# Understanding Python decorators optimizing a recursive Fibonacci implementation

Posted on Mon 31 August 2015 in Programmazione, Python

A **decorator** is a Python function that takes a *function object* as
an argument and returns a function as a *value*. Here is an example of
decorator definition:

```
def foo(function):
# make a new function
def new_function():
# some code
return new_function
```

To apply a decorator to an existing function, you just need to put
**@*decorator_name*** in the line before its definition, like this
example:

```
@foo
def hello():
print 'Hello World'
```

This decorator doesn't do anything, so let's think about a more concrete problem we could solve using decorators.

## Fibonacci sequence

By definition, the first two numbers in the
**Fibonacci** sequence
are either 1 and 1 or 0 and 1. All the other numbers are the sum of the
previous two numbers of the sequence. Example:

- 0, 1: the third number is 1
- 0, 1, 1: the fourth number is 2
- 0, 1, 1, 2: the fifth number is 3
- 0, 1, 1, 2, 3: the sixth number is 5
- etc...

If we wanted to give a **math definition** of the sequence, we could
describe it in this way:

**F(0): 0****F(1): 1****F(n): F(n-1) + F(n-2)**

In **Python** we could have a **recursive function** like the following
one:

```
def fib(n):
if n < 2:
return n
else:
return fib(n-1) + fib(n-2)
```

What's the problem with this implementation? The code works as expected,
but it's very **inefficient**. The next picture will explain what
happens when we will try, for example, to calculate the 5th number of
the sequence:

Fib(5) is Fib(4) + Fib(3), but Fib(4) itself is Fib(3) + Fib(2), and... the picture just tell us that we have calculated Fib(3) 2 times, Fib(2) 3 times, Fib(1) 5 times! Why are we repeating the same operation every time if we already calculated the result?

## Memoization

In computing,

memoizationis an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again.

We need to store values of the sequence we have already calculated and get them later when we need them. Let's implement a simple memoization decorator:

```
def memoize(function):
cache = {}
def decorated_function(*args):
if args in cache:
return cache[args]
else:
val = function(*args)
cache[args] = val
return val
return decorated_function
```

The decorator defines a dict at the beginning that is used as a cache. When we want to find the n number of the sequence, it first checks if the value was already calculated and that value is returned instead of being calculated again. If the value is not found, then the original function is being called and then the value is store in the cache, then returned to the caller.

## Using the memoize decorator

How much this decorator can speed up our fib method? Let's try to
benchmark the execution using Python
**timeit** module.

```
# First example, not using the memoize decorator
import timeit
def fib(n):
if n < 2:
return n
else:
return fib(n-1) + fib(n-2)
t1 = timeit.Timer("fib(35)", "from __main__ import fib")
print t1.timeit(1)
```

The required time to calculate the 35th number of the Fibonacci sequence
on my laptop is: **4.73480010033**

```
# Second example, using the memoize decorator
import timeit
from memoize import memoize # For convenience I put my decorator
# in a module named memoize.py
@memoize
def fib(n):
if n < 2:
return n
else:
return fib(n-1) + fib(n-2)
t1 = timeit.Timer("fib(35)", "from __main__ import fib")
print t1.timeit(1)
```

The required time to calculate the 35th number of the Fibonacci sequence
on my laptop is: **0.000133037567139**

Quite faster, don't you think? I will let you try how long does it take
to calculate the 60th number of the sequence with and without using the
decorator. **Hint:** grab a cup of coffee before beginning!