声明
本文基于Python2.7语言，给出判断列表是否已排序的多种方法，并在作者的Windows XP主机(Pentium G630 2.7GHz主频2GB内存)上对比和分析其性能表现。
一. 问题提出
Haskell培训老师提出一个问题：如何判断列表是否已经排序？
排序与否实际只是相邻元素间的某种二元关系，即a->a->Bool。所以第一步可以把二元组列表找出来；第二步是把这个函数作用于每个元组，然后用and操作。老师给出的实现代码如下：
pair lst = zip lst ( tail lst )sorted lst predict = and [ predict x y | (x,y) <- pair lst]
Haskell中，等号前面是函数的名称和参数，后面是函数的定义体。pair函数将列表lst错位一下(tail除去列表的第一个元素)后，和原列表在zip的作用下形成前后相邻元素二元组列表。predict函数接受两个数字，根据大小返回True或False。and对类型为[Bool]的列表中所有元素求与，其语义类似Python的all()函数。
随后，老师请大家思考性能问题。作者提出，若准确性要求不高，可生成三组随机数排序后作为下标，提取原列表相应的三组元素组成三个新的子列表("采样")。若判断三个子列表遵循同样的排序规则时，则认为原列表已排序。当列表很大且前段已排序时，选取适当数目的随机数，可在保障一定准确率的同时显著地降低运算规模。
此外，实际应用中还应考虑数据来源。例如，Python语言的os.listdir()方法在Windows系统中返回的列表条目满足字母序，在Linux系统中则返回乱序条目。因此，若判断系统平台(os.platform)为win32，则条目必然已经排序。
为对比验证随机采样方式的可行性，作者先在网上搜集判断列表排序的现有方法，主要参考Stack Overflow网站上"Pythonic way to check if a list is sorted or not"问题的答案，并在本文第二节给出相关的代码示例。注意，本文所述的"排序"不要求严格排序，即相邻元素允许相等。例如，[1,2,2,3]视为升序，[3,3,2,2]视为降序。
二. 代码实现
本节判断列表排序的函数名格式为IsListSorted_XXX()。为简洁起见，除代码片段及其输出外，一律以_XXX()指代。
2.1 guess
def IsListSorted_guess(lst):listLen = len(lst)if listLen <= 1:return True#由首个元素和末尾元素猜测可能的排序规则if lst[0] == lst[-1]: #列表元素相同for elem in lst:if elem != lst[0]: return Falseelif lst[0] < lst[-1]: #列表元素升序for i, elem in enumerate(lst[1:]):if elem < lst[i]: return Falseelse: #列表元素降序for i, elem in enumerate(lst[1:]):if elem > lst[i]: return Falsereturn True
_guess()是最通用的实现，几乎与语言无关。值得注意的是，该函数内会猜测给定列表可能的排序规则，因此无需外部调用者指明排序规则。
2.2 sorted
def IsListSorted_sorted(lst):return sorted(lst) == lst or sorted(lst, reverse=True) == lst
_sorted()使用Python内置函数sorted()。由于sorted()会对未排序的列表排序，_sorted()函数主要适用于已排序列表。
若想判断列表未排序后再对其排序，不如直接调用列表的sort()方法，因为该方法内部会判断列表是否排序。对于已排序列表，该方法的时间复杂度为线性阶O(n)――判断为O(n)而排序为O(nlgn)。
2.3 for-loop
def IsListSorted_forloop(lst, key=lambda x, y: x <= y):for i, elem in enumerate(lst[1:]): #注意，enumerate默认迭代下标从0开始if not key(lst[i], elem): #if elem > lst[i]更快，但通用性差return Falsereturn True
无论列表是否已排序，本函数的时间复杂度均为线性阶O(n)。注意，参数key表明缺省的排序规则为升序。
2.4 all
def IsListSorted_allenumk(lst, key=lambda x, y: x <= y):return all(key(lst[i], elem) for i, elem in enumerate(lst[1:]))import operatordef IsListSorted_allenumo(lst, oCmp=operator.le):return all(oCmp(lst[i], elem) for i, elem in enumerate(lst[1:]))def IsListSorted_allenumd(lst):return all((lst[i] <= elem) for i, elem in enumerate(lst[1:]))def IsListSorted_allxran(lst, key=lambda x,y: x <= y):return all(key(lst[i],lst[i+1]) for i in xrange(len(lst)-1))def IsListSorted_allzip(lst, key=lambda x,y: x <= y):from itertools import izip #Python 3中zip返回生成器(generator)，而izip被废弃return all(key(a, b) for (a, b) in izip(lst[:-1],lst[1:]))
