On these last months I've been solving some problems (such as some HMMs algorithms) which the best solutions involves some kind of dynamic programming. Some of them are quite simple to implement, but their recursive formulation are far more intuitive. The problem is that even in functional languages, the recursive functions aren't well handled unless you some mechanism like tail call, which aren't intuitive as we would like to. The simplest example that comes in my mind is the fibonacci function which is usually defined as:

fib(0) = 1

fib(1) = 1

fib(n) = fib(n-1) + fib(n-2)

As we know, almost all the languages compilers and interpreters use the call stack to call the recursives cases on functions being executed. We can analyze the following C fibonacci version:

int fib(n)

{

if (n == 0 || n == 1)

return 1;

else

return fib(n-1) + fib(n-2);

}

It is really simple to understand when contrasted with the definition. But, if we make a trace of the the program (even with a small input value) we'll have something like the following evaluation order:

fib(6) = fib(5) + fib(4)

fib(5) = fib(4) + fib(3)

fib(4) = fib(3) + fib(2)

fib(3) = fib(2) + fib(1)

fib(2) = fib(1) + fib(0)

fib(3) = fib(2) + fib(1)

fib(2) = fib(1) + fib(0)

fib(4) = fib(3) + fib(2)

fib(3) = fib(2) + fib(1)

fib(2) = fib(1) + fib(0)

fib(3) = fib(2) + fib(1)

fib(2) = fib(1) + fib(0)

Call stack for fib(6) |

As we can see, there is a repetition of the calculation of fib 4 to 1, many times and is something we can avoid. In fact, the complexity of this solution has a exponencial computational complexity because for each n from input we branch it in 2 until it reachs 0 or 1 approximately n times, leading us into a O(2

^{n}) complexity. A simple way to avoid it, is converting into a interactive form:

int fib(int n)

{

int current = 1;

int previous = 1;

int i;

for (i = 1; i < n; i++) {

` int temp = current; // XXX: nasty`

` current += previous;`

` previous = temp;`

` }`

` return current;`

`}`

` `

The same result is achieved by using tail call for functional languages.

As you can see, it obfuscates the intuitive definition given in as the recursive formulation. But we still have a problem whenever we calculate fib(n), we have to recalculate it's previous results even if they was previously calculated. If this function is used many times in our program it will take a lot of processing re-computing many of the values. We can avoid this by using the dynamic programming, which keeps the record of previously calculated results. The drawback of this technique is the memory usage, which for large entries can become a bottleneck. However, processing usually is a more rare computer resource. A C implementation (not the most elegant) for it is:

// XXX: kids, don't do this at home

int fib_results[10000];

int last_fib;

int fib(int n)

{

if (n <= last_fib

` return fib_results[n];`

` int current = fib_results[last_fib-1];`

` int previous = fib_results[last_fib-2];`

` for (; last_fib < n; last_fib++) {`

` int temp = current;`

` current += previous;`

` fib_results[last_fib] = current;`

` previous = temp;`

` }`

` return current;`

`}`

`int main()`

`{`

` fib_results[0] = 1;`

` fib_results[1] = 1;`

` last_fib = 1;`

` // ... other stuff ...`

` return 0;`

`} `

As we can see, dynamic programming isn't too hard to implement. On the other hand, reading a code it's a though task to do unless you are already familiar with the algorithm.

If we extract what is dynamic programming fundamental concept, which is "store pre-computed results", we find a regularity in every recursive function which we can be transformed into a dynamic programming one. One of the reasons I love python because it's easy to use meta-programming concepts, and that's what I will use to transform recursive functions into it's dynamic form in a ridiculous easy way using function decorators.

Function decorators (or annotations in Java) are a form of meta-programming for functions. It extends functions with some functionalities, such as debugging, tracing, adding meta-data to the function, synchronization or memoization (not memorization) of values, which is a great way of optimizing recursive functions by caching their results (if you have enough memory available). One possible implementation of memoitized decorator in python is the following:

def cached(function):

cache = {}

def wrapper(*args):

if args in cache:

` `

`return cache[args]`

` else:`

` `

` result = function(*args)`

` `

` `

` cache[args] = result`

` `

` `

` return result`

` `

`return wrapper`

Note that I'm not using kwargs because they're not hashable, such as the tuple args, and will add a few complexity in the example. See also that we a have a function that returns another function, which uses a given one to calculate results and store them in a cache. To cache our fib function we may use the following code:

@cached

def fib(n):

if n == 0 or n == 1:

return 1

else:

return fib(n-1) + fib(n-2)

# or in a not so clean version:

def normal_fib(n):

if n == 0 or n == 1:

return 1

else:

return fib(n-1) + fib(n-2)

fib = cached(normal_fib)

This technique is really useful to improve your code performance in a really easy. On the other hand, it isn't the best solution for almost all the cases. Many times code a dynamic programming method (if performance is crucial) will be necessary. Is also important to notice that I didn't used any cache memory management policy, which is important to economize memory. Most appropriate cache data structures (such as numpy arrays for integer arguments) also are welcome. The python 3.2 version added the lru_cache decorator into the functools module to make this cache mechanism. If you are already using this version, it's smarter to use it instead of implementing your one. Here is how it must be used:

# Python > 3.2

import functools

@functools.lru_cached(max_size=500) # uses a fixed size cache to avoid memory usage explosion

def fib(n)

...

This technique is very useful not only for economize the CPU resources but also network (such as caching SQL query results), other IO operations (such as disk reading) and even user interaction input in a straightforward way.

As you wrote that the problem lies in the fact that even in functional languages, the recursive functions are not handled properly.I got this point but what is tail call.Is it what you explained in example of preserving old computation.

ReplyDeleteeSignature

I like your blog, I read this blog please update more content on hacking, further check it once at

ReplyDeletepython online training