Dynamic Programming is one way which can be used as an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of subproblems so that we do not have to re-compute them when needed later. This simple optimization reduces time complexities from exponential to polynomial. In this article, a method to use dictionaries of python to implement dynamic programming has been discussed.
In order to understand the implementation of the dynamic programming in python, lets visualize it using the Fibonacci numbers problem.
In mathematical terms, the sequence of Fibonacci numbers is defined by the recurrence relation:
Fn = Fn-1 + Fn-2
with seed values:
F0 = 0 and F1 = 1
Examples:
Input: N = 9
Output: 34
Explanation:
9th number in the Fibonacci series is 34.Input: N = 2
Output: 1
Explanation:
2nd number in the Fibonacci series is 1.
Below is the implementation of the naive approach:
Python3
# Function to find nth Fibonacci number def Fibonacci(n): # Corner case if n<0: print("Incorrect input") # Base case elif n == 0: return 0 elif n == 1: return 1 # Recursive case else: return Fibonacci(n-1)+Fibonacci(n-2) print(Fibonacci(9)) |
34
Clearly, the above approach has exponential time complexity. In order to store the previously computed results, let us use the dictionary class of python.
Approach: The idea is to customize the __missing__ method of the dictionary class. This method is executed when the user tries to access a key which is not in the dictionary. We will use our own function definition to rewrite this method.
Below is the implementation of the above approach:
Python3
# Python program to customize the # __missing__ method of the # dictionary class in pythonclass Fibonacci(dict): # Function signature of the # __missing__ function in # python def __missing__(self, n): # Base case if n<= 1: # Storing the value in the # dictionary before returning self[n] = n return n # Storing the value in the dictionary # before returning the value val = self[n] = self[n-1] + self[n-2] return valif __name__ == "__main__": # Create an instance of the class Fib = Fibonacci() N = Fib[9] print(N) |
34
The above method can also be implemented by using a decorator in python.
Decorator is a very powerful and useful tool in python since it allows programmers to modify the behavior of function or class. Decorators allow us to wrap another function in order to extend the behavior of the wrapped function, without permanently modifying it. Here, memoization is used to implement a decorator.
Below is the implementation of the above approach:
Python3
# Python program to find the nth Fibonacci# number with memoization using decoratorsfrom inspect import signature# Defining a decorator class memoize(dict): # Initializing function def __init__(self, func): self.func = func self.signature = signature(func) # Missing method to store the # Fibonacci numbers in a # Dictionary def __missing__(self, key): (arg, kwarg) = key self[key] = val = self.func(*arg, **dict(kwarg)) return val def __call__(self, *arg, **kwarg): key = self.signature.bind(*arg, **kwarg) return self[key.args, frozenset(key.kwargs.items())]# Function to find the n-th Fibonacci # number using the above defined # decorator@memoizedef Fibonacci(n): # Corner case if n<0: print("Incorrect input") # Base cases elif n == 0: return 0 elif n == 1: return 1 # Recursive case else: return Fibonacci(n-1)+Fibonacci(n-2)if __name__ == "__main__": print(Fibonacci(9)) |
34
