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Memoization fibonacci algorithm in python
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Closed 2 years ago.
I'm working on a problem in codewars that wants you to memoize The Fibonacci sequence. My solution so far has been:
def fibonacci(n):
return fibonacci_helper(n, dict())
def fibonacci_helper(n, fib_nums):
if n in [0, 1]:
return fib_nums.setdefault(n, n)
fib1 = fib_nums.setdefault(n - 1, fibonacci_helper(n - 1, fib_nums))
fib2 = fib_nums.setdefault(n - 2, fibonacci_helper(n - 2, fib_nums))
return fib_nums.setdefault(n, fib1 + fib2)
It works reasonably well for small values of n, but slows down significantly beyond the 30 mark, which made me wonder — is this solution even memoized? How would I get this type of solution working fast enough for large values of n?
Your function isn't memoized (at least not effectively) because you call fibonacci_helper regardless of whether you already have a memoized value. This is because setdefault doesn't do any magic that would prevent the arguments from being evaluated before they're passed into the function -- you make the recursive call before the dict has checked to see whether it contains the value.
The point of memoization is to be careful to avoid doing the computation (in this case a lengthy recursive call) in cases where you already know the answer.
The way to fix this implementation would be something like:
def fibonacci(n):
return fibonacci_helper(n, {0: 0, 1: 1})
def fibonacci_helper(n, fib_nums):
if n not in fib_nums:
fib1 = fibonacci_helper(n-1, fib_nums)
fib2 = fibonacci_helper(n-2, fib_nums)
fib_nums[n] = fib1 + fib2
return fib_nums[n]
If you're allowed to not reinvent the wheel, you could also just use functools.lru_cache, which adds memoization to any function through the magic of decorators:
from functools import lru_cache
#lru_cache
def fibonacci(n):
if n in {0, 1}:
return n
return fibonacci(n-1) + fibonacci(n-2)
You'll find that this is very fast for even very high values:
>>> fibonacci(300)
222232244629420445529739893461909967206666939096499764990979600
but if you define the exact same function without the #lru_cache it gets very slow because it's not benefitting from the cache.
>>> fibonacci(300)
(very very long wait)
You're close. The point of "a memo" is to save calls, but you're making recursive calls regardless of whether the result for an argument has already been memorized. So you're not actually saving the work of calling. Simplest is to define the cache outside the function, and simply return at once if the argument is in the cache:
fib_cache = {0 : 0, 1 : 1}
def fib(n):
if n in fib_cache:
return fib_cache[n]
fib_cache[n] = result = fib(n-1) + fib(n-2)
return result
Then the cache will persist across top-level calls too.
But now there's another problem ;-) If the argument is large enough (say, 30000), you're likely to get a RecursionError (too many levels of recursive calls). That's not due to using a cache, it's just inherent in very deep recursion.
You can worm around that too, by exploiting the cache to call smaller arguments first, working your way up to the actual argument. For example, insert this after the if block:
for i in range(100, n, 100):
fib(i)
This ensures that recursion never has to go more than 100 levels deep to find an argument already memorized in the cache. I thought I'd mention that because hardly anyone ever does when answering a "memo" question. But memos are in fact a way not just to greatly speed some kinds of recursive algorithms, but also to apply them to some problems that "recurse too deep" without a memo constructed to limit the max call depth.
I am coming from c
The concept of first class function is interesting and exiting.
However, I am struggling to find a practical use-case to returning a function.
I have seen the examples of building a function that returns print a grating...
Hello = grating('Hello')
Hi = grating('Hi')
But why is this better then just using
grating('Hello') and grating('Hi')?
Consider:
def func_a():
def func_b():
do sonthing
return somthing
return func_b
When this is better then:
def func_b():
do sonthing
return somthing
def func_a():
res = func_b()
return (res)
can someone point me to a real world useful example?
Thanks!
Your examples aren't helpful as written. But there are other cases where it's useful, e.g. decorators (which are functions that are called on functions and return functions, to modify the behavior of the function they're called on for future callers) and closures. The closure case can be a convenience (e.g. some other part of your code doesn't want to have to pass 10 arguments on every call when eight of them are always the same value; this is usually covered by functools.partial, but closures also work), or it can be a caching time saver. For an example of the latter, imagine a stupid function that tests which of a set of numbers less than some bound are prime by computing all primes up to that bound, then filtering the inputs to those in the set of primes.