lambda表达式与operator运算符速度相当，前者简单灵活，后者略为高效(实测并不一定)。但两者速度均不如列表元素直接比较(可能存在调用开销)。亦即，_allenumd()快于_allenumo()快于_allenumk()。
若使用lambda表达式指示排序规则，更改规则时只需要改变x和y之间的比较运算符；若使用operator模块指示排序规则，更改规则时需要改变对象比较方法。具体地，lt(x, y)等效于x < y，le(x, y)等效于x <= y，eq(x, y)等效于x == y，ne(x, y)等效于x != y，gt(x, y)等效于x > y，ge(x, y)等效于x >= y。例如，_allenumo()函数若要严格升序可设置oCmp=operator.lt。
此外，由all()函数的帮助信息可知，_allenumk()其实是_forloop()的等效形式。
2.5 numpy
def IsListSorted_numpy(arr, key=lambda dif: dif >= 0):import numpytry:if arr.dtype.kind == 'u': #无符号整数数组执行np.diff时存在underflow风险arr = numpy.int64(lst)except AttributeError:pass #无dtype属性，非数组return (key(numpy.diff(arr))).all() #numpy.diff(x)返回相邻数组元素的差值构成的数组
NumPy是用于科学计算的Python基础包，可存储和处理大型矩阵。它包含一个强大的N维数组对象，比Python自身的嵌套列表结构(nested list structure)高效得多。第三节的实测数据表明，_numpy()处理大型列表时性能非常出色。
在Windows系统中可通过pip install numpy命令安装NumPy包，不建议登录官网下载文件自行安装。
2.6 reduce
def IsListSorted_reduce(iterable, key=lambda x, y: x <= y):cmpFunc = lambda x, y: y if key(x, y) else float('inf')return reduce(cmpFunc, iterable, .0) < float('inf') reduce实现是all实现的变体。累加器(accumulator)中仅存储最后一个检查的列表元素，或者Infinity(若任一元素小于前个元素值)。
前面2.1~2.5小节涉及下标操作的函数适用于列表等可迭代对象(Iterable)。对于通用迭代器(Iterator)对象，即可以作用于next()函数或方法的对象，可使用_reduce()及后面除_rand()外各小节的函数。迭代器的计算是惰性的，只有在需要返回下一个数据时才会计算，以避免不必要的计算。而且，迭代器方式无需像列表那样切片为两个迭代对象。
2.7 imap
def IsListSorted_itermap(iterable, key=lambda x, y: x <= y):from itertools import imap, teea, b = tee(iterable) #为单个iterable创建两个独立的iteratornext(b, None)return all(imap(key, a, b))
2.8 izip
def IsListSorted_iterzip(iterable, key=lambda x, y: x <= y):from itertools import tee, izipa, b = tee(iterable)next(b, None)return all(key(x, y) for x, y in izip(a, b))def pairwise(iterable):from itertools import tee, izipa, b = tee(iterable)next(b, None)return izip(a, b) #"s -> (s0,s1), (s1,s2), (s2, s3), ..."def IsListSorted_iterzipf(iterable, key=lambda x, y: x <= y):return all(key(a, b) for a, b in pairwise(iterable))
第三节的实测数据表明，虽然存在外部函数调用，_iterzipf()却比_iterzip()略为高效。
2.9 fast
def IsListSorted_fastd(lst):it = iter(lst)try:prev = it.next()except StopIteration:return Truefor cur in it:if prev > cur:return Falseprev = curreturn Truedef IsListSorted_fastk(lst, key=lambda x, y: x <= y):it = iter(lst)try:prev = it.next()except StopIteration:return Truefor cur in it:if not key(prev, cur):return Falseprev = curreturn True
_fastd()和_fastk()是Stack Overflow网站回答里据称执行最快的。实测数据表明，在列表未排序时，它们的性能表现确实优异。
2.10 random
