422 0.000 {range}
1111820 0.108 0.000 0.108 0.000 {len}
1 0.000 0.000 14.137 14.137 <string>:1(<module>)
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
现在使用 find_sum_realDP:
736 function calls (7 primitive calls) in 0.007 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
730/1 0.007 0.000 0.007 0.007 <ipython-input-155-4a624e5a99b7>:9(find_sum_realDP_do)
1 0.000 0.000 0.007 0.007 <ipython-input-155-4a624e5a99b7>:1(find_sum_realDP)
1 0.000 0.000 0.000 0.000 {numpy.core.multiarray.zeros}
1 0.000 0.000 0.007 0.007 <string>:1(<module>)
2 0.000 0.000 0.000 0.000 {len}
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
所以我们的调用次数不到 1/1000，运行时间不到 1/2000。当然，您使用的列表越大，DP 算法的效果就越好。在我的电脑上，sum_to_find 为 15 和从 1 到 8 的 600 个随机数列表运行，realDP 只需要 0.09 秒，函数调用不到 10,000 次；正是在这一点上，我使用的 64 位整数开始溢出，我们遇到了各种其他问题。不用说，在计算机停止运行之前，递归算法永远无法处理接近该大小的列表，无论是内部 Material 分解还是宇宙热寂。
关于python - 关于DP的性能讨论，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/18267848/