9.7. itertools
— Functions creating iterators for efficient looping¶
버전 2.3에 추가.
This module implements a number of iterator building blocks inspired by constructs from APL, Haskell, and SML. Each has been recast in a form suitable for Python.
The module standardizes a core set of fast, memory efficient tools that are useful by themselves or in combination. Together, they form an 《iterator algebra》 making it possible to construct specialized tools succinctly and efficiently in pure Python.
For instance, SML provides a tabulation tool: tabulate(f)
which produces a
sequence f(0), f(1), ...
. The same effect can be achieved in Python
by combining imap()
and count()
to form imap(f, count())
.
These tools and their builtin counterparts also work well with the highspeed
functions in the operator
module. For example, the multiplication
operator can be mapped across two vectors to form an efficient dotproduct:
sum(imap(operator.mul, vector1, vector2))
.
Infinite Iterators:
Iterator 
Arguments 
Results 
Example 

start, [step] 
start, start+step, start+2*step, … 


p 
p0, p1, … plast, p0, p1, … 


elem [,n] 
elem, elem, elem, … endlessly or up to n times 

Iterators terminating on the shortest input sequence:
Iterator 
Arguments 
Results 
Example 

p, q, … 
p0, p1, … plast, q0, q1, … 


data, selectors 
(d[0] if s[0]), (d[1] if s[1]), … 


pred, seq 
seq[n], seq[n+1], starting when pred fails 


iterable[, keyfunc] 
subiterators grouped by value of keyfunc(v) 

pred, seq 
elements of seq where pred(elem) is true 


pred, seq 
elements of seq where pred(elem) is false 


seq, [start,] stop [, step] 
elements from seq[start:stop:step] 


func, p, q, … 
func(p0, q0), func(p1, q1), … 


func, seq 
func(*seq[0]), func(*seq[1]), … 


it, n 
it1, it2, … itn splits one iterator into n 

pred, seq 
seq[0], seq[1], until pred fails 


p, q, … 
(p[0], q[0]), (p[1], q[1]), … 


p, q, … 
(p[0], q[0]), (p[1], q[1]), … 

Combinatoric generators:
Iterator 
Arguments 
Results 

p, q, … [repeat=1] 
cartesian product, equivalent to a nested forloop 

p[, r] 
rlength tuples, all possible orderings, no repeated elements 

p, r 
rlength tuples, in sorted order, no repeated elements 

p, r 
rlength tuples, in sorted order, with repeated elements 












9.7.1. Itertool functions¶
The following module functions all construct and return iterators. Some provide streams of infinite length, so they should only be accessed by functions or loops that truncate the stream.

itertools.
chain
(*iterables)¶ Make an iterator that returns elements from the first iterable until it is exhausted, then proceeds to the next iterable, until all of the iterables are exhausted. Used for treating consecutive sequences as a single sequence. Roughly equivalent to:
def chain(*iterables): # chain('ABC', 'DEF') > A B C D E F for it in iterables: for element in it: yield element

classmethod
chain.
from_iterable
(iterable)¶ Alternate constructor for
chain()
. Gets chained inputs from a single iterable argument that is evaluated lazily. Roughly equivalent to:def from_iterable(iterables): # chain.from_iterable(['ABC', 'DEF']) > A B C D E F for it in iterables: for element in it: yield element
버전 2.6에 추가.

itertools.
combinations
(iterable, r)¶ Return r length subsequences of elements from the input iterable.
Combinations are emitted in lexicographic sort order. So, if the input iterable is sorted, the combination tuples will be produced in sorted order.
Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeat values in each combination.
Roughly equivalent to:
def combinations(iterable, r): # combinations('ABCD', 2) > AB AC AD BC BD CD # combinations(range(4), 3) > 012 013 023 123 pool = tuple(iterable) n = len(pool) if r > n: return indices = range(r) yield tuple(pool[i] for i in indices) while True: for i in reversed(range(r)): if indices[i] != i + n  r: break else: return indices[i] += 1 for j in range(i+1, r): indices[j] = indices[j1] + 1 yield tuple(pool[i] for i in indices)
The code for
combinations()
can be also expressed as a subsequence ofpermutations()
after filtering entries where the elements are not in sorted order (according to their position in the input pool):def combinations(iterable, r): pool = tuple(iterable) n = len(pool) for indices in permutations(range(n), r): if sorted(indices) == list(indices): yield tuple(pool[i] for i in indices)
The number of items returned is
n! / r! / (nr)!
when0 <= r <= n
or zero whenr > n
.버전 2.6에 추가.

