Have you ever heard of a problem statement that already has a predefined answer? If not, then you should know about the knapsack problem.

The Knapsack basically optimises the characteristics of the elements present within the given set of problem statements.

The problem statement in the Knapsack problem deals with characteristics of the elements like weight and value. 

Quite similar to the Knapsack problem we have the 0 1 Knapsack problem where the programmer has to figure out whether the set of elements is completely filled or empty.

And, since a Knapsack problem already has a predefined answer, the programmer would have to match the outcome with the output statement.

Analysing elements in a data table is definitely the most important aspect of programming. Apart from the Knapsack problem, we also have the set matrix zeroes concept that generates similar output.

What is a Knapsack problem?

Before learning the differences between the Knapsack and the 0 1 Knapsack problem, let us briefly discuss their definitions.

In simple terms the Knapsack problem is used to optimize the input statement and the results of a given set of problems.

The programmer is provided a predefined output that they have to match or level up using the algorithm for the Knapsack problem.

Usually the weight and value of the given set of problems are determined by using the logic for Knapsack. 

The aim here is to attain greater than the exact value or equal value to the predefined output.

What is a 0 1 Knapsack problem?

Staying across the borders of the Knapsack problem, the 0 1 Knapsack problem deals with figuring out whether a given set of problems is either empty or filled completely.

For instance, you have been provided two objects, each weighing 1kg and 2 kgs respectively. In the context of a 0 1 Knapsack problem, you can only pick both items if you equal the combining weight of both.

What we mean to say is that, you would have to drop 1 kg weight from the second item in order to pick both the first and second, or you cannot pick either.

With that you now have a general understanding of the definitions for the Knapsack and the 0 1 Knapsack problems. Let us now get along further and discuss the differences.

What are the differences between the Knapsack and the 0-1 Knapsack problems?

Take a look at the differences between the Knapsack and the 0-1 Knapsack problem:

Approaches

While the Knapt problem is solved using the greedy approach, the 0-1 Knapsack problem is solved using the dynamic programming approach.

These approaches can be better understood by the help of the following examples:

Greedy approach

This approach optimises using the set of elements or items presented in a problem statement that are readily available to solve the problem. Hence the name greedy approach.

The algorithm for the greedy approach will never reverse itself even if the choice or decision was wrong.

For instance, we have three different components D, E and F in a given set of array with values 1, 2 and 3 respectively. You will have to start with placing the D and E components within the Knapsack. Till now we have 3 as total value of the Knapsack. Finally we will consider the value F within the existing Knapsack and the whole set will be complete.

This greedy approach will be used within the fractional Knapsack problem. Moving forward to the 0 1 Knapsack problem, we will briefly take an example for the Dynamic programming approach.

Dynamic Programming Approach

The dynamic programming approach comparative follows a similar approach like brute force. If you are a beginner level programmer, it is important to learn the aspects of brute force.

This approach mainly focuses on the divide and conquer approach. The program will keep on dividing the elements present within the program until each element has been reduced to a singular correspondent.

Hence, in the context of the 0 1 Knapsack problem, this approach will divide each element D, E and F. Further it will consider each element separately and you will be able to input the value for each separately.

For instance, iterating on the same example as earlier we have three components, D, E and F with 1, 2 and 3 as the values. In the context of Dynamic Program Approach, the first two approaches will be considered and this will make the total value as 3. Now since we have to derive the output statement as 5 we will not be able to take the value of F into consideration.

Value of components

In context of the 0 1 Knapsack problem, the program will only consider the elements if the output statement matches the value of the elements.

On the other hand, in the context of the fractional Knapsack problem, the value of the elements will still be fractionally considered even if the output statement is not related to the value.

Weight of the elements

The 0 1 Knapsack problem considers the weight of the subsets or elements in either less than or equal to the total value.

The fractional Knapsack problem considers the exact value of the weight of the elements.

The fundamentals of the basic differences between the 0 1 Knapsack and the Fractional Knapsack Problems are closely based on how the weight and value of the elements are treated in each concept.

Now, we will take a similar yet different approach and discuss the set matrix zeroes problem statement.

What is the approach for setting matrices to zero?

The setting of matrices to zero problem is more commonly applied while solving the Sudokos found in newspapers.

In order to set a matrix to zero in a grid problem, try changing the values of each row and column so that the value of the middle grid will return to 0.

You can use the brute force algorithm as discussed in the previous section to solve the grid problem.

Winding Up

There are absolutely no problem statements that cannot be solved by the use of programming concepts. 

From the 0 1 Knapsack problem to setting the matrices to zero, algorithms such as brute force and recursive approach can be applied in any scenario.