If you write it as:
def get_primes(*args):
maxtotest = max(args)
primes = sieve_of_eratosthenes(maxtotest) # (expensive) Produce set of primes up to maxtotest
return primes.intersection(args)
then you're redoing the Sieve of Eratosthenes every call, which swamps the cost of a set intersection. If you implement it as a closure, you might do:
def make_get_primes(initialmax=1000):
primes = sieve_of_eratosthenes(initialmax) # (expensive) Produce set of primes up to maxtotest
currentmax = initialmax
def get_primes(*args):
nonlocal currentmax
maxtotest = max(args)
if maxtotest > currentmax:
primes.update(partial_sieve_of_eratosthenes(currentmax, maxtotest)) # (less expensive) Fill in additional primes not sieved
currentmax = maxtotest
return primes.intersection(args)
return get_primes
Now, if you need a tester for a while, you can do:
get_primes = make_get_primes()
and each call to get_primes is cheap (essentially free if the cached primes already cover you, and cheaper if it has to compute more).
Imagine you wanted to pass a function that would choose a function based on parameters:
def compare(a, b):
...
def anticompare(a, b): # Compare but backwards
...
def get_comparator(reverse):
if reverse: return anticompare
else: return compare
def sort(arr, reverse=false):
comparator = get_comparator(reverse)
...
Obviously this is mildly contrived, but it separates the logic of choosing a comparator from the comparator functions themselves.
A perfect example of a function returning a function is a Python decorator:
def foo(func):
def bar():
return func().upper()
return bar
#foo
def hello_world():
return "Hello World!"
print(hello_world())
The decorator is the #foo symbol above the hello_world() function declaration. In essence, we tell the interpreter that whenever it sees the mention of hello_world(), to actually call foo(hello_world()), therefore, the output of the code above is:
HELLO WORLD!
The idea of a function that returns a function in Python is for decorators, which is a design pattern that allows to add new functionality to an existing object.
I recommend you to read about decorators
On example would be creating a timer function. Let's say we want a function that accepts another function, and times how long it takes to complete:
import time
def timeit(func, args):
st = time.time()
func(args)
return str('Took ' + str(time.time() - st) + ' seconds to complete.'
timeit(lambda x: x*10, 10)
# Took 1.9073486328125e-06 seconds to complete
Edit: Using a function that returns a function
import time
def timeit(func):
st = time.time()
func(10)
return str('Took ' + str(time.time() - st) + ' seconds to complete.')
def make_func(x):
return lambda x: x
timeit(make_func(10))
# Took 1.90734863281e-06 seconds to complete.
I just started Python and I've got no idea what memoization is and how to use it. Also, may I have a simplified example?
Memoization effectively refers to remembering ("memoization" → "memorandum" → to be remembered) results of method calls based on the method inputs and then returning the remembered result rather than computing the result again. You can think of it as a cache for method results. For further details, see page 387 for the definition in Introduction To Algorithms (3e), Cormen et al.
A simple example for computing factorials using memoization in Python would be something like this:
factorial_memo = {}
def factorial(k):
if k < 2: return 1
if k not in factorial_memo:
factorial_memo[k] = k * factorial(k-1)
return factorial_memo[k]
You can get more complicated and encapsulate the memoization process into a class:
class Memoize:
def __init__(self, f):
self.f = f
self.memo = {}
def __call__(self, *args):
if not args in self.memo:
self.memo[args] = self.f(*args)
#Warning: You may wish to do a deepcopy here if returning objects
return self.memo[args]
Then:
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
factorial = Memoize(factorial)
A feature known as "decorators" was added in Python 2.4 which allow you to now simply write the following to accomplish the same thing:
#Memoize
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
The Python Decorator Library has a similar decorator called memoized that is slightly more robust than the Memoize class shown here.
functools.cache decorator:
Python 3.9 released a new function functools.cache. It caches in memory the result of a functional called with a particular set of arguments, which is memoization. It's easy to use:
import functools
import time
#functools.cache
def calculate_double(num):
time.sleep(1) # sleep for 1 second to simulate a slow calculation
return num * 2
The first time you call caculate_double(5), it will take a second and return 10. The second time you call the function with the same argument calculate_double(5), it will return 10 instantly.
Adding the cache decorator ensures that if the function has been called recently for a particular value, it will not recompute that value, but use a cached previous result. In this case, it leads to a tremendous speed improvement, while the code is not cluttered with the details of caching.
