Introduction

• Gate Smashers video on recursive equation of 0-1 Knapsack

• Importance of knowing the recursive equation for competitive exams, college, university exams, and interviews

Recursive Equation of 0-1 Knapsack

• Brute force method of considering or not considering each object

• Total time complexity: 2^n

• Dynamic programming approach to reduce time complexity

Recursive Equation

• n: number of objects, m: total weight of knapsack

• Pn: profit of nth object, Wn: weight of nth object

• Two cases: consider nth object or not consider nth object

Case 1: Consider nth object

• Remaining objects: n-1

• Remaining weight: m-Wn

• Profit: Pn

Case 2: Do not consider nth object

• Remaining objects: n-1

• Remaining weight: m

• Profit: 0

Maximum of the two cases is the output

Additional Conditions

• Termination condition: if weight of an object is more than total knapsack weight, remove it

• Termination condition: if no elements or remaining weight is 0, output is 0

Example and Recursion Tree

• Next, discuss example and recursion tree

• Also discuss time complexity Chapter 1: Introduction to Recursive Equations and Recurrence Relations

• Recursive equation and recurrence relation explained

• Request for confirmation of understanding

Chapter 2: Explaining the Recursion Tree with an Example

• Example of 0-1 knapsack problem with 4 objects and weight 4

• Objects and their weights explained

• Consideration and non-consideration of objects in the knapsack

• Recursive calls and their effects on the number of objects and weight of the knapsack

Chapter 3: Termination Condition and Optimal Result

• Termination condition when either the number of objects or the weight of the knapsack becomes 0

• Recursive calls returning 0 at the termination condition

• Adding the maximum result obtained from the recursive calls to get the optimal result

Chapter 4: Time Complexity Analysis

• Brute force method and its time complexity of 2^n

• Introduction to dynamic programming and its purpose of avoiding repeated calculations

• Identification of repeated subproblems in the example

• Calculation of the total number of unique and repeated problems

• Reduction of time complexity from 2^n to n times m (or n times w) through dynamic programming

Chapter 1: Storing Data in a Table

• Data is stored in a table

• The size of the table needs to be determined

Supporting details

• The table is used to store the value of every subproblem

• Storing the value of subproblems allows for direct use without re-evaluation

• The size of the table is determined by (n + 1) times (m + 1)

• The additional 1 is added because the table includes the 0th level

Chapter 2: Determining Table Size

• The size of the table is (n + 1) times (m + 1)

Supporting details

• The reason for adding 1 is to account for the 0th level

• The table starts from (n, m) and goes down to (0, 0)

Chapter 3: Example Table Size

• An example table size is 5x5, which is 25 in total

Supporting details

• The table can be made for any given values of n and m

• In this case, a table of size 5x5 is used to store all the values

Chapter 4: Time Complexity

• The time complexity can be expressed as O(n * m)

Supporting details

• The time complexity is determined by the size of the table

• In this case, the time complexity is O(5 * 5) or O(25)

Chapter 5: Space Complexity

• The space required is determined by the size of the table

Supporting details

• In normal cases, the space complexity is 2^n

• However, with dynamic programming, the space complexity is reduced to (n + 1) * (m + 1)

Chapter 6: Benefits of Dynamic Programming

• Dynamic programming saves time

Supporting details

• By storing values in a table, dynamic programming avoids re-evaluating subproblems

• This leads to improved time efficiency

Thank you.