import randomdef IsListSorted_rand(lst, randNum=3, randLen=100):listLen = len(lst)if listLen <= 1:return True#由首个元素和末尾元素猜测可能的排序规则if lst[0] < lst[-1]: #列表元素升序key = lambda dif: dif >= 0else: #列表元素降序或相等key = lambda dif: dif <= 0threshold, sortedFlag = 10000, Trueimport numpyif listLen <= threshold or listLen <= randLen*2 or not randNum:return (key(numpy.diff(numpy.array(lst)))).all()from random import samplefor i in range(randNum):sortedRandList = sorted(sample(xrange(listLen), randLen))flag = (key(numpy.diff(numpy.array([lst[x] for x in sortedRandList])))).all()sortedFlag = sortedFlag and flagreturn sortedFlag
_rand()借助随机采样降低运算规模，并融入其他判断函数的优点。例如，猜测列表可能的排序规则，并在随机采样不适合时使用相对快速的判断方式，如NumPy。
通过line_profiler分析可知，第20行和第21行与randLen有关，但两者耗时接近。因此randLen应小于listLen的一半，以抵消sorted开销。除内部限制外，用户可以调节随机序列个数和长度，如定制单个但较长的序列。
注意，_rand()不适用于存在微量异常数据的长列表。因为这些数据很可能被随机采样遗漏，从而影响判断结果的准确性。
三. 性能分析
本节借助Python标准库random模块，生成各种随机列表，以对比和分析上节列表排序判断函数的性能。
可通过sample()、randint()等方法生成随机列表。例如：
>>>import random>>> random.sample(range(10), 10); random.sample(range(10), 5)[9, 1, 6, 3, 0, 8, 4, 2, 7, 5][1, 2, 5, 6, 0]>>> rand = [random.randint(1,10) for i in range(10)]; rand[7, 3, 7, 5, 7, 2, 4, 4, 9, 8]>>> random.sample(rand, 5); random.sample(rand, 5)[4, 7, 7, 9, 8][3, 9, 2, 5, 7]>>> randGen = (random.randint(1,10) for i in range(10)); randGen<generator object <genexpr> at 0x0192C878>
sample()方法从列表中随机选择数字，结合range()函数可生产不含重复元素的随机列表；而randint()方法结合列表解析生成的随机列表可能包含重复元素。Python文档规定，首次导入random模块时使用当前系统时间作为种子初始化随机数生成器。因此，本文并未显式地调用seed()方法设置种子。
为度量性能表现，定义如下计时装饰器：
from time import clockTimeList = []def FuncTimer(repeats=1000):def decorator(func):def wrapper(*args, **kwargs):try:startTime = clock()for i in xrange(repeats):ret = func(*args, **kwargs)except Exception, e:print '%s() Error: %s!' %(func.__name__, e)ret = Nonefinally: #当目标函数发生异常时，仍旧输出计时信息endTime = clock()timeElasped = (endTime-startTime)*1000.0msg = '%21s(): %s =>Time Elasped: %.3f msec, repeated %d time(s).' /%(func.__name__, ret, timeElasped, repeats)global TimeList; TimeList.append([timeElasped, msg])return retreturn wrapperreturn decoratordef ReportTimer():global TimeList; TimeList.sort(key=lambda x:x[0])for entry in TimeList:print entry[1]TimeList = [] #Flush
该装饰器允许对输出进行排序，以便更直观地观察性能。基于该装饰器，下文将分别测试不同的排序场景。注意，第二节各函数头部需添加FuncTimer()装饰。
3.1 列表前段乱序
测试代码如下：
def TestHeadUnorderedList():TEST_NAME = 'HeadUnorderedList'; scale = int(1e5)List = random.sample(xrange(scale), scale) + range(scale)print 'Test 1: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_guess(List)IsListSorted_sorted(List)IsListSorted_allenumk(List)IsListSorted_allenumo(List)IsListSorted_allenumd(List)IsListSorted_allxran(List)IsListSorted_allzip(List)IsListSorted_forloop(List)IsListSorted_itermap(List)IsListSorted_iterzipf(List)IsListSorted_iterzip(List)IsListSorted_reduce(List)IsListSorted_numpy(numpy.array(List)) #若不先转换为数组，则耗时骤增IsListSorted_fastd(List)IsListSorted_fastk(List)ReportTimer()
