itertools — Functions creating iterators for efficient looping

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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 map() and count() to form map(f,count()).

Infinite iterators:

Iterator	Arguments	Results	Example
count()	[start[, step]]	start, start+step, start+2*step, …	count(10)→1011121314...
cycle()	p	p0, p1, … plast, p0, p1, …	cycle('ABCD')→ABCDABCD...
repeat()	elem [,n]	elem, elem, elem, … endlessly or up to n times	repeat(10,3)→101010

Iterators terminating on the shortest input sequence:

Iterator	Arguments	Results	Example
accumulate()	p [,func]	p0, p0+p1, p0+p1+p2, …	accumulate([1,2,3,4,5])→1361015
batched()	p, n	(p0, p1, …, p_n-1), …	batched('ABCDEFG',n=3)→ABCDEFG
chain()	p, q, …	p0, p1, … plast, q0, q1, …	chain('ABC','DEF')→ABCDEF
chain.from_iterable()	iterable	p0, p1, … plast, q0, q1, …	chain.from_iterable(['ABC','DEF'])→ABCDEF
compress()	data, selectors	(d[0] if s[0]), (d[1] if s[1]), …	compress('ABCDEF',[1,0,1,0,1,1])→ACEF
dropwhile()	predicate, seq	seq[n], seq[n+1], starting when predicate fails	dropwhile(lambdax:x<5,[1,4,6,3,8])→638
filterfalse()	predicate, seq	elements of seq where predicate(elem) fails	filterfalse(lambdax:x<5,[1,4,6,3,8])→68
groupby()	iterable[, key]	sub-iterators grouped by value of key(v)	groupby(['A','B','DEF'],len)→(1,AB)(3,DEF)
islice()	seq, [start,] stop [, step]	elements from seq[start:stop:step]	islice('ABCDEFG',2,None)→CDEFG
pairwise()	iterable	(p[0], p[1]), (p[1], p[2])	pairwise('ABCDEFG')→ABBCCDDEEFFG
starmap()	func, seq	func(*seq[0]), func(*seq[1]), …	starmap(pow,[(2,5),(3,2),(10,3)])→3291000
takewhile()	predicate, seq	seq[0], seq[1], until predicate fails	takewhile(lambdax:x<5,[1,4,6,3,8])→14
tee()	it, n	it1, it2, … itn splits one iterator into n
zip_longest()	p, q, …	(p[0], q[0]), (p[1], q[1]), …	zip_longest('ABCD','xy',fillvalue='-')→AxByC-D-

Combinatoric iterators:

Iterator	Arguments	Results
product()	p, q, … [repeat=1]	cartesian product, equivalent to a nested for-loop
permutations()	p[, r]	r-length tuples, all possible orderings, no repeated elements
combinations()	p, r	r-length tuples, in sorted order, no repeated elements
combinations_with_replacement()	p, r	r-length tuples, in sorted order, with repeated elements

Examples	Results
product('ABCD',repeat=2)	AAABACADBABBBCBDCACBCCCDDADBDCDD
permutations('ABCD',2)	ABACADBABCBDCACBCDDADBDC
combinations('ABCD',2)	ABACADBCBDCD
combinations_with_replacement('ABCD', 2)	AAABACADBBBCBDCCCDDD

Itertool Functions

The following 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.accumulate(iterable[, function, *, initial=None])

Make an iterator that returns accumulated sums or accumulated results from other binary functions.

The function defaults to addition. The function should accept two arguments, an accumulated total and a value from the iterable.

If an initial value is provided, the accumulation will start with that value and the output will have one more element than the input iterable.