itertools.
combinations_with_replacement
(iterable, r)¶ Return r length subsequences of elements from the input iterable allowing individual elements to be repeated more than once.
Combinations are emitted in lexicographic sort order. So, if the input iterable is sorted, the combination tuples will be produced in sorted order.
Elements are treated as unique based on their position, not on their value. So if the input elements are unique, the generated combinations will also be unique.
Roughly equivalent to:
def combinations_with_replacement(iterable, r): # combinations_with_replacement('ABC', 2) > AA AB AC BB BC CC pool = tuple(iterable) n = len(pool) if not n and r: return indices = [0] * r yield tuple(pool[i] for i in indices) while True: for i in reversed(range(r)): if indices[i] != n  1: break else: return indices[i:] = [indices[i] + 1] * (r  i) yield tuple(pool[i] for i in indices)
The code for
combinations_with_replacement()
can be also expressed as a subsequence ofproduct()
after filtering entries where the elements are not in sorted order (according to their position in the input pool):def combinations_with_replacement(iterable, r): pool = tuple(iterable) n = len(pool) for indices in product(range(n), repeat=r): if sorted(indices) == list(indices): yield tuple(pool[i] for i in indices)
The number of items returned is
(n+r1)! / r! / (n1)!
whenn > 0
.버전 2.7에 추가.

itertools.
compress
(data, selectors)¶ Make an iterator that filters elements from data returning only those that have a corresponding element in selectors that evaluates to
True
. Stops when either the data or selectors iterables has been exhausted. Roughly equivalent to:def compress(data, selectors): # compress('ABCDEF', [1,0,1,0,1,1]) > A C E F return (d for d, s in izip(data, selectors) if s)
버전 2.7에 추가.

itertools.
count
(start=0, step=1)¶ Make an iterator that returns evenly spaced values starting with n. Often used as an argument to
imap()
to generate consecutive data points. Also, used withizip()
to add sequence numbers. Equivalent to:def count(start=0, step=1): # count(10) > 10 11 12 13 14 ... # count(2.5, 0.5) > 2.5 3.0 3.5 ... n = start while True: yield n n += step
When counting with floating point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as:
(start + step * i for i in count())
.버전 2.7에서 변경: added step argument and allowed noninteger arguments.

itertools.
cycle
(iterable)¶ Make an iterator returning elements from the iterable and saving a copy of each. When the iterable is exhausted, return elements from the saved copy. Repeats indefinitely. Roughly equivalent to:
def cycle(iterable): # cycle('ABCD') > A B C D A B C D A B C D ... saved = [] for element in iterable: yield element saved.append(element) while saved: for element in saved: yield element
Note, this member of the toolkit may require significant auxiliary storage (depending on the length of the iterable).

itertools.
dropwhile
(predicate, iterable)¶ Make an iterator that drops elements from the iterable as long as the predicate is true; afterwards, returns every element. Note, the iterator does not produce any output until the predicate first becomes false, so it may have a lengthy startup time. Roughly equivalent to:
def dropwhile(predicate, iterable): # dropwhile(lambda x: x<5, [1,4,6,4,1]) > 6 4 1 iterable = iter(iterable) for x in iterable: if not predicate(x): yield x break for x in iterable: yield x