(Edit: the previous example calculated a fibonacci number using recursion, but I changed the example to prevent confusion, hence the old comments.)
functools.lru_cache decorator:
If you need to support older versions of Python, functools.lru_cache works in Python 3.2+. By default, it only caches the 128 most recently used calls, but you can set the maxsize to None to indicate that the cache should never expire:
#functools.lru_cache(maxsize=None)
def calculate_double(num):
# etc
The other answers cover what it is quite well. I'm not repeating that. Just some points that might be useful to you.
Usually, memoisation is an operation you can apply on any function that computes something (expensive) and returns a value. Because of this, it's often implemented as a decorator. The implementation is straightforward and it would be something like this
memoised_function = memoise(actual_function)
or expressed as a decorator
#memoise
def actual_function(arg1, arg2):
#body
I've found this extremely useful
from functools import wraps
def memoize(function):
memo = {}
#wraps(function)
def wrapper(*args):
# add the new key to dict if it doesn't exist already
if args not in memo:
memo[args] = function(*args)
return memo[args]
return wrapper
#memoize
def fibonacci(n):
if n < 2: return n
return fibonacci(n - 1) + fibonacci(n - 2)
fibonacci(25)
Memoization is keeping the results of expensive calculations and returning the cached result rather than continuously recalculating it.
Here's an example:
def doSomeExpensiveCalculation(self, input):
if input not in self.cache:
<do expensive calculation>
self.cache[input] = result
return self.cache[input]
A more complete description can be found in the wikipedia entry on memoization.
Let's not forget the built-in hasattr function, for those who want to hand-craft. That way you can keep the mem cache inside the function definition (as opposed to a global).
def fact(n):
if not hasattr(fact, 'mem'):
fact.mem = {1: 1}
if not n in fact.mem:
fact.mem[n] = n * fact(n - 1)
return fact.mem[n]
Memoization is basically saving the results of past operations done with recursive algorithms in order to reduce the need to traverse the recursion tree if the same calculation is required at a later stage.
see http://scriptbucket.wordpress.com/2012/12/11/introduction-to-memoization/
Fibonacci Memoization example in Python:
fibcache = {}
def fib(num):
if num in fibcache:
return fibcache[num]
else:
fibcache[num] = num if num < 2 else fib(num-1) + fib(num-2)
return fibcache[num]
Memoization is the conversion of functions into data structures. Usually one wants the conversion to occur incrementally and lazily (on demand of a given domain element--or "key"). In lazy functional languages, this lazy conversion can happen automatically, and thus memoization can be implemented without (explicit) side-effects.
Well I should answer the first part first: what's memoization?
It's just a method to trade memory for time. Think of Multiplication Table.
Using mutable object as default value in Python is usually considered bad. But if use it wisely, it can actually be useful to implement a memoization.
Here's an example adapted from http://docs.python.org/2/faq/design.html#why-are-default-values-shared-between-objects
Using a mutable dict in the function definition, the intermediate computed results can be cached (e.g. when calculating factorial(10) after calculate factorial(9), we can reuse all the intermediate results)
def factorial(n, _cache={1:1}):
try:
return _cache[n]
except IndexError:
_cache[n] = factorial(n-1)*n
return _cache[n]
Here is a solution that will work with list or dict type arguments without whining:
def memoize(fn):
"""returns a memoized version of any function that can be called
with the same list of arguments.
Usage: foo = memoize(foo)"""
def handle_item(x):
if isinstance(x, dict):
return make_tuple(sorted(x.items()))
elif hasattr(x, '__iter__'):
return make_tuple(x)
else:
return x
def make_tuple(L):
return tuple(handle_item(x) for x in L)
def foo(*args, **kwargs):
items_cache = make_tuple(sorted(kwargs.items()))
args_cache = make_tuple(args)
if (args_cache, items_cache) not in foo.past_calls:
foo.past_calls[(args_cache, items_cache)] = fn(*args,**kwargs)
return foo.past_calls[(args_cache, items_cache)]
foo.past_calls = {}
foo.__name__ = 'memoized_' + fn.__name__
return foo
Note that this approach can be naturally extended to any object by implementing your own hash function as a special case in handle_item. For example, to make this approach work for a function that takes a set as an input argument, you could add to handle_item:
if is_instance(x, set):
return make_tuple(sorted(list(x)))
Solution that works with both positional and keyword arguments independently of order in which keyword args were passed (using inspect.getargspec):
import inspect
import functools
def memoize(fn):
cache = fn.cache = {}
#functools.wraps(fn)
def memoizer(*args, **kwargs):
kwargs.update(dict(zip(inspect.getargspec(fn).args, args)))
key = tuple(kwargs.get(k, None) for k in inspect.getargspec(fn).args)
if key not in cache:
cache[key] = fn(**kwargs)
return cache[key]
return memoizer
Similar question: Identifying equivalent varargs function calls for memoization in Python
Just wanted to add to the answers already provided, the Python decorator library has some simple yet useful implementations that can also memoize "unhashable types", unlike functools.lru_cache.
