Greedy algorithms work by making locally optimal choices at each step to arrive at a global optimal solution. They require that the problem exhibits the greedy choice property and optimal substructure. Examples that can be solved with greedy algorithms include fractional knapsack problem, minimum spanning tree, and activity selection. The fractional knapsack problem is solved greedily by sorting items by value/weight ratio and filling the knapsack completely. The 0/1 knapsack problem differs in that items are indivisible.

1. Greedy Algorithms

2. Greedy Algorithm Introduction
• "Greedy Method finds out of many options,
but you have to choose the best option."
• In this method, we have to find out the best
method/option out of many present ways.
• In this approach/method we focus on the first
stage and decide the output, don't think
about the future.

3. A greedy algorithm works if a problem exhibits the
following two properties:
Greedy Choice Property: A globally optimal solution can be
reached at by creating a locally optimal solution. In other
words, an optimal solution can be obtained by creating
"greedy" choices.
Optimal substructure: Optimal solutions contain optimal
subsolutions. In other words, answers to subproblems of an
optimal solution are optimal.
Example:
• machine scheduling
• Fractional Knapsack Problem
• Minimum Spanning Tree
• Huffman Code
• Job Sequencing
• Activity Selection Problem

4. Steps for achieving a Greedy Algorithm
• Feasible: Here we check whether it satisfies all
possible constraints or not, to obtain at least
one solution to our problems.
• Local Optimal Choice: In this, the choice
should be the optimum which is selected from
the currently available
• Unalterable: Once the decision is made, at
any subsequence step that option is not
altered.

5. Knapsack Problem-
You are given the following-
• A knapsack (kind of shoulder bag) with limited weight capacity.
• Few items each having some weight and value.
The problem states-
Which items should be placed into the knapsack such that-
• The value or profit obtained by putting the items into the knapsack is maximum.
• And the weight limit of the knapsack does not exceed.

7. Knapsack Problem Variants-
Knapsack problem has the following two variants-
• Fractional Knapsack Problem
• 0/1 Knapsack Problem

8. Fractional Knapsack Problem-
In Fractional Knapsack Problem,
• As the name suggests, items are divisible here.
• We can even put the fraction of any item into the
knapsack if taking the complete item is not
possible.
• It is solved using Greedy Method.

9. Fractional Knapsack Problem Using
Greedy Method-
Fractional knapsack problem is solved using greedy method in the
following steps-
Step-01:
For each item, compute its value / weight ratio.
Step-02:
Arrange all the items in decreasing order of their value / weight
ratio.
Step-03:
Start putting the items into the knapsack beginning from the item
with the highest ratio.Put as many items as you can into the
knapsack.

10. 0/1 Knapsack Problem-
In 0/1 Knapsack Problem,
• As the name suggests, items are indivisible here.
• We can not take the fraction of any item.
• We have to either take an item completely or
leave it completely.
• It is solved using dynamic programming
approach.

11. • 3. You are given a knapsack that can carry a
maximum weight of 60. There are 4 items with
weights {20, 30, 40, 70} and values {70, 80, 90,
200}. What is the maximum value of the items
you can carry using the knapsack?
a) 160
b) 200
c) 170
d) 90

12. • 5. What is the time complexity of the brute
force algorithm used to solve the Knapsack
problem?
a) O(n)
b) O(n!)
c) O(2n)
d) O(n3)