Roughly equivalent to:

defaccumulate(iterable,function=operator.add,*,initial=None):'Return running totals'# accumulate([1,2,3,4,5]) → 1 3 6 10 15# accumulate([1,2,3,4,5], initial=100) → 100 101 103 106 110 115# accumulate([1,2,3,4,5], operator.mul) → 1 2 6 24 120iterator=iter(iterable)total=initialifinitialisNone:try:total=next(iterator)exceptStopIteration:returnyieldtotalforelementiniterator:total=function(total,element)yieldtotal

To compute a running minimum, set function to min(). For a running maximum, set function to max(). Or for a running product, set function to operator.mul(). To build an amortization table, accumulate the interest and apply payments:

>>> data=[3,4,6,2,1,9,0,7,5,8]>>> list(accumulate(data,max))# running maximum[3, 4, 6, 6, 6, 9, 9, 9, 9, 9]>>> list(accumulate(data,operator.mul))# running product[3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]# Amortize a 5% loan of 1000 with 10 annual payments of 90>>> update=lambdabalance,payment:round(balance*1.05)-payment>>> list(accumulate(repeat(90,10),update,initial=1_000))[1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

See functools.reduce() for a similar function that returns only the final accumulated value.

Added in version 3.2.

Changed in version 3.3: Added the optional function parameter.

Changed in version 3.8: Added the optional initial parameter.

itertools.batched(iterable, n, *, strict=False)

Batch data from the iterable into tuples of length n. The last batch may be shorter than n.

If strict is true, will raise a ValueError if the final batch is shorter than n.

Loops over the input iterable and accumulates data into tuples up to size n. The input is consumed lazily, just enough to fill a batch. The result is yielded as soon as the batch is full or when the input iterable is exhausted:

>>> flattened_data=['roses','red','violets','blue','sugar','sweet']>>> unflattened=list(batched(flattened_data,2))>>> unflattened[('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

Roughly equivalent to:

defbatched(iterable,n,*,strict=False):# batched('ABCDEFG', 3) → ABC DEF Gifn<1:raiseValueError('n must be at least one')iterator=iter(iterable)whilebatch:=tuple(islice(iterator,n)):ifstrictandlen(batch)!=n:raiseValueError('batched(): incomplete batch')yieldbatch

Added in version 3.12.

Changed in version 3.13: Added the strict option.

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. This combines multiple data sources into a single iterator. Roughly equivalent to:

defchain(*iterables):# chain('ABC', 'DEF') → A B C D E Fforiterableiniterables:yield fromiterable

classmethodchain.from_iterable(iterable)

Alternate constructor for chain(). Gets chained inputs from a single iterable argument that is evaluated lazily. Roughly equivalent to:

deffrom_iterable(iterables):# chain.from_iterable(['ABC', 'DEF']) → A B C D E Fforiterableiniterables:yield fromiterable

itertools.combinations(iterable, r)

Return r length subsequences of elements from the input iterable.

The output is a subsequence of product() keeping only entries that are subsequences of the iterable. The length of the output is given by math.comb() which computes n!/r!/(n-r)! when 0≤r≤n or zero when r>n.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within each combination.

Roughly equivalent to:

defcombinations(iterable,r):# combinations('ABCD', 2) → AB AC AD BC BD CD# combinations(range(4), 3) → 012 013 023 123pool=tuple(iterable)n=len(pool)ifr>n:returnindices=list(range(r))yieldtuple(pool[i]foriinindices)whileTrue:foriinreversed(range(r)):ifindices[i]!=i+n-r:breakelse:returnindices[i]+=1forjinrange(i+1,r):indices[j]=indices[j-1]+1yieldtuple(pool[i]foriinindices)

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.

The output is a subsequence of product() that keeps only entries that are subsequences (with possible repeated elements) of the iterable. The number of subsequence returned is (n+r-1)!/r!/(n-1)! when n>0.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. if the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, the generated combinations will also be unique.

Roughly equivalent to:

defcombinations_with_replacement(iterable,r):# combinations_with_replacement('ABC', 2) → AA AB AC BB BC CCpool=tuple(iterable)n=len(pool)ifnotnandr:returnindices=[0]*ryieldtuple(pool[i]foriinindices)whileTrue:foriinreversed(range(r)):ifindices[i]!=n-1:breakelse:returnindices[i:]=[indices[i]+1]*(r-i)yieldtuple(pool[i]foriinindices)

Added in version 3.1.

itertools.compress(data, selectors)