运行输出如下：
Test 1: HeadUnorderedList, list len: 200IsListSorted_fastd(): False =>Time Elasped: 0.757 msec, repeated 1000 time(s).IsListSorted_fastk(): False =>Time Elasped: 1.091 msec, repeated 1000 time(s).IsListSorted_forloop(): False =>Time Elasped: 2.080 msec, repeated 1000 time(s).IsListSorted_guess(): False =>Time Elasped: 2.123 msec, repeated 1000 time(s).IsListSorted_allxran(): False =>Time Elasped: 2.255 msec, repeated 1000 time(s).IsListSorted_allenumd(): False =>Time Elasped: 2.672 msec, repeated 1000 time(s).IsListSorted_allenumo(): False =>Time Elasped: 3.021 msec, repeated 1000 time(s).IsListSorted_allenumk(): False =>Time Elasped: 3.207 msec, repeated 1000 time(s).IsListSorted_itermap(): False =>Time Elasped: 5.845 msec, repeated 1000 time(s).IsListSorted_allzip(): False =>Time Elasped: 7.793 msec, repeated 1000 time(s).IsListSorted_iterzip(): False =>Time Elasped: 9.667 msec, repeated 1000 time(s).IsListSorted_iterzipf(): False =>Time Elasped: 9.969 msec, repeated 1000 time(s).IsListSorted_numpy(): False =>Time Elasped: 16.203 msec, repeated 1000 time(s).IsListSorted_sorted(): False =>Time Elasped: 45.742 msec, repeated 1000 time(s).IsListSorted_reduce(): False =>Time Elasped: 145.447 msec, repeated 1000 time(s).Test 1: HeadUnorderedList, list len: 200000IsListSorted_fastd(): False =>Time Elasped: 0.717 msec, repeated 1000 time(s).IsListSorted_fastk(): False =>Time Elasped: 0.876 msec, repeated 1000 time(s).IsListSorted_allxran(): False =>Time Elasped: 2.104 msec, repeated 1000 time(s).IsListSorted_itermap(): False =>Time Elasped: 6.062 msec, repeated 1000 time(s).IsListSorted_iterzip(): False =>Time Elasped: 7.244 msec, repeated 1000 time(s).IsListSorted_iterzipf(): False =>Time Elasped: 8.491 msec, repeated 1000 time(s).IsListSorted_numpy(): False =>Time Elasped: 801.916 msec, repeated 1000 time(s).IsListSorted_forloop(): False =>Time Elasped: 2924.755 msec, repeated 1000 time(s).IsListSorted_guess(): False =>Time Elasped: 2991.756 msec, repeated 1000 time(s).IsListSorted_allenumo(): False =>Time Elasped: 3025.864 msec, repeated 1000 time(s).IsListSorted_allenumk(): False =>Time Elasped: 3062.792 msec, repeated 1000 time(s).IsListSorted_allenumd(): False =>Time Elasped: 3190.896 msec, repeated 1000 time(s).IsListSorted_allzip(): False =>Time Elasped: 6586.183 msec, repeated 1000 time(s).IsListSorted_sorted(): False =>Time Elasped: 119974.955 msec, repeated 1000 time(s).IsListSorted_reduce(): False =>Time Elasped: 154747.659 msec, repeated 1000 time(s).
