Greedy Algorithm: Napsack Problem

Introduction to Napsack Problem

• The Napsack Problem is a problem that comes under the Greedy Technique.
• It is one of the main and most important problems in Greedy Technique.

Problem Statement

• The Napsack Problem involves finding the optimal solution to maximize profit.
• The problem is given a bag with a certain capacity, and several objects with their respective profits and weights.
• The goal is to find the optimal combination of objects to put in the bag to maximize profit.

Greedy Approach for Profit

• The first approach is to be greedy about profit.
• This involves selecting the object with the highest profit first.
• The object with the highest profit is put in the bag, and the remaining capacity is calculated.
• The process is repeated until the bag is full or there are no more objects to add.

Greedy Approach for Weight

• The second approach is to be greedy about weight.
• This involves selecting the object with the lowest weight first.
• The object with the lowest weight is put in the bag, and the remaining capacity is calculated.
• The process is repeated until the bag is full or there are no more objects to add.

Greedy Approach for Both Profit and Weight

• The third approach is to be greedy about both profit and weight.
• This involves calculating the profit by weight ratio for each object.
• The objects are then sorted based on their profit by weight ratio.
• The object with the highest profit by weight ratio is selected first, and the process is repeated until the bag is full or there are no more objects to add.

Time Complexity of Knapsack Problem

• The time complexity of the Knapsack Problem is O(n log n).
• This is because the algorithm involves calculating the profit by weight ratio for each object, which takes O(n) time, and then sorting the objects, which takes O(n log n) time.
• The overall time complexity is dominated by the sorting step, which is O(n log n).

Introduction to Knapsack Problem

• The Knapsack Problem is a classic problem in Greedy Technique, a key strategy in algorithm design.

Problem Statement

• The problem involves finding the optimal solution to maximize profit given a bag with a certain capacity and several objects with their respective profits and weights.
• The goal is to find the optimal combination of objects to put in the bag to maximize profit.

Greedy Approach

• There are three greedy approaches to solve the Knapsack Problem:

Greedy Approach for Profit

• This approach involves selecting the object with the highest profit first.
• The object with the highest profit is put in the bag, and the remaining capacity is calculated.
• The process is repeated until the bag is full or there are no more objects to add.

Greedy Approach for Weight

• This approach involves selecting the object with the lowest weight first.
• The object with the lowest weight is put in the bag, and the remaining capacity is calculated.
• The process is repeated until the bag is full or there are no more objects to add.

Greedy Approach for Both Profit and Weight

• This approach involves calculating the profit by weight ratio for each object.
• The objects are then sorted based on their profit by weight ratio.
• The object with the highest profit by weight ratio is selected first, and the process is repeated until the bag is full or there are no more objects to add.

Time Complexity of Knapsack Problem

• The time complexity of the Knapsack Problem is O(n log n).
• The algorithm involves calculating the profit by weight ratio for each object, which takes O(n) time.
• The sorting step, which takes O(n log n) time, dominates the overall time complexity.