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Computational complexity theory

What is computational complexity theory?

Computational complexity theory is a branch of theoretical computer science. It’s main goal is to classify and compare the practical difficulty involved in solving computational problems about finite combinatorial objects.

It seeks to determine the amount of resources that it will take to solve a specific problem. It also aims to explain why some problems are undecidable or intractable. 

What are computational problems?

Computational problems are tasks that are solved by computers. These problems can be considered infinite collections of instances or cases along with a set of solutions (which may be empty) for every instance or case.

These problems are one of the main things studied in theoretical computer science.

The types of computational problems are:

1. Decision problems

In a decision problem, the answer for every question is either ‘yes’ or ‘no’. These problems are usually presented as the sets of all the instances for which the answer is ‘yes.’


2. Search problems

The answers in a search problem can be arbitrary strings. Search problems can be represented as search relations (relations that consist of all the instance-solution pairs).


3. Counting problems

These problems ask for the number of solutions for a particular search problem. They are represented by a function f from {0,1}* to the nonnegative integers.


4. Optimization problems

Such problems seek to find the best possible solution from the set of all the possible solutions to a search problem. These problems can be represented by their search relations.


5. Function problems

In these problems, for every input, there is a single output expected. These outputs are far more complex than the outputs for decision problems. The answer can’t just be ‘yes’ or ‘no’.

What is computational complexity?

Computational complexity refers to the complexity of an algorithm. It is the amount of resources that are needed to run the algorithm and specifically focuses on time and memory requirements. 

The computational complexity of a problem is essentially the complexity of the best algorithms that make it possible to solve the problem.

What are the best, worst, and average-case complexity?

These are three different ways to measure the time complexity (or any other complexity measure) of different inputs of the same size. 

1. Best-case complexity

This is the complexity involved in solving a problem for the best input of size n.


2. Average-case complexity

This is the complexity involved in solving a problem on average. It is only defined regarding a probability distribution over the inputs.


3. Worst-case complexity

This is the complexity involved in solving a problem for the worst input of size n.

An amortized analysis takes the costly and the less costly operations together over the whole series of operations of the algorithm.

What are complexity classes?

Complexity classes are sets of problems of related complexity. Simple complexity classes are defined by the type of computational problem, the model of computation, and The resource or resources that are being bounded, and the bound.

To see how complexity classes are defined, here is the definition for the complexity class DTIME(f(n)).
The set of decision problems that can be solved using a deterministic Turing machine within time f(n).


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Computational complexity theory

October 14, 2020

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What is computational complexity theory?

Computational complexity theory is a branch of theoretical computer science. It’s main goal is to classify and compare the practical difficulty involved in solving computational problems about finite combinatorial objects.

It seeks to determine the amount of resources that it will take to solve a specific problem. It also aims to explain why some problems are undecidable or intractable. 

What are computational problems?

Computational problems are tasks that are solved by computers. These problems can be considered infinite collections of instances or cases along with a set of solutions (which may be empty) for every instance or case.

These problems are one of the main things studied in theoretical computer science.

The types of computational problems are:

1. Decision problems

In a decision problem, the answer for every question is either ‘yes’ or ‘no’. These problems are usually presented as the sets of all the instances for which the answer is ‘yes.’


2. Search problems

The answers in a search problem can be arbitrary strings. Search problems can be represented as search relations (relations that consist of all the instance-solution pairs).


3. Counting problems

These problems ask for the number of solutions for a particular search problem. They are represented by a function f from {0,1}* to the nonnegative integers.


4. Optimization problems

Such problems seek to find the best possible solution from the set of all the possible solutions to a search problem. These problems can be represented by their search relations.


5. Function problems

In these problems, for every input, there is a single output expected. These outputs are far more complex than the outputs for decision problems. The answer can’t just be ‘yes’ or ‘no’.

What is computational complexity?

Computational complexity refers to the complexity of an algorithm. It is the amount of resources that are needed to run the algorithm and specifically focuses on time and memory requirements. 

The computational complexity of a problem is essentially the complexity of the best algorithms that make it possible to solve the problem.

What are the best, worst, and average-case complexity?

These are three different ways to measure the time complexity (or any other complexity measure) of different inputs of the same size. 

1. Best-case complexity

This is the complexity involved in solving a problem for the best input of size n.


2. Average-case complexity

This is the complexity involved in solving a problem on average. It is only defined regarding a probability distribution over the inputs.


3. Worst-case complexity

This is the complexity involved in solving a problem for the worst input of size n.

An amortized analysis takes the costly and the less costly operations together over the whole series of operations of the algorithm.

What are complexity classes?

Complexity classes are sets of problems of related complexity. Simple complexity classes are defined by the type of computational problem, the model of computation, and The resource or resources that are being bounded, and the bound.

To see how complexity classes are defined, here is the definition for the complexity class DTIME(f(n)).
The set of decision problems that can be solved using a deterministic Turing machine within time f(n).


Thanks for reading! We hope you found this helpful.

Ready to level-up your business? Click here.

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