可见，对于前段乱序的列表，无论其长短_fastd()和_fastk()的表现均为最佳。对于未排序列表，_sorted()需要进行排序，故性能非常差。然而，_reduce()性能最差。
实际上除_guess()和_sorted()外，其他函数都按升序检查列表。为安全起见，可仿照_guess()实现，先猜测排序方式，再进一步检查。
因为短列表耗时差异大多可以忽略，后续测试将不再包含短列表(但作者确实测试过)，仅关注长列表。除非特别说明，列表长度为10万级，重复计时1000次。
3.2 列表中段乱序
测试代码及结果如下：
def TestMiddUnorderedList():TEST_NAME = 'MiddUnorderedList'; scale = int(1e5)List = range(scale) + random.sample(xrange(scale), scale) + range(scale)print 'Test 2: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_numpy(numpy.array(List)) # 1572.295 msec IsListSorted_guess(List) # 14540.637 msecIsListSorted_itermap(List) # 21013.096 msecIsListSorted_fastk(List) # 23840.582 msecIsListSorted_allxran(List) # 31014.215 msecIsListSorted_iterzip(List) # 33386.059 msecIsListSorted_forloop(List) # 34228.006 msecIsListSorted_allenumk(List) # 34416.802 msecIsListSorted_allzip(List) # 42370.528 msecIsListSorted_sorted(List) # 142592.756 msecIsListSorted_reduce(List) # 187514.967 msecReportTimer()
为节省篇幅，已根据运行输出调整函数的调用顺序。下文也作如此处理。
3.3 列表后段乱序
测试代码及结果如下：
def TestTailUnorderedList():TEST_NAME = 'TailUnorderedList'; scale = int(1e5)List = range(scale, 0, -1) + random.sample(xrange(scale), scale)print 'Test 3: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_numpy(numpy.array(List), key=lambda dif: dif <= 0) # 980.789 msec IsListSorted_guess(List) # 13273.862 msecIsListSorted_itermap(List, key=lambda x, y: x >= y) # 21603.315 msecIsListSorted_fastk(List, key=lambda x, y: x >= y) # 24183.548 msecIsListSorted_allxran(List, key=lambda x, y: x >= y) # 32850.254 msecIsListSorted_forloop(List, key=lambda x, y: x >= y) # 33918.848 msecIsListSorted_iterzip(List, key=lambda x, y: x >= y) # 34505.809 msecIsListSorted_allenumk(List, key=lambda x, y: x >= y) # 35631.625 msecIsListSorted_allzip(List, key=lambda x, y: x >= y) # 40076.330 msecReportTimer()
3.4 列表完全乱序
测试代码及结果如下：
def TestUnorderedList():TEST_NAME = 'UnorderedList'; scale = int(1e5)List = random.sample(xrange(scale), scale)print 'Test 4: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_fastk(List) # 0.856 msecIsListSorted_allxran(List) # 3.438 msecIsListSorted_iterzip(List) # 7.233 msecIsListSorted_itermap(List) # 7.595 msecIsListSorted_numpy(numpy.array(List)) # 207.222 msecIsListSorted_allenumk(List) # 953.423 msecIsListSorted_guess(List) # 1023.575 msecIsListSorted_forloop(List) # 1076.986 msecIsListSorted_allzip(List) # 2062.821 msecReportTimer()
3.5 列表元素相同
测试代码及结果如下：
```python def TestSameElemList(): TEST_NAME = 'SameElemList'; scale = int(1e5) List = [5]*scale print 'Test 5: %s, list len: %d' %(TEST_NAME, len(List)) IsListSorted_numpy(numpy.array(List)) # 209.324 msec IsListSorted_sorted(List) # 2760.139 msec IsListSorted_guess(List) # 5843.942 msec IsListSorted_itermap(List) # 20609.704 msec IsListSorted_fastk(List) # 23035.760 msec IsListSorted_forloop(List) # 29043.206 msec IsListSorted_allenumk(List) # 29553.716 msec IsListSorted_allxran(List) # 30348.549 msec IsListSorted_iterzip(List) # 32806.217 msec ReportTimer()