cache = {}
def fib(n):
if n <= 1:
return n
else:
if n not in cache:
cache[n] = fib(n-1) + fib(n-2)
return cache[n]
If speed is a consideration:
#functools.cache and #functools.lru_cache(maxsize=None) are equally fast, taking 0.122 seconds (best of 15 runs) to loop a million times on my system
a global cache variable is quite a lot slower, taking 0.180 seconds (best of 15 runs) to loop a million times on my system
a self.cache class variable is a bit slower still, taking 0.214 seconds (best of 15 runs) to loop a million times on my system
The latter two are implemented similar to how it is described in the currently top-voted answer.
This is without memory exhaustion prevention, i.e. I did not add code in the class or global methods to limit that cache's size, this is really the barebones implementation. The lru_cache method has that for free, if you need this.
One open question for me would be how to unit test something that has a functools decorator. Is it possible to empty the cache somehow? Unit tests seem like they would be cleanest using the class method (where you can instantiate a new class for each test) or, secondarily, the global variable method (since you can do yourimportedmodule.cachevariable = {} to empty it).
I just started Python and I've got no idea what memoization is and how to use it. Also, may I have a simplified example?
Memoization effectively refers to remembering ("memoization" → "memorandum" → to be remembered) results of method calls based on the method inputs and then returning the remembered result rather than computing the result again. You can think of it as a cache for method results. For further details, see page 387 for the definition in Introduction To Algorithms (3e), Cormen et al.
A simple example for computing factorials using memoization in Python would be something like this:
factorial_memo = {}
def factorial(k):
if k < 2: return 1
if k not in factorial_memo:
factorial_memo[k] = k * factorial(k-1)
return factorial_memo[k]
You can get more complicated and encapsulate the memoization process into a class:
class Memoize:
def __init__(self, f):
self.f = f
self.memo = {}
def __call__(self, *args):
if not args in self.memo:
self.memo[args] = self.f(*args)
#Warning: You may wish to do a deepcopy here if returning objects
return self.memo[args]
Then:
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
factorial = Memoize(factorial)
A feature known as "decorators" was added in Python 2.4 which allow you to now simply write the following to accomplish the same thing:
#Memoize
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
The Python Decorator Library has a similar decorator called memoized that is slightly more robust than the Memoize class shown here.
functools.cache decorator:
Python 3.9 released a new function functools.cache. It caches in memory the result of a functional called with a particular set of arguments, which is memoization. It's easy to use:
import functools
import time
#functools.cache
def calculate_double(num):
time.sleep(1) # sleep for 1 second to simulate a slow calculation
return num * 2
The first time you call caculate_double(5), it will take a second and return 10. The second time you call the function with the same argument calculate_double(5), it will return 10 instantly.
Adding the cache decorator ensures that if the function has been called recently for a particular value, it will not recompute that value, but use a cached previous result. In this case, it leads to a tremendous speed improvement, while the code is not cluttered with the details of caching.
(Edit: the previous example calculated a fibonacci number using recursion, but I changed the example to prevent confusion, hence the old comments.)
functools.lru_cache decorator:
If you need to support older versions of Python, functools.lru_cache works in Python 3.2+. By default, it only caches the 128 most recently used calls, but you can set the maxsize to None to indicate that the cache should never expire:
#functools.lru_cache(maxsize=None)
def calculate_double(num):
# etc
The other answers cover what it is quite well. I'm not repeating that. Just some points that might be useful to you.
Usually, memoisation is an operation you can apply on any function that computes something (expensive) and returns a value. Because of this, it's often implemented as a decorator. The implementation is straightforward and it would be something like this
memoised_function = memoise(actual_function)
or expressed as a decorator
#memoise
def actual_function(arg1, arg2):
#body
I've found this extremely useful
from functools import wraps
def memoize(function):
memo = {}
#wraps(function)
def wrapper(*args):
# add the new key to dict if it doesn't exist already
if args not in memo:
memo[args] = function(*args)
return memo[args]
return wrapper
#memoize
def fibonacci(n):
if n < 2: return n
return fibonacci(n - 1) + fibonacci(n - 2)
fibonacci(25)
Memoization is keeping the results of expensive calculations and returning the cached result rather than continuously recalculating it.
