Dynamic Programming Algorithm

In this article Dynamic Programming Algorithm is an algorithm design technique used to solve complex problems by breaking them into smaller overlapping subproblems.

General Concept of Dynamic Programming

What is Dynamic Programming?

Dynamic Programming (DP) is an algorithm design technique used to solve complex problems by breaking them into smaller overlapping subproblems.
Each subproblem is solved only once, and its result is stored for future use, thereby avoiding repeated computations.

This technique significantly improves efficiency compared to naive recursive solutions.

Need for Dynamic Programming

In many recursive algorithms, the same subproblem is solved multiple times, which leads to:

• Increased time complexity
• Redundant computations

Dynamic Programming helps because:

• Unnecessary repetition is avoided
• Previously computed results are reused
• Both time and memory efficiency are improved

Main Features of Dynamic Programming

For a problem to be solved using Dynamic Programming, it must satisfy two key properties:

(i) Overlapping Subproblems

• The same subproblems appear repeatedly during computation
• Example: Fibonacci Series
  • While computing fib(5), values of fib(3) and fib(2) are calculated multiple times

(ii) Optimal Substructure

• The optimal solution of a problem can be constructed from the optimal solutions of its subproblems
• Examples:
  • Shortest Path Problem
  • Knapsack Problem

How Does Dynamic Programming Work?

Dynamic Programming follows these steps:

1. Divide the main problem into smaller subproblems
2. Solve each subproblem
3. Store the result of each subproblem in a table or array
4. Use the stored results to build the final solution

Two Main Approaches of Dynamic Programming

(i) Top-Down Approach (Memoization)

• Uses recursion
• Stores results of subproblems after computation
• Reuses stored results when the same subproblem occurs again
• Memoization means remembering previously computed values

(ii) Bottom-Up Approach (Tabulation)

• Starts by solving the smallest subproblems first
• Gradually builds the solution to the larger problem
• Does not use recursion
• Generally more efficient in terms of performance

Advantages of Dynamic Programming

• Reduces time complexity
• Avoids repeated recursive calls
• Guarantees optimal solutions
• Efficiently solves large and complex problems

Disadvantages of Dynamic Programming

• Requires extra memory (space complexity increases)
• Not applicable to all problems
• Problem formulation and analysis can be challenging

Applications of Dynamic Programming

Dynamic Programming is widely used in:

• Fibonacci Series
• 0/1 Knapsack Problem
• Matrix Chain Multiplication
• Longest Common Subsequence (LCS)
• Shortest Path Algorithms

Conclusion

Dynamic Programming is a powerful and efficient algorithm design technique.
When a problem exhibits overlapping subproblems and optimal substructure, Dynamic Programming provides an optimal solution with improved time and space efficiency.

It plays a crucial role in solving real-world computational problems in computer science.

Some More: 

• POP- Introduction to Programming Using ‘C’
• DS – Data structure Using C
• OOP – Object Oriented Programming 
• Java Programming
• DBMS – Database Management System
• RDBMS – Relational Database Management System
• https://defineinfoloop.blogspot.com/?m=1