3.6 列表升序
测试代码及结果如下：
def TestAscendingList():TEST_NAME = 'AscendingList'; scale = int(1e5)List = range(scale)print 'Test 6: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_numpy(numpy.array(List)) # 209.217 msecIsListSorted_sorted(List) # 2845.166 msecIsListSorted_fastd(List) # 5977.520 msecIsListSorted_guess(List) # 10408.204 msecIsListSorted_allenumd(List) # 15812.754 msecIsListSorted_itermap(List) # 21244.476 msecIsListSorted_fastk(List) # 23900.196 msecIsListSorted_allenumo(List) # 28607.724 msecIsListSorted_forloop(List) # 30075.538 msecIsListSorted_allenumk(List) # 30274.017 msecIsListSorted_allxran(List) # 31126.404 msecIsListSorted_reduce(List) # 32940.108 msecIsListSorted_iterzip(List) # 34188.312 msecIsListSorted_iterzipf(List) # 34425.097 msecIsListSorted_allzip(List) # 37967.447 msecReportTimer()
可见，列表已排序时，_sorted()的性能较占优势。
3.7 列表降序
剔除不支持降序的函数，测试代码及结果如下：
def TestDescendingList():TEST_NAME = 'DescendingList'; scale = int(1e2)List = range(scale, 0, -1)print 'Test 7: %s, list len: %d' %(TEST_NAME, len(List))IsListSorted_numpy(numpy.array(List), key=lambda dif: dif <= 0) # 209.318 msecIsListSorted_sorted(List) # 5707.067 msecIsListSorted_guess(List) # 10549.928 msecIsListSorted_itermap(List, key=lambda x, y: x >= y) # 21529.547 msecIsListSorted_fastk(List, key=lambda x, y: x >= y) # 24264.465 msecimport operator; IsListSorted_allenumo(List, oCmp=operator.ge) # 28093.035 msecIsListSorted_forloop(List, key=lambda x, y: x >= y) # 30745.943 msecIsListSorted_allenumk(List, key=lambda x, y: x >= y) # 32696.205 msecIsListSorted_allxran(List, key=lambda x, y: x >= y) # 33431.473 msecIsListSorted_allzip(List, key=lambda x, y: x >= y) # 34837.019 msecIsListSorted_iterzip(List, key=lambda x, y: x >= y) # 35237.475 msecIsListSorted_reduce(List, key=lambda x, y: x >= y) # 37035.270 msecReportTimer()
3.8 迭代器测试
参数为列表的函数，需要先将列表���过iter()函数转换为迭代器。注意，当iterable参数为iterator时，只能计时一次，因为该次执行将用尽迭代器。
测试代码如下：
def TestIter():TEST_NAME = 'Iter'; scale = int(1e7)List = range(scale) #升序# List = random.sample(xrange(scale), scale) #乱序print 'Test 8: %s, list len: %d' %(TEST_NAME, len(List))iterL = iter(List); IsListSorted_guess(list(iterL))iterL = iter(List); IsListSorted_sorted(iterL)iterL = iter(List); IsListSorted_itermap(iterL)iterL = iter(List); IsListSorted_iterzipf(iterL)iterL = iter(List); IsListSorted_iterzip(iterL)iterL = iter(List); IsListSorted_reduce(iterL)iterL = iter(List); IsListSorted_fastd(iterL)iterL = iter(List); IsListSorted_fastk(iterL, key=lambda x, y: x >= y)ReportTimer()
运行结果如下：