Make an iterator that returns elements from data where the corresponding element in selectors is true. Stops when either the data or selectors iterables have been exhausted. Roughly equivalent to:

defcompress(data,selectors):# compress('ABCDEF', [1,0,1,0,1,1]) → A C E Freturn(datumfordatum,selectorinzip(data,selectors)ifselector)

Added in version 3.1.

itertools.count(start=0, step=1)

Make an iterator that returns evenly spaced values beginning with start. Can be used with map() to generate consecutive data points or with zip() to add sequence numbers. Roughly equivalent to:

defcount(start=0,step=1):# count(10) → 10 11 12 13 14 ...# count(2.5, 0.5) → 2.5 3.0 3.5 ...n=startwhileTrue:yieldnn+=step

When counting with floating-point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: (start+step*iforiincount()).

Changed in version 3.1: Added step argument and allowed non-integer 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:

defcycle(iterable):# cycle('ABCD') → A B C D A B C D A B C D ...saved=[]forelementiniterable:yieldelementsaved.append(element)whilesaved:forelementinsaved:yieldelement

This itertool 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 while the predicate is true and afterwards returns every element. Roughly equivalent to:

defdropwhile(predicate,iterable):# dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8iterator=iter(iterable)forxiniterator:ifnotpredicate(x):yieldxbreakforxiniterator:yieldx

Note this does not produce any output until the predicate first becomes false, so this itertool may have a lengthy start-up time.

itertools.filterfalse(predicate, iterable)

Make an iterator that filters elements from the iterable returning only those for which the predicate returns a false value. If predicate is None, returns the items that are false. Roughly equivalent to:

deffilterfalse(predicate,iterable):# filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8ifpredicateisNone:predicate=boolforxiniterable:ifnotpredicate(x):yieldx

itertools.groupby(iterable, key=None)

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 the uniq 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 the groupby() 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)fork,gingroupby(data,keyfunc):groups.append(list(g))# Store group iterator as a listuniquekeys.append(k)

groupby() is roughly equivalent to:

defgroupby(iterable,key=None):# [k for k, g in groupby('AAAABBBCCDAABBB')] → A B C D A B# [list(g) for k, g in groupby('AAAABBBCCD')] → AAAA BBB CC Dkeyfunc=(lambdax:x)ifkeyisNoneelsekeyiterator=iter(iterable)exhausted=Falsedef_grouper(target_key):nonlocalcurr_value,curr_key,exhaustedyieldcurr_valueforcurr_valueiniterator:curr_key=keyfunc(curr_value)ifcurr_key!=target_key:returnyieldcurr_valueexhausted=Truetry:curr_value=next(iterator)exceptStopIteration:returncurr_key=keyfunc(curr_value)whilenotexhausted:target_key=curr_keycurr_group=_grouper(target_key)yieldcurr_key,curr_groupifcurr_key==target_key:for_incurr_group:pass

itertools.islice(iterable, stop)

itertools.islice(iterable, start, stop[, step])

Make an iterator that returns selected elements from the iterable. Works like sequence slicing but does not support negative values for start, stop, or step.

If start is zero or None, iteration starts at zero. Otherwise, elements from the iterable are skipped until start is reached.

If stop is None, iteration continues until the input is exhausted, if at all. Otherwise, it stops at the specified position.

If step is None, the step defaults to one. Elements are returned consecutively unless step is set higher than one which results in items being skipped.

Roughly equivalent to:

defislice(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 Gs=slice(*args)start=0ifs.startisNoneelses.startstop=s.stopstep=1ifs.stepisNoneelses.stepifstart<0or(stopisnotNoneandstop<0)orstep<=0:raiseValueErrorindices=count()ifstopisNoneelserange(max(start,stop))next_i=startfori,elementinzip(indices,iterable):ifi==next_i:yieldelementnext_i+=step

If the input is an iterator, then fully consuming the islice advances the input iterator by max(start,stop) steps regardless of the step value.

itertools.pairwise(iterable)

Return successive overlapping pairs taken from the input iterable.

The number of 2-tuples in the output iterator will be one fewer than the number of inputs. It will be empty if the input iterable has fewer than two values.