itertools.
groupby
(iterable[, key])¶ Make an iterator that returns consecutive keys and groups from the iterable. The key is a function computing a key value for each element. If not specified or is
None
, key defaults to an identity function and returns the element unchanged. Generally, the iterable needs to already be sorted on the same key function.The operation of
groupby()
is similar to theuniq
filter in Unix. It generates a break or new group every time the value of the key function changes (which is why it is usually necessary to have sorted the data using the same key function). That behavior differs from SQL’s GROUP BY which aggregates common elements regardless of their input order.The returned group is itself an iterator that shares the underlying iterable with
groupby()
. Because the source is shared, when thegroupby()
object is advanced, the previous group is no longer visible. So, if that data is needed later, it should be stored as a list:groups = [] uniquekeys = [] data = sorted(data, key=keyfunc) for k, g in groupby(data, keyfunc): groups.append(list(g)) # Store group iterator as a list uniquekeys.append(k)
groupby()
is roughly equivalent to:class groupby(object): # [k for k, g in groupby('AAAABBBCCDAABBB')] > A B C D A B # [list(g) for k, g in groupby('AAAABBBCCD')] > AAAA BBB CC D def __init__(self, iterable, key=None): if key is None: key = lambda x: x self.keyfunc = key self.it = iter(iterable) self.tgtkey = self.currkey = self.currvalue = object() def __iter__(self): return self def next(self): while self.currkey == self.tgtkey: self.currvalue = next(self.it) # Exit on StopIteration self.currkey = self.keyfunc(self.currvalue) self.tgtkey = self.currkey return (self.currkey, self._grouper(self.tgtkey)) def _grouper(self, tgtkey): while self.currkey == tgtkey: yield self.currvalue self.currvalue = next(self.it) # Exit on StopIteration self.currkey = self.keyfunc(self.currvalue)
버전 2.4에 추가.

itertools.
ifilter
(predicate, iterable)¶ Make an iterator that filters elements from iterable returning only those for which the predicate is
True
. If predicate isNone
, return the items that are true. Roughly equivalent to:def ifilter(predicate, iterable): # ifilter(lambda x: x%2, range(10)) > 1 3 5 7 9 if predicate is None: predicate = bool for x in iterable: if predicate(x): yield x

itertools.
ifilterfalse
(predicate, iterable)¶ Make an iterator that filters elements from iterable returning only those for which the predicate is
False
. If predicate isNone
, return the items that are false. Roughly equivalent to:def ifilterfalse(predicate, iterable): # ifilterfalse(lambda x: x%2, range(10)) > 0 2 4 6 8 if predicate is None: predicate = bool for x in iterable: if not predicate(x): yield x

itertools.
imap
(function, *iterables)¶ Make an iterator that computes the function using arguments from each of the iterables. If function is set to
None
, thenimap()
returns the arguments as a tuple. Likemap()
but stops when the shortest iterable is exhausted instead of filling inNone
for shorter iterables. The reason for the difference is that infinite iterator arguments are typically an error formap()
(because the output is fully evaluated) but represent a common and useful way of supplying arguments toimap()
. Roughly equivalent to:def imap(function, *iterables): # imap(pow, (2,3,10), (5,2,3)) > 32 9 1000 iterables = map(iter, iterables) while True: args = [next(it) for it in iterables] if function is None: yield tuple(args) else: yield function(*args)

itertools.
islice
(iterable, stop)¶ 
itertools.
islice
(iterable, start, stop[, step]) Make an iterator that returns selected elements from the iterable. If start is nonzero, then elements from the iterable are skipped until start is reached. Afterward, elements are returned consecutively unless step is set higher than one which results in items being skipped. If stop is
None
, then iteration continues until the iterator is exhausted, if at all; otherwise, it stops at the specified position. Unlike regular slicing,islice()
does not support negative values for start, stop, or step. Can be used to extract related fields from data where the internal structure has been flattened (for example, a multiline report may list a name field on every third line). Roughly equivalent to:def islice(iterable, *args): # islice('ABCDEFG', 2) > A B # islice('ABCDEFG', 2, 4) > C D # islice('ABCDEFG', 2, None) > C D E F G # islice('ABCDEFG', 0, None, 2) > A C E G s = slice(*args) start, stop, step = s.start or 0, s.stop or sys.maxint, s.step or 1 it = iter(xrange(start, stop, step))) try: nexti = next(it) except StopIteration: # Consume *iterable* up to the *start* position. for i, element in izip(xrange(start), iterable): pass return try: for i, element in enumerate(iterable): if i == nexti: yield element nexti = next(it) except StopIteration: # Consume to *stop*. for i, element in izip(xrange(i + 1, stop), iterable): pass
If start is
None
, then iteration starts at zero. If step isNone
, then the step defaults to one.버전 2.5에서 변경: accept
None
values for default start and step.