Test 8: Iter, list len: 10000000 ---升序IsListSorted_fastd(): True =>Time Elasped: 611.028 msec, repeated 1 time(s).IsListSorted_sorted(): False =>Time Elasped: 721.751 msec, repeated 1 time(s).IsListSorted_guess(): True =>Time Elasped: 1142.065 msec, repeated 1 time(s).IsListSorted_itermap(): True =>Time Elasped: 2097.696 msec, repeated 1 time(s).IsListSorted_fastk(): True =>Time Elasped: 2337.233 msec, repeated 1 time(s).IsListSorted_reduce(): True =>Time Elasped: 3307.361 msec, repeated 1 time(s).IsListSorted_iterzipf(): True =>Time Elasped: 3354.034 msec, repeated 1 time(s).IsListSorted_iterzip(): True =>Time Elasped: 3442.520 msec, repeated 1 time(s).Test 8: Iter, list len: 10000000 ---乱序IsListSorted_fastk(): False =>Time Elasped: 0.004 msec, repeated 1 time(s).IsListSorted_fastd(): False =>Time Elasped: 0.010 msec, repeated 1 time(s).IsListSorted_iterzip(): False =>Time Elasped: 0.015 msec, repeated 1 time(s).IsListSorted_itermap(): False =>Time Elasped: 0.055 msec, repeated 1 time(s).IsListSorted_iterzipf(): False =>Time Elasped: 0.062 msec, repeated 1 time(s).IsListSorted_guess(): False =>Time Elasped: 736.810 msec, repeated 1 time(s).IsListSorted_reduce(): False =>Time Elasped: 8919.611 msec, repeated 1 time(s).IsListSorted_sorted(): False =>Time Elasped: 12273.018 msec, repeated 1 time(s).
其中，_itermap()、_iterzip()、_iterzipf()、_reduce()、_fastd()、_fastk()可直接判断迭代器是否已排序。其他函数需将迭代器转换为列表后才能处理。_sorted()虽然接受迭代器参数，但sorted()返回列表，故无法正确判断迭代器顺序。
3.9 随机采样测试
综合以上测试，可知_fastk()和_numpy()性能较为突出，而且_rand()内置numpy方式。因此，以_fastk()和_numpy()为参照对象，测试代码如下(计时1次)：
def TestRandList():scale = int(1e6)List = random.sample(xrange(scale), scale) + range(scale)print 'Test 1: %s, list len: %d' %('HeadUnorderedList', len(List))IsListSorted_fastk(List)IsListSorted_numpy(numpy.array(List))IsListSorted_rand(List, randNum=1)ReportTimer()List = range(scale) + random.sample(xrange(scale), scale) + range(scale)print 'Test 2: %s, list len: %d' %('MiddUnorderedList', len(List))IsListSorted_fastk(List)IsListSorted_numpy(numpy.array(List))IsListSorted_rand(List, randNum=1)ReportTimer()List = range(scale, 0, -1) + random.sample(xrange(scale), scale)print 'Test 3: %s, list len: %d' %('TailUnorderedList', len(List))IsListSorted_fastk(List, key=lambda x, y: x >= y)IsListSorted_numpy(numpy.array(List), key=lambda dif: dif <= 0)IsListSorted_rand(List, randNum=1)ReportTimer()List = [random.randint(1,scale) for i in xrange(scale)] #random.sample(xrange(scale), scale)print 'Test 4: %s, list len: %d' %('UnorderedList', len(List))IsListSorted_fastk(List)IsListSorted_numpy(numpy.array(List))IsListSorted_rand(List, randNum=1)ReportTimer()List = [5]*scaleprint 'Test 5: %s, list len: %d' %('SameElemList', len(List))IsListSorted_fastk(List)IsListSorted_numpy(numpy.array(List))IsListSorted_rand(List, randNum=1)ReportTimer()List = range(scale)print 'Test 6: %s, list len: %d' %('AscendingList', len(List))IsListSorted_fastk(List)IsListSorted_numpy(numpy.array(List))IsListSorted_rand(List, randNum=1)ReportTimer()List = range(scale, 0, -1)print 'Test 7: %s, list len: %d' %('DescendingList', len(List))IsListSorted_fastk(List, key=lambda x, y: x >= y)IsListSorted_numpy(numpy.array(List), key=lambda dif: dif <= 0)IsListSorted_rand(List, randNum=1)ReportTimer()List = range(scale, 0, -1); List[scale/2]=0print 'Test 8: %s, list len: %d' %('MiddleNotchList', len(List))IsListSorted_fastk(List, key=lambda x, y: x >= y)IsListSorted_numpy(numpy.array(List), key=lambda dif: dif <= 0)IsListSorted_rand(List, randNum=1)IsListSorted_rand(List, randNum=1, randLen=scale/2)ReportTimer()运行输出如下：
Test 1: HeadUnorderedList, list len: 2000000IsListSorted_fastk(): False =>Time Elasped: 0.095 msec, repeated 1 time(s).IsListSorted_rand(): False =>Time Elasped: 0.347 msec, repeated 1 time(s).IsListSorted_numpy(): False =>Time Elasped: 7.893 msec, repeated 1 time(s).Test 2: MiddUnorderedList, list len: 3000000IsListSorted_rand(): False =>Time Elasped: 0.427 msec, repeated 1 time(s).IsListSorted_numpy(): False =>Time Elasped: 11.868 msec, repeated 1 time(s).IsListSorted_fastk(): False =>Time Elasped: 210.842 msec, repeated 1 time(s).Test 3: TailUnorderedList, list len: 2000000IsListSorted_rand(): False =>Time Elasped: 0.355 msec, repeated 1 time(s).IsListSorted_numpy(): False =>Time Elasped: 7.548 msec, repeated 1 time(s).IsListSorted_fastk(): False =>Time Elasped: 280.416 msec, repeated 1 time(s).Test 4: UnorderedList, list len: 1000000IsListSorted_fastk(): False =>Time Elasped: 0.074 msec, repeated 1 time(s).IsListSorted_rand(): False =>Time Elasped: 0.388 msec, repeated 1 time(s).IsListSorted_numpy(): False =>Time Elasped: 3.757 msec, repeated 1 time(s).Test 5: SameElemList, list len: 1000000IsListSorted_rand(): True =>Time Elasped: 0.304 msec, repeated 1 time(s).IsListSorted_numpy(): True =>Time Elasped: 3.955 msec, repeated 1 time(s).IsListSorted_fastk(): True =>Time Elasped: 210.977 msec, repeated 1 time(s).Test 6: AscendingList, list len: 1000000IsListSorted_rand(): True =>Time Elasped: 0.299 msec, repeated 1 time(s).IsListSorted_numpy(): True =>Time Elasped: 4.822 msec, repeated 1 time(s).IsListSorted_fastk(): True =>Time Elasped: 214.171 msec, repeated 1 time(s).Test 7: DescendingList, list len: 1000000IsListSorted_rand(): True =>Time Elasped: 0.336 msec, repeated 1 time(s).IsListSorted_numpy(): True =>Time Elasped: 3.867 msec, repeated 1 time(s).IsListSorted_fastk(): True =>Time Elasped: 279.322 msec, repeated 1 time(s).Test 8: MiddleNotchList, list len: 1000000IsListSorted_rand(): True =>Time Elasped: 0.387 msec, repeated 1 time(s).IsListSorted_numpy(): False =>Time Elasped: 4.792 msec, repeated 1 time(s).IsListSorted_rand(): False =>Time Elasped: 78.903 msec, repeated 1 time(s).IsListSorted_fastk(): False =>Time Elasped: 110.444 msec, repeated 1 time(s).
可见，在绝大部分测试场景中，_rand()性能均为最佳，且不失正确率。注意测试8，当降序列表中间某个元素被置0(开槽)时，随机采样很容易遗漏该元素，导致误判。然而，这种场景在实际使用中非常罕见。
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