Wednesday, September 15, 2010


Computers are very fast machines. But the computation time
also depend on the algorithm we choose. It can happen that
if a proper algorithm is not chosen, it may take a very long
time for execution.

On writing a simple recursive fibonacci generator function for
the nth element, fib(n), we will get fine results, but only for low
values of n. This is because fib(n) has exponential complexity.
Such functions grow very fast, so fast that even for small
values of `n' (where `n' is the size of the input), our program
will take a long time to finish.

Memoization is the process by which we speed up the situation.
The underlying principle is to create an associative array, say  
dict, which contain keys = n, and values = fib(n). Each time a
fib(n) is called, the associative array will be checked whether it
contain the key 'n'. If yes, just return dict[n]. Otherwise, create
a new key: value pair for dict. Thus, the number of recursion will
be drastically reduced, with this look-in-the-lookup-table policy.

Ofcourse, memoization is an upgrade to time compplexity, but
will consume space.
The implementation in Python looked like:

def memoized_fib(x):
if dict.has_key(x):
return dict[x]
dict[x] = fib(x)
return dict[x]

def fib(n):
if n == 0: return 0
elif n == 1: return 1
else: return memoized_fib(n-2) + memoized_fib(n-1)

 dict = {0: 0, 1: 1}

In Python, the recursion limit is fixed by the interpreter itself. It can
be found in:

Now, set a very high limit, and run the code. If its high enough, u will
deplete the stack memory size (8 MB) of the C backend of your Python
virtual machine, leading to a segmentation fault there, which is
propagated back to your Python interpreter, crashing Python itself!

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