itertools.
izip
(*iterables)¶ Make an iterator that aggregates elements from each of the iterables. Like
zip()
except that it returns an iterator instead of a list. Used for lockstep iteration over several iterables at a time. Roughly equivalent to:def izip(*iterables): # izip('ABCD', 'xy') > Ax By iterators = map(iter, iterables) while iterators: yield tuple(map(next, iterators))
버전 2.4에서 변경: When no iterables are specified, returns a zero length iterator instead of raising a
TypeError
exception.The lefttoright evaluation order of the iterables is guaranteed. This makes possible an idiom for clustering a data series into nlength groups using
izip(*[iter(s)]*n)
.izip()
should only be used with unequal length inputs when you don’t care about trailing, unmatched values from the longer iterables. If those values are important, useizip_longest()
instead.

itertools.
izip_longest
(*iterables[, fillvalue])¶ Make an iterator that aggregates elements from each of the iterables. If the iterables are of uneven length, missing values are filledin with fillvalue. Iteration continues until the longest iterable is exhausted. Roughly equivalent to:
class ZipExhausted(Exception): pass def izip_longest(*args, **kwds): # izip_longest('ABCD', 'xy', fillvalue='') > Ax By C D fillvalue = kwds.get('fillvalue') counter = [len(args)  1] def sentinel(): if not counter[0]: raise ZipExhausted counter[0] = 1 yield fillvalue fillers = repeat(fillvalue) iterators = [chain(it, sentinel(), fillers) for it in args] try: while iterators: yield tuple(map(next, iterators)) except ZipExhausted: pass
If one of the iterables is potentially infinite, then the
izip_longest()
function should be wrapped with something that limits the number of calls (for exampleislice()
ortakewhile()
). If not specified, fillvalue defaults toNone
.버전 2.6에 추가.

itertools.
permutations
(iterable[, r])¶ Return successive r length permutations of elements in the iterable.
If r is not specified or is
None
, then r defaults to the length of the iterable and all possible fulllength permutations are generated.Permutations are emitted in lexicographic sort order. So, if the input iterable is sorted, the permutation tuples will be produced in sorted order.
Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeat values in each permutation.
Roughly equivalent to:
def permutations(iterable, r=None): # permutations('ABCD', 2) > AB AC AD BA BC BD CA CB CD DA DB DC # permutations(range(3)) > 012 021 102 120 201 210 pool = tuple(iterable) n = len(pool) r = n if r is None else r if r > n: return indices = range(n) cycles = range(n, nr, 1) yield tuple(pool[i] for i in indices[:r]) while n: for i in reversed(range(r)): cycles[i] = 1 if cycles[i] == 0: indices[i:] = indices[i+1:] + indices[i:i+1] cycles[i] = n  i else: j = cycles[i] indices[i], indices[j] = indices[j], indices[i] yield tuple(pool[i] for i in indices[:r]) break else: return
The code for
permutations()
can be also expressed as a subsequence ofproduct()
, filtered to exclude entries with repeated elements (those from the same position in the input pool):def permutations(iterable, r=None): pool = tuple(iterable) n = len(pool) r = n if r is None else r for indices in product(range(n), repeat=r): if len(set(indices)) == r: yield tuple(pool[i] for i in indices)
The number of items returned is
n! / (nr)!
when0 <= r <= n
or zero whenr > n
.버전 2.6에 추가.

itertools.
product
(*iterables[, repeat])¶ Cartesian product of input iterables.
Roughly equivalent to nested forloops in a generator expression. For example,
product(A, B)
returns the same as((x,y) for x in A for y in B)
.The nested loops cycle like an odometer with the rightmost element advancing on every iteration. This pattern creates a lexicographic ordering so that if the input’s iterables are sorted, the product tuples are emitted in sorted order.
To compute the product of an iterable with itself, specify the number of repetitions with the optional repeat keyword argument. For example,
product(A, repeat=4)
means the same asproduct(A, A, A, A)
.This function is roughly equivalent to the following code, except that the actual implementation does not build up intermediate results in memory:
def product(*args, **kwds): # product('ABCD', 'xy') > Ax Ay Bx By Cx Cy Dx Dy # product(range(2), repeat=3) > 000 001 010 011 100 101 110 111 pools = map(tuple, args) * kwds.get('repeat', 1) result = [[]] for pool in pools: result = [x+[y] for x in result for y in pool] for prod in result: yield tuple(prod)
버전 2.6에 추가.