Roughly equivalent to:

defpairwise(iterable):# pairwise('ABCDEFG') → AB BC CD DE EF FGiterator=iter(iterable)a=next(iterator,None)forbiniterator:yielda,ba=b

Added in version 3.10.

itertools.permutations(iterable, r=None)

Return successive r length permutations of elements from the iterable.

If r is not specified or is None, then r defaults to the length of the iterable and all possible full-length permutations are generated.

The output is a subsequence of product() where entries with repeated elements have been filtered out. The length of the output is given by math.perm() which computes n!/(n-r)! when 0≤r≤n or zero when r>n.

The permutation tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within a permutation.

Roughly equivalent to:

defpermutations(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 210pool=tuple(iterable)n=len(pool)r=nifrisNoneelserifr>n:returnindices=list(range(n))cycles=list(range(n,n-r,-1))yieldtuple(pool[i]foriinindices[:r])whilen:foriinreversed(range(r)):cycles[i]-=1ifcycles[i]==0:indices[i:]=indices[i+1:]+indices[i:i+1]cycles[i]=n-ielse:j=cycles[i]indices[i],indices[-j]=indices[-j],indices[i]yieldtuple(pool[i]foriinindices[:r])breakelse:return

itertools.product(*iterables, repeat=1)

Cartesian product of the input iterables.

Roughly equivalent to nested for-loops in a generator expression. For example, product(A,B) returns the same as ((x,y)forxinAforyinB).

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 as product(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:

defproduct(*iterables,repeat=1):# product('ABCD', 'xy') → Ax Ay Bx By Cx Cy Dx Dy# product(range(2), repeat=3) → 000 001 010 011 100 101 110 111ifrepeat<0:raiseValueError('repeat argument cannot be negative')pools=[tuple(pool)forpooliniterables]*repeatresult=[[]]forpoolinpools:result=[x+[y]forxinresultforyinpool]forprodinresult:yieldtuple(prod)

Before product() runs, it completely consumes the input iterables, keeping pools of values in memory to generate the products. Accordingly, it is only useful with finite inputs.

itertools.repeat(object[, times])

Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified.

Roughly equivalent to:

defrepeat(object,times=None):# repeat(10, 3) → 10 10 10iftimesisNone:whileTrue:yieldobjectelse:foriinrange(times):yieldobject

A common use for repeat is to supply a stream of constant values to map or zip:

>>> list(map(pow,range(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 map() when argument parameters have already been “pre-zipped” into tuples.

The difference between map() and starmap() parallels the distinction between function(a,b) and function(*c). Roughly equivalent to:

defstarmap(function,iterable):# starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000forargsiniterable:yieldfunction(*args)

itertools.takewhile(predicate, iterable)

Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:

deftakewhile(predicate,iterable):# takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4forxiniterable:ifnotpredicate(x):breakyieldx

Note, the element that first fails the predicate condition is consumed from the input iterator and there is no way to access it. This could be an issue if an application wants to further consume the input iterator after takewhile has been run to exhaustion. To work around this problem, consider using more-itertools before_and_after() instead.

itertools.tee(iterable, n=2)

Return n independent iterators from a single iterable.

Roughly equivalent to:

deftee(iterable,n=2):ifn<0:raiseValueErrorifn==0:return()iterator=_tee(iterable)result=[iterator]for_inrange(n-1):result.append(_tee(iterator))returntuple(result)class_tee:def__init__(self,iterable):it=iter(iterable)ifisinstance(it,_tee):self.iterator=it.iteratorself.link=it.linkelse:self.iterator=itself.link=[None,None]def__iter__(self):returnselfdef__next__(self):link=self.linkiflink[1]isNone:link[0]=next(self.iterator)link[1]=[None,None]value,self.link=linkreturnvalue

When the input iterable is already a tee iterator object, all members of the return tuple are constructed as if they had been produced by the upstream tee() call. This “flattening step” allows nested tee() calls to share the same underlying data chain and to have a single update step rather than a chain of calls.

