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L-5.3: 0/1 Knapsack Problem |Dynamic Programming |Recursive Equation |Recursion Tree Time Complexity

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.

L-5.3: 0/1 Knapsack Problem |Dynamic Programming |Recursive Equation |Recursion Tree Time Complexity

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.