Dynamic Programming is a commonly used algorithmic technique used to optimize recursive solutions when same subproblems are called again.

• The core idea behind DP is to store solutions to subproblems so that each is solved only once.
• To solve DP problems, we first write a recursive solution in a way that there are overlapping subproblems in the recursion tree (the recursive function is called with the same parameters multiple times)
• To make sure that a recursive value is computed only once (to improve time taken by algorithm), we store results of the recursive calls.
• There are two ways to store the results, one is top down (or memoization) and other is bottom up (or tabulation).

Approaches of Dynamic Programming (DP) in Python

Dynamic programming in Python can be achieved using two approaches:

1. Top-Down Approach (Memoization):

In the top-down approach, also known as memoization, we keep the solution recursive and add a memoization table to avoid repeated calls of same subproblems.

• Before making any recursive call, we first check if the memoization table already has solution for it.
• After the recursive call is over, we store the solution in the memoization table.

2. Bottom-Up Approach (Tabulation):

In the bottom-up approach, also known as tabulation, we start with the smallest subproblems and gradually build up to the final solution.

• We write an iterative solution (avoid recursion overhead) and build the solution in bottom-up manner.
• We use a dp table where we first fill the solution for base cases and then fill the remaining entries of the table using recursive formula.
• We only use recursive formula on table entries and do not make recursive calls.

read more about - When to Use Dynamic Programming (DP)?

Example of Dynamic Programming (DP)

Example 1: Consider the problem of finding the Fibonacci sequence:

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

Brute Force Approach

To find the nth Fibonacci number using a brute force approach, you would simply add the (n-1)th and (n-2)th Fibonacci numbers.

# Python program to find 
# fibonacci number using recursion.

# Function to find nth fibonacci number
def fib(n):
    if n <= 1:
        return n
    return fib(n - 1) + fib(n - 2)

if __name__ == "__main__":
    n = 5
    print(fib(n))

5

Below is the recursion tree of the above recursive solution.

How will Dynamic Programming (DP) Work?

Let’s us now see the above recursion tree with overlapping subproblems highlighted with same color. We can clearly see that that recursive solution is doing a lot work again and again which is causing the time complexity to be exponential. Imagine time taken for computing a large Fibonacci number.

• Identify Subproblems: Divide the main problem into smaller, independent subproblems, i.e., F(n-1) and F(n-2)
• Store Solutions: Solve each subproblem and store the solution in a table or array so that we do not have to recompute the same again.
• Build Up Solutions: Use the stored solutions to build up the solution to the main problem. For F(n), look up F(n-1) and F(n-2) in the table and add them.
• Avoid Recomputation: By storing solutions, DP ensures that each subproblem (for example, F(2)) is solved only once, reducing computation time.

Using Memoization Approach – O(n) Time and O(n) Space

To achieve this in our example we simply take an memo array initialized to -1. As we make a recursive call, we first check if the value stored in the memo array corresponding to that position is -1. The value -1 indicates that we haven’t calculated it yet and have to recursively compute it. The output must be stored in the memo array so that, next time, if the same value is encountered, it can be directly used from the memo array.   

# Python program to find
# fibonacci number using memoization.
def fibRec(n, memo):
  
    # Base case
    if n <= 1:
        return n

    # To check if output already exists
    if memo[n] != -1:
        return memo[n]

    # Calculate and save output for future use
    memo[n] = fibRec(n - 1, memo) + \
              fibRec(n - 2, memo)
    return memo[n]

def fib(n):
    memo = [-1] * (n + 1)
    return fibRec(n, memo)

n = 5
print(fib(n))

5

Using Tabulation Approach – O(n) Time and O(n) Space

In this approach, we use an array of size (n + 1), often called dp[], to store Fibonacci numbers. The array is initialized with base values at the appropriate indices, such as dp[0] = 0 and dp[1] = 1. Then, we iteratively calculate Fibonacci values from dp[2] to dp[n] by using the relation dp[i] = dp[i-1] + dp[i-2]. This allows us to efficiently compute Fibonacci numbers in a loop. Finally, the value at dp[n] gives the Fibonacci number for the input n, as each index holds the answer for its corresponding Fibonacci number.

# Python program to find
# fibonacci number using tabulation.
def fibo(n):
    dp = [0] * (n + 1)

    # Storing the independent values in dp
    dp[0] = 0
    dp[1] = 1

    # Using the bottom-up approach
    for i in range(2, n + 1):
        dp[i] = dp[i - 1] + dp[i - 2]
    
    return dp[n]

n = 5
print(fibo(n))

5

Using Space Optimised Approach – O(n) Time and O(1) Space

In the above code, we can see that the current state of any fibonacci number depends only on the previous two values. So we do not need to store the whole table of size n+1 but instead of that we can only store the previous two values.

# Python program to find
# fibonacci number using space optimised.
def fibo(n):
    prevPrev, prev, curr = 0, 1, 1

    # Using the bottom-up approach
    for i in range(2, n + 1):
        curr = prev + prevPrev
        prevPrev = prev
        prev = curr

    return curr

n = 5
print(fibo(n))

5

Common Algorithms that Use DP:

• Longest Common Subsequence (LCS): This is used in day to day life to find difference between two files (diff utility)
• Edit Distance : Checks how close to strings are. Can we be useful in implementing Google’s did you mean type feature.
• Longest Increasing Subsequence : There are plenty of variations of this problem that arise in real world.
• Bellman–Ford Shortest Path: Finds the shortest path from a given source to all other vertices.
• Floyd Warshall : Finds shortest path from every pair of vertices.
• Knapsack Problem: Determines the maximum value of items that can be placed in a knapsack with a given capacity.
• Matrix Chain Multiplication: Optimizes the order of matrix multiplication to minimize the number of operations.
• Fibonacci Sequence: Calculates the nth Fibonacci number.