Here's an example:
def doSomeExpensiveCalculation(self, input):
if input not in self.cache:
<do expensive calculation>
self.cache[input] = result
return self.cache[input]
A more complete description can be found in the wikipedia entry on memoization.
Let's not forget the built-in hasattr function, for those who want to hand-craft. That way you can keep the mem cache inside the function definition (as opposed to a global).
def fact(n):
if not hasattr(fact, 'mem'):
fact.mem = {1: 1}
if not n in fact.mem:
fact.mem[n] = n * fact(n - 1)
return fact.mem[n]
Memoization is basically saving the results of past operations done with recursive algorithms in order to reduce the need to traverse the recursion tree if the same calculation is required at a later stage.
see http://scriptbucket.wordpress.com/2012/12/11/introduction-to-memoization/
Fibonacci Memoization example in Python:
fibcache = {}
def fib(num):
if num in fibcache:
return fibcache[num]
else:
fibcache[num] = num if num < 2 else fib(num-1) + fib(num-2)
return fibcache[num]
Memoization is the conversion of functions into data structures. Usually one wants the conversion to occur incrementally and lazily (on demand of a given domain element--or "key"). In lazy functional languages, this lazy conversion can happen automatically, and thus memoization can be implemented without (explicit) side-effects.
Well I should answer the first part first: what's memoization?
It's just a method to trade memory for time. Think of Multiplication Table.
Using mutable object as default value in Python is usually considered bad. But if use it wisely, it can actually be useful to implement a memoization.
Here's an example adapted from http://docs.python.org/2/faq/design.html#why-are-default-values-shared-between-objects
Using a mutable dict in the function definition, the intermediate computed results can be cached (e.g. when calculating factorial(10) after calculate factorial(9), we can reuse all the intermediate results)
def factorial(n, _cache={1:1}):
try:
return _cache[n]
except IndexError:
_cache[n] = factorial(n-1)*n
return _cache[n]
Here is a solution that will work with list or dict type arguments without whining:
def memoize(fn):
"""returns a memoized version of any function that can be called
with the same list of arguments.
Usage: foo = memoize(foo)"""
def handle_item(x):
if isinstance(x, dict):
return make_tuple(sorted(x.items()))
elif hasattr(x, '__iter__'):
return make_tuple(x)
else:
return x
def make_tuple(L):
return tuple(handle_item(x) for x in L)
def foo(*args, **kwargs):
items_cache = make_tuple(sorted(kwargs.items()))
args_cache = make_tuple(args)
if (args_cache, items_cache) not in foo.past_calls:
foo.past_calls[(args_cache, items_cache)] = fn(*args,**kwargs)
return foo.past_calls[(args_cache, items_cache)]
foo.past_calls = {}
foo.__name__ = 'memoized_' + fn.__name__
return foo
Note that this approach can be naturally extended to any object by implementing your own hash function as a special case in handle_item. For example, to make this approach work for a function that takes a set as an input argument, you could add to handle_item:
if is_instance(x, set):
return make_tuple(sorted(list(x)))
Solution that works with both positional and keyword arguments independently of order in which keyword args were passed (using inspect.getargspec):
import inspect
import functools
def memoize(fn):
cache = fn.cache = {}
#functools.wraps(fn)
def memoizer(*args, **kwargs):
kwargs.update(dict(zip(inspect.getargspec(fn).args, args)))
key = tuple(kwargs.get(k, None) for k in inspect.getargspec(fn).args)
if key not in cache:
cache[key] = fn(**kwargs)
return cache[key]
return memoizer
Similar question: Identifying equivalent varargs function calls for memoization in Python
Just wanted to add to the answers already provided, the Python decorator library has some simple yet useful implementations that can also memoize "unhashable types", unlike functools.lru_cache.
cache = {}
def fib(n):
if n <= 1:
return n
else:
if n not in cache:
cache[n] = fib(n-1) + fib(n-2)
return cache[n]
If speed is a consideration:
#functools.cache and #functools.lru_cache(maxsize=None) are equally fast, taking 0.122 seconds (best of 15 runs) to loop a million times on my system
a global cache variable is quite a lot slower, taking 0.180 seconds (best of 15 runs) to loop a million times on my system
a self.cache class variable is a bit slower still, taking 0.214 seconds (best of 15 runs) to loop a million times on my system
The latter two are implemented similar to how it is described in the currently top-voted answer.