The flattening property makes tee iterators efficiently peekable:

deflookahead(tee_iterator):"Return the next value without moving the input forward"[forked_iterator]=tee(tee_iterator,1)returnnext(forked_iterator)

>>> iterator=iter('abcdef')>>> [iterator]=tee(iterator,1)# Make the input peekable>>> next(iterator)# Move the iterator forward'a'>>> lookahead(iterator)# Check next value'b'>>> next(iterator)# Continue moving forward'b'

tee iterators are not threadsafe. A RuntimeError may be raised when simultaneously using iterators returned by the same tee() 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 of tee().

itertools.zip_longest(*iterables, fillvalue=None)

Make an iterator that aggregates elements from each of the iterables.

If the iterables are of uneven length, missing values are filled-in with fillvalue. If not specified, fillvalue defaults to None.

Iteration continues until the longest iterable is exhausted.

Roughly equivalent to:

defzip_longest(*iterables,fillvalue=None):# zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-iterators=list(map(iter,iterables))num_active=len(iterators)ifnotnum_active:returnwhileTrue:values=[]fori,iteratorinenumerate(iterators):try:value=next(iterator)exceptStopIteration:num_active-=1ifnotnum_active:returniterators[i]=repeat(fillvalue)value=fillvaluevalues.append(value)yieldtuple(values)

If one of the iterables is potentially infinite, then the zip_longest() function should be wrapped with something that limits the number of calls (for example islice() or takewhile()).

Itertools Recipes

This section shows recipes for creating an extended toolset using the existing itertools as building blocks.

The primary purpose of the itertools recipes is educational. The recipes show various ways of thinking about individual tools — for example, that chain.from_iterable is related to the concept of flattening. The recipes also give ideas about ways that the tools can be combined — for example, how starmap() and repeat() can work together. The recipes also show patterns for using itertools with the operator and collections modules as well as with the built-in itertools such as map(), filter(), reversed(), and enumerate().

A secondary purpose of the recipes is to serve as an incubator. The accumulate(), compress(), and pairwise() itertools started out as recipes. Currently, the sliding_window(), iter_index(), and sieve() recipes are being tested to see whether they prove their worth.

Substantially all of these recipes and many, many others can be installed from the more-itertools project found on the Python Package Index:

python-mpipinstallmore-itertools

Many of the recipes offer the same high performance as the underlying toolset. 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. High speed is retained by preferring “vectorized” building blocks over the use of for-loops and generators which incur interpreter overhead.