itertools.
repeat
(object[, times])¶ Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified. Used as argument to
imap()
for invariant function parameters. Also used withizip()
to create constant fields in a tuple record. Roughly equivalent to:def repeat(object, times=None): # repeat(10, 3) > 10 10 10 if times is None: while True: yield object else: for i in xrange(times): yield object
A common use for repeat is to supply a stream of constant values to imap or zip:
>>> list(imap(pow, xrange(10), repeat(2))) [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

itertools.
starmap
(function, iterable)¶ Make an iterator that computes the function using arguments obtained from the iterable. Used instead of
imap()
when argument parameters are already grouped in tuples from a single iterable (the data has been 《prezipped》). The difference betweenimap()
andstarmap()
parallels the distinction betweenfunction(a,b)
andfunction(*c)
. Roughly equivalent to:def starmap(function, iterable): # starmap(pow, [(2,5), (3,2), (10,3)]) > 32 9 1000 for args in iterable: yield function(*args)
버전 2.6에서 변경: Previously,
starmap()
required the function arguments to be tuples. Now, any iterable is allowed.

itertools.
takewhile
(predicate, iterable)¶ Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:
def takewhile(predicate, iterable): # takewhile(lambda x: x<5, [1,4,6,4,1]) > 1 4 for x in iterable: if predicate(x): yield x else: break

