Lecture 102: Introduction to Dynamic Programming (DP)

Introduction

• Start of a new series on Dynamic Programming (DP)
• Importance of understanding DP for advanced algorithms and interviews
• Comparison with previous topics like Pointers and Linked Lists in C++ which were tough but covered thoroughly

Goals of the Series

• Develop a command over DP through various lectures and examples
• Equip students to solve DP-related questions in tech interviews, especially for companies like Google

Prerequisites

• Before starting DP, ensure a strong understanding of Recursion and Array-based problems.
• Recommendation to review previous lectures on Recursion and Arrays.

Understanding DP

• Definition: Breaking down problems to solve and reuse the solutions of subproblems for efficiency.
• Quote: "To shift the past is to repeat."
• Motivation for why DP solves overlapping subproblems efficiently compared to brute-force methods.

Approaches in DP

• Top-down (Memoization):

  • Use recursion and store results of subproblems
  • Example: Fibonacci sequence using recursion and memoization

• Bottom-up (Tabulation):

  • Solve subproblems first and use their results to build up solutions to bigger problems
  • Example: Iteratively solving the Fibonacci sequence

Optimizations

• Space Optimization:
  • Reduce space complexity by utilizing only essential previous states rather than storing all states
  • Example: Keeping track of just two previous numbers in the Fibonacci sequence instead of the entire table

Implementation Steps

1. Top-down (Memoization):

• Store subproblem results in a table/array to avoid redundant calculations
• Implement recursive function with memoization

2. Bottom-up (Tabulation):

• Initialize base cases
• Use iteration to fill out the table based on previously computed states

3. Space Optimization:

• Iterate while maintaining only essential states for the solution in variables

Example Problem: Fibonacci Sequence

• Recursive Memoization Approach:

  • Define base cases: Fib(0)=0, Fib(1)=1
  • Recursive relation: Fib(n) = Fib(n-1) + Fib(n-2)

• Tabulation Approach:

  • Initialize array for storage
  • Iteratively fill the array based on the relation

• Space-Optimized Approach:

  • Use variables to keep track of the last two computed values instead of an entire array

Conclusion

• Recap of DP’s importance and methods
• Encourage practice through problem-solving
• Reminder to subscribe, like, and share the channel for more content