fromcollectionsimportCounter,dequefromcontextlibimportsuppressfromfunctoolsimportreducefrommathimportcomb,prod,sumprod,isqrtfromoperatorimportitemgetter,getitem,mul,negdeftake(n,iterable):"Return first n items of the iterable as a list."returnlist(islice(iterable,n))defprepend(value,iterable):"Prepend a single value in front of an iterable."# prepend(1, [2, 3, 4]) → 1 2 3 4returnchain([value],iterable)deftabulate(function,start=0):"Return function(0), function(1), ..."returnmap(function,count(start))defrepeatfunc(function,times=None,*args):"Repeat calls to a function with specified arguments."iftimesisNone:returnstarmap(function,repeat(args))returnstarmap(function,repeat(args,times))defflatten(list_of_lists):"Flatten one level of nesting."returnchain.from_iterable(list_of_lists)defncycles(iterable,n):"Returns the sequence elements n times."returnchain.from_iterable(repeat(tuple(iterable),n))defloops(n):"Loop n times. Like range(n) but without creating integers."# for _ in loops(100): ...returnrepeat(None,n)deftail(n,iterable):"Return an iterator over the last n items."# tail(3, 'ABCDEFG') → E F Greturniter(deque(iterable,maxlen=n))defconsume(iterator,n=None):"Advance the iterator n-steps ahead. If n is None, consume entirely."# Use functions that consume iterators at C speed.ifnisNone:deque(iterator,maxlen=0)else:next(islice(iterator,n,n),None)defnth(iterable,n,default=None):"Returns the nth item or a default value."returnnext(islice(iterable,n,None),default)defquantify(iterable,predicate=bool):"Given a predicate that returns True or False, count the True results."returnsum(map(predicate,iterable))deffirst_true(iterable,default=False,predicate=None):"Returns the first true value or the *default* if there is no true value."# first_true([a,b,c], x) → a or b or c or x# first_true([a,b], x, f) → a if f(a) else b if f(b) else xreturnnext(filter(predicate,iterable),default)defall_equal(iterable,key=None):"Returns True if all the elements are equal to each other."# all_equal('4٤௪౪໔', key=int) → Truereturnlen(take(2,groupby(iterable,key)))<=1defunique_justseen(iterable,key=None):"Yield unique elements, preserving order. Remember only the element just seen."# unique_justseen('AAAABBBCCDAABBB') → A B C D A B# unique_justseen('ABBcCAD', str.casefold) → A B c A DifkeyisNone:returnmap(itemgetter(0),groupby(iterable))returnmap(next,map(itemgetter(1),groupby(iterable,key)))defunique_everseen(iterable,key=None):"Yield unique elements, preserving order. Remember all elements ever seen."# unique_everseen('AAAABBBCCDAABBB') → A B C D# unique_everseen('ABBcCAD', str.casefold) → A B c Dseen=set()ifkeyisNone:forelementinfilterfalse(seen.__contains__,iterable):seen.add(element)yieldelementelse:forelementiniterable:k=key(element)ifknotinseen:seen.add(k)yieldelementdefunique(iterable,key=None,reverse=False):"Yield unique elements in sorted order. Supports unhashable inputs."# unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4]sequenced=sorted(iterable,key=key,reverse=reverse)returnunique_justseen(sequenced,key=key)defsliding_window(iterable,n):"Collect data into overlapping fixed-length chunks or blocks."# sliding_window('ABCDEFG', 4) → ABCD BCDE CDEF DEFGiterator=iter(iterable)window=deque(islice(iterator,n-1),maxlen=n)forxiniterator:window.append(x)yieldtuple(window)defgrouper(iterable,n,*,incomplete='fill',fillvalue=None):"Collect data into non-overlapping fixed-length chunks or blocks."# grouper('ABCDEFG', 3, fillvalue='x') → ABC DEF Gxx# grouper('ABCDEFG', 3, incomplete='strict') → ABC DEF ValueError# grouper('ABCDEFG', 3, incomplete='ignore') → ABC DEFiterators=[iter(iterable)]*nmatchincomplete:case'fill':returnzip_longest(*iterators,fillvalue=fillvalue)case'strict':returnzip(*iterators,strict=True)case'ignore':returnzip(*iterators)case_:raiseValueError('Expected fill, strict, or ignore')defroundrobin(*iterables):"Visit input iterables in a cycle until each is exhausted."# roundrobin('ABC', 'D', 'EF') → A D E B F C# Algorithm credited to George Sakkisiterators=map(iter,iterables)fornum_activeinrange(len(iterables),0,-1):iterators=cycle(islice(iterators,num_active))yield frommap(next,iterators)defsubslices(seq):"Return all contiguous non-empty subslices of a sequence."# subslices('ABCD') → A AB ABC ABCD B BC BCD C CD Dslices=starmap(slice,combinations(range(len(seq)+1),2))returnmap(getitem,repeat(seq),slices)defiter_index(iterable,value,start=0,stop=None):"Return indices where a value occurs in a sequence or iterable."# iter_index('AABCADEAF', 'A') → 0 1 4 7seq_index=getattr(iterable,'index',None)ifseq_indexisNone:iterator=islice(iterable,start,stop)fori,elementinenumerate(iterator,start):ifelementisvalueorelement==value:yieldielse:stop=len(iterable)ifstopisNoneelsestopi=startwithsuppress(ValueError):whileTrue:yield(i:=seq_index(value,i,stop))i+=1defiter_except(function,exception,first=None):"Convert a call-until-exception interface to an iterator interface."# iter_except(d.popitem, KeyError) → non-blocking dictionary iteratorwithsuppress(exception):iffirstisnotNone:yieldfirst()whileTrue:yieldfunction()