itertools.
tee
(iterable[, n=2])¶ Return n independent iterators from a single iterable. Roughly equivalent to:
def tee(iterable, n=2): it = iter(iterable) deques = [collections.deque() for i in range(n)] def gen(mydeque): while True: if not mydeque: # when the local deque is empty newval = next(it) # fetch a new value and for d in deques: # load it to all the deques d.append(newval) yield mydeque.popleft() return tuple(gen(d) for d in deques)
Once
tee()
has made a split, the original iterable should not be used anywhere else; otherwise, the iterable could get advanced without the tee objects being informed.tee
iterators are not threadsafe. ARuntimeError
may be raised when using simultaneously iterators returned by the sametee()
call, even if the original iterable is threadsafe.This itertool may require significant auxiliary storage (depending on how much temporary data needs to be stored). In general, if one iterator uses most or all of the data before another iterator starts, it is faster to use
list()
instead oftee()
.버전 2.4에 추가.
9.7.2. Recipes¶
This section shows recipes for creating an extended toolset using the existing itertools as building blocks.
The extended tools offer the same high performance as the underlying toolset. The superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style which helps eliminate temporary variables. High speed is retained by preferring 《vectorized》 building blocks over the use of forloops and generators which incur interpreter overhead.
def take(n, iterable):
"Return first n items of the iterable as a list"
return list(islice(iterable, n))
def tabulate(function, start=0):
"Return function(0), function(1), ..."
return imap(function, count(start))
def consume(iterator, n=None):
"Advance the iterator nsteps ahead. If n is None, consume entirely."
# Use functions that consume iterators at C speed.
if n is None:
# feed the entire iterator into a zerolength deque
collections.deque(iterator, maxlen=0)
else:
# advance to the empty slice starting at position n
next(islice(iterator, n, n), None)
def nth(iterable, n, default=None):
"Returns the nth item or a default value"
return next(islice(iterable, n, None), default)
def all_equal(iterable):
"Returns True if all the elements are equal to each other"
g = groupby(iterable)
return next(g, True) and not next(g, False)
def quantify(iterable, pred=bool):
"Count how many times the predicate is true"
return sum(imap(pred, iterable))
def padnone(iterable):
"""Returns the sequence elements and then returns None indefinitely.
Useful for emulating the behavior of the builtin map() function.
"""
return chain(iterable, repeat(None))
def ncycles(iterable, n):
"Returns the sequence elements n times"
return chain.from_iterable(repeat(tuple(iterable), n))
def dotproduct(vec1, vec2):
return sum(imap(operator.mul, vec1, vec2))
def flatten(listOfLists):
"Flatten one level of nesting"
return chain.from_iterable(listOfLists)
def repeatfunc(func, times=None, *args):
"""Repeat calls to func with specified arguments.
Example: repeatfunc(random.random)
"""
if times is None:
return starmap(func, repeat(args))
return starmap(func, repeat(args, times))
def pairwise(iterable):
"s > (s0,s1), (s1,s2), (s2, s3), ..."
a, b = tee(iterable)
next(b, None)
return izip(a, b)
def grouper(iterable, n, fillvalue=None):
"Collect data into fixedlength chunks or blocks"
# grouper('ABCDEFG', 3, 'x') > ABC DEF Gxx
args = [iter(iterable)] * n
return izip_longest(fillvalue=fillvalue, *args)
def roundrobin(*iterables):
"roundrobin('ABC', 'D', 'EF') > A D E B F C"
# Recipe credited to George Sakkis
pending = len(iterables)
nexts = cycle(iter(it).next for it in iterables)
while pending:
try:
for next in nexts:
yield next()
except StopIteration:
pending = 1
nexts = cycle(islice(nexts, pending))
def powerset(iterable):
"powerset([1,2,3]) > () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
s = list(iterable)
return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))
def unique_everseen(iterable, key=None):
"List unique elements, preserving order. Remember all elements ever seen."
# unique_everseen('AAAABBBCCDAABBB') > A B C D
# unique_everseen('ABBCcAD', str.lower) > A B C D
seen = set()
seen_add = seen.add
if key is None:
for element in ifilterfalse(seen.__contains__, iterable):
seen_add(element)
yield element
else:
for element in iterable:
k = key(element)
if k not in seen:
seen_add(k)
yield element
def unique_justseen(iterable, key=None):
"List unique elements, preserving order. Remember only the element just seen."
# unique_justseen('AAAABBBCCDAABBB') > A B C D A B
# unique_justseen('ABBCcAD', str.lower) > A B C A D
return imap(next, imap(itemgetter(1), groupby(iterable, key)))
def iter_except(func, exception, first=None):
""" Call a function repeatedly until an exception is raised.
Converts a calluntilexception interface to an iterator interface.
Like __builtin__.iter(func, sentinel) but uses an exception instead
of a sentinel to end the loop.
Examples:
bsddbiter = iter_except(db.next, bsddb.error, db.first)
heapiter = iter_except(functools.partial(heappop, h), IndexError)
dictiter = iter_except(d.popitem, KeyError)
dequeiter = iter_except(d.popleft, IndexError)
queueiter = iter_except(q.get_nowait, Queue.Empty)
setiter = iter_except(s.pop, KeyError)
"""
try:
if first is not None:
yield first()
while 1:
yield func()
except exception:
pass
def random_product(*args, **kwds):
"Random selection from itertools.product(*args, **kwds)"
pools = map(tuple, args) * kwds.get('repeat', 1)
return tuple(random.choice(pool) for pool in pools)
def random_permutation(iterable, r=None):
"Random selection from itertools.permutations(iterable, r)"
pool = tuple(iterable)
r = len(pool) if r is None else r
return tuple(random.sample(pool, r))
def random_combination(iterable, r):
"Random selection from itertools.combinations(iterable, r)"
pool = tuple(iterable)
n = len(pool)
indices = sorted(random.sample(xrange(n), r))
return tuple(pool[i] for i in indices)
def random_combination_with_replacement(iterable, r):
"Random selection from itertools.combinations_with_replacement(iterable, r)"
pool = tuple(iterable)
n = len(pool)
indices = sorted(random.randrange(n) for i in xrange(r))
return tuple(pool[i] for i in indices)
def tee_lookahead(t, i):
"""Inspect the ith upcomping value from a tee object
while leaving the tee object at its current position.
Raise an IndexError if the underlying iterator doesn't
have enough values.
"""
for value in islice(t.__copy__(), i, None):
return value
raise IndexError(i)
Note, many of the above recipes can be optimized by replacing global lookups with local variables defined as default values. For example, the dotproduct recipe can be written as:
def dotproduct(vec1, vec2, sum=sum, imap=imap, mul=operator.mul):
return sum(imap(mul, vec1, vec2))