This is without memory exhaustion prevention, i.e. I did not add code in the class or global methods to limit that cache's size, this is really the barebones implementation. The lru_cache method has that for free, if you need this.
One open question for me would be how to unit test something that has a functools decorator. Is it possible to empty the cache somehow? Unit tests seem like they would be cleanest using the class method (where you can instantiate a new class for each test) or, secondarily, the global variable method (since you can do yourimportedmodule.cachevariable = {} to empty it).
I just started Python and I've got no idea what memoization is and how to use it. Also, may I have a simplified example?
Memoization effectively refers to remembering ("memoization" → "memorandum" → to be remembered) results of method calls based on the method inputs and then returning the remembered result rather than computing the result again. You can think of it as a cache for method results. For further details, see page 387 for the definition in Introduction To Algorithms (3e), Cormen et al.
A simple example for computing factorials using memoization in Python would be something like this:
factorial_memo = {}
def factorial(k):
if k < 2: return 1
if k not in factorial_memo:
factorial_memo[k] = k * factorial(k-1)
return factorial_memo[k]
You can get more complicated and encapsulate the memoization process into a class:
class Memoize:
def __init__(self, f):
self.f = f
self.memo = {}
def __call__(self, *args):
if not args in self.memo:
self.memo[args] = self.f(*args)
#Warning: You may wish to do a deepcopy here if returning objects
return self.memo[args]
Then:
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
factorial = Memoize(factorial)
A feature known as "decorators" was added in Python 2.4 which allow you to now simply write the following to accomplish the same thing:
#Memoize
def factorial(k):
if k < 2: return 1
return k * factorial(k - 1)
The Python Decorator Library has a similar decorator called memoized that is slightly more robust than the Memoize class shown here.
functools.cache decorator:
Python 3.9 released a new function functools.cache. It caches in memory the result of a functional called with a particular set of arguments, which is memoization. It's easy to use:
import functools
import time
#functools.cache
def calculate_double(num):
time.sleep(1) # sleep for 1 second to simulate a slow calculation
return num * 2
The first time you call caculate_double(5), it will take a second and return 10. The second time you call the function with the same argument calculate_double(5), it will return 10 instantly.
Adding the cache decorator ensures that if the function has been called recently for a particular value, it will not recompute that value, but use a cached previous result. In this case, it leads to a tremendous speed improvement, while the code is not cluttered with the details of caching.
(Edit: the previous example calculated a fibonacci number using recursion, but I changed the example to prevent confusion, hence the old comments.)
functools.lru_cache decorator:
If you need to support older versions of Python, functools.lru_cache works in Python 3.2+. By default, it only caches the 128 most recently used calls, but you can set the maxsize to None to indicate that the cache should never expire:
#functools.lru_cache(maxsize=None)
def calculate_double(num):
# etc
The other answers cover what it is quite well. I'm not repeating that. Just some points that might be useful to you.
Usually, memoisation is an operation you can apply on any function that computes something (expensive) and returns a value. Because of this, it's often implemented as a decorator. The implementation is straightforward and it would be something like this
memoised_function = memoise(actual_function)
or expressed as a decorator
#memoise
def actual_function(arg1, arg2):
#body
I've found this extremely useful
from functools import wraps
def memoize(function):
memo = {}
#wraps(function)
def wrapper(*args):
# add the new key to dict if it doesn't exist already
if args not in memo:
memo[args] = function(*args)
return memo[args]
return wrapper
#memoize
def fibonacci(n):
if n < 2: return n
return fibonacci(n - 1) + fibonacci(n - 2)
fibonacci(25)
Memoization is keeping the results of expensive calculations and returning the cached result rather than continuously recalculating it.
Here's an example:
def doSomeExpensiveCalculation(self, input):
if input not in self.cache:
<do expensive calculation>
self.cache[input] = result
return self.cache[input]
A more complete description can be found in the wikipedia entry on memoization.