The following recipes have a more mathematical flavor:

defmultinomial(*counts):"Number of distinct arrangements of a multiset."# Counter('abracadabra').values() → 5 2 2 1 1# multinomial(5, 2, 2, 1, 1) → 83160returnprod(map(comb,accumulate(counts),counts))defpowerset(iterable):"Subsequences of the iterable from shortest to longest."# powerset([1,2,3]) → () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)s=list(iterable)returnchain.from_iterable(combinations(s,r)forrinrange(len(s)+1))defsum_of_squares(iterable):"Add up the squares of the input values."# sum_of_squares([10, 20, 30]) → 1400returnsumprod(*tee(iterable))defreshape(matrix,columns):"Reshape a 2-D matrix to have a given number of columns."# reshape([(0, 1), (2, 3), (4, 5)], 3) → (0, 1, 2), (3, 4, 5)returnbatched(chain.from_iterable(matrix),columns,strict=True)deftranspose(matrix):"Swap the rows and columns of a 2-D matrix."# transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33)returnzip(*matrix,strict=True)defmatmul(m1,m2):"Multiply two matrices."# matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) → (49, 80), (41, 60)n=len(m2[0])returnbatched(starmap(sumprod,product(m1,transpose(m2))),n)defconvolve(signal,kernel):"""Discrete linear convolution of two iterables. Equivalent to polynomial multiplication. Convolutions are mathematically commutative; however, the inputs are evaluated differently. The signal is consumed lazily and can be infinite. The kernel is fully consumed before the calculations begin. Article: https://betterexplained.com/articles/intuitive-convolution/ Video: https://www.youtube.com/watch?v=KuXjwB4LzSA """# convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60# convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)# convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate# convolve(data, [1, -2, 1]) → 2nd derivative estimatekernel=tuple(kernel)[::-1]n=len(kernel)padded_signal=chain(repeat(0,n-1),signal,repeat(0,n-1))windowed_signal=sliding_window(padded_signal,n)returnmap(sumprod,repeat(kernel),windowed_signal)defpolynomial_from_roots(roots):"""Compute a polynomial's coefficients from its roots. (x - 5) (x + 4) (x - 3) expands to: x³ -4x² -17x + 60 """# polynomial_from_roots([5, -4, 3]) → [1, -4, -17, 60]factors=zip(repeat(1),map(neg,roots))returnlist(reduce(convolve,factors,[1]))defpolynomial_eval(coefficients,x):"""Evaluate a polynomial at a specific value. Computes with better numeric stability than Horner's method. """# Evaluate x³ -4x² -17x + 60 at x = 5# polynomial_eval([1, -4, -17, 60], x=5) → 0n=len(coefficients)ifnotn:returntype(x)(0)powers=map(pow,repeat(x),reversed(range(n)))returnsumprod(coefficients,powers)defpolynomial_derivative(coefficients):"""Compute the first derivative of a polynomial. f(x) = x³ -4x² -17x + 60 f'(x) = 3x² -8x -17 """# polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]n=len(coefficients)powers=reversed(range(1,n))returnlist(map(mul,coefficients,powers))defsieve(n):"Primes less than n."# sieve(30) → 2 3 5 7 11 13 17 19 23 29ifn>2:yield2data=bytearray((0,1))*(n//2)forpiniter_index(data,1,start=3,stop=isqrt(n)+1):data[p*p:n:p+p]=bytes(len(range(p*p,n,p+p)))yield fromiter_index(data,1,start=3)deffactor(n):"Prime factors of n."# factor(99) → 3 3 11# factor(1_000_000_000_000_007) → 47 59 360620266859# factor(1_000_000_000_000_403) → 1000000000000403forprimeinsieve(isqrt(n)+1):whilenotn%prime:yieldprimen//=primeifn==1:returnifn>1:yieldndefis_prime(n):"Return True if n is prime."# is_prime(1_000_000_000_000_403) → Truereturnn>1andnext(factor(n))==ndeftotient(n):"Count of natural numbers up to n that are coprime to n."# https://mathworld.wolfram.com/TotientFunction.html# totient(12) → 4 because len([1, 5, 7, 11]) == 4forprimeinset(factor(n)):n-=n//primereturnn