Let's not forget the built-in hasattr function, for those who want to hand-craft. That way you can keep the mem cache inside the function definition (as opposed to a global).
def fact(n):
if not hasattr(fact, 'mem'):
fact.mem = {1: 1}
if not n in fact.mem:
fact.mem[n] = n * fact(n - 1)
return fact.mem[n]
Memoization is basically saving the results of past operations done with recursive algorithms in order to reduce the need to traverse the recursion tree if the same calculation is required at a later stage.
see http://scriptbucket.wordpress.com/2012/12/11/introduction-to-memoization/
Fibonacci Memoization example in Python:
fibcache = {}
def fib(num):
if num in fibcache:
return fibcache[num]
else:
fibcache[num] = num if num < 2 else fib(num-1) + fib(num-2)
return fibcache[num]
Memoization is the conversion of functions into data structures. Usually one wants the conversion to occur incrementally and lazily (on demand of a given domain element--or "key"). In lazy functional languages, this lazy conversion can happen automatically, and thus memoization can be implemented without (explicit) side-effects.
Well I should answer the first part first: what's memoization?
It's just a method to trade memory for time. Think of Multiplication Table.
Using mutable object as default value in Python is usually considered bad. But if use it wisely, it can actually be useful to implement a memoization.
Here's an example adapted from http://docs.python.org/2/faq/design.html#why-are-default-values-shared-between-objects
Using a mutable dict in the function definition, the intermediate computed results can be cached (e.g. when calculating factorial(10) after calculate factorial(9), we can reuse all the intermediate results)
def factorial(n, _cache={1:1}):
try:
return _cache[n]
except IndexError:
_cache[n] = factorial(n-1)*n
return _cache[n]
Here is a solution that will work with list or dict type arguments without whining:
def memoize(fn):
"""returns a memoized version of any function that can be called
with the same list of arguments.
Usage: foo = memoize(foo)"""
def handle_item(x):
if isinstance(x, dict):
return make_tuple(sorted(x.items()))
elif hasattr(x, '__iter__'):
return make_tuple(x)
else:
return x
def make_tuple(L):
return tuple(handle_item(x) for x in L)
def foo(*args, **kwargs):
items_cache = make_tuple(sorted(kwargs.items()))
args_cache = make_tuple(args)
if (args_cache, items_cache) not in foo.past_calls:
foo.past_calls[(args_cache, items_cache)] = fn(*args,**kwargs)
return foo.past_calls[(args_cache, items_cache)]
foo.past_calls = {}
foo.__name__ = 'memoized_' + fn.__name__
return foo
Note that this approach can be naturally extended to any object by implementing your own hash function as a special case in handle_item. For example, to make this approach work for a function that takes a set as an input argument, you could add to handle_item:
if is_instance(x, set):
return make_tuple(sorted(list(x)))
Solution that works with both positional and keyword arguments independently of order in which keyword args were passed (using inspect.getargspec):
import inspect
import functools
def memoize(fn):
cache = fn.cache = {}
#functools.wraps(fn)
def memoizer(*args, **kwargs):
kwargs.update(dict(zip(inspect.getargspec(fn).args, args)))
key = tuple(kwargs.get(k, None) for k in inspect.getargspec(fn).args)
if key not in cache:
cache[key] = fn(**kwargs)
return cache[key]
return memoizer
Similar question: Identifying equivalent varargs function calls for memoization in Python
Just wanted to add to the answers already provided, the Python decorator library has some simple yet useful implementations that can also memoize "unhashable types", unlike functools.lru_cache.
cache = {}
def fib(n):
if n <= 1:
return n
else:
if n not in cache:
cache[n] = fib(n-1) + fib(n-2)
return cache[n]
If speed is a consideration:
#functools.cache and #functools.lru_cache(maxsize=None) are equally fast, taking 0.122 seconds (best of 15 runs) to loop a million times on my system
a global cache variable is quite a lot slower, taking 0.180 seconds (best of 15 runs) to loop a million times on my system
a self.cache class variable is a bit slower still, taking 0.214 seconds (best of 15 runs) to loop a million times on my system
The latter two are implemented similar to how it is described in the currently top-voted answer.
This is without memory exhaustion prevention, i.e. I did not add code in the class or global methods to limit that cache's size, this is really the barebones implementation. The lru_cache method has that for free, if you need this.
One open question for me would be how to unit test something that has a functools decorator. Is it possible to empty the cache somehow? Unit tests seem like they would be cleanest using the class method (where you can instantiate a new class for each test) or, secondarily, the global variable method (since you can do yourimportedmodule.cachevariable = {} to empty it).