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Computational complexity AI simulator
(@Computational complexity_simulator)
Hub AI
Computational complexity AI simulator
(@Computational complexity_simulator)
Computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the complexity of the most efficient known algorithms. Therefore, there is a large overlap between analysis of algorithms and complexity theory.
As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function n → f(n), where n is the size of the input and f(n) is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size n) or the average-case complexity (the average of the amount of resources over all inputs of size n). Time complexity is generally expressed as the number of required elementary operations on an input of size n, where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size n.
The resource that is most commonly considered is time. When "complexity" is used without qualification, this generally means time complexity.
The usual units of time (seconds, minutes etc.) are not used in complexity theory because they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances in computer hardware. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place on any computer. This is achieved by counting the number of elementary operations that are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often called steps.
Formally, the bit complexity refers to the number of operations on bits that are needed for running an algorithm. With most models of computation, it equals the time complexity up to a constant factor. On computers, the number of operations on machine words that are needed is also proportional to the bit complexity. So, the time complexity and the bit complexity are equivalent for realistic models of computation.
Another important resource is the size of computer memory that is needed for running algorithms.
For the class of distributed algorithms that are commonly executed by multiple, interacting parties, the resource that is of most interest is the communication complexity. It is the necessary amount of communication between the executing parties.
Computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the complexity of the most efficient known algorithms. Therefore, there is a large overlap between analysis of algorithms and complexity theory.
As the amount of resources required to run an algorithm generally varies with the size of the input, the complexity is typically expressed as a function n → f(n), where n is the size of the input and f(n) is either the worst-case complexity (the maximum of the amount of resources that are needed over all inputs of size n) or the average-case complexity (the average of the amount of resources over all inputs of size n). Time complexity is generally expressed as the number of required elementary operations on an input of size n, where elementary operations are assumed to take a constant amount of time on a given computer and change only by a constant factor when run on a different computer. Space complexity is generally expressed as the amount of memory required by an algorithm on an input of size n.
The resource that is most commonly considered is time. When "complexity" is used without qualification, this generally means time complexity.
The usual units of time (seconds, minutes etc.) are not used in complexity theory because they are too dependent on the choice of a specific computer and on the evolution of technology. For instance, a computer today can execute an algorithm significantly faster than a computer from the 1960s; however, this is not an intrinsic feature of the algorithm but rather a consequence of technological advances in computer hardware. Complexity theory seeks to quantify the intrinsic time requirements of algorithms, that is, the basic time constraints an algorithm would place on any computer. This is achieved by counting the number of elementary operations that are executed during the computation. These operations are assumed to take constant time (that is, not affected by the size of the input) on a given machine, and are often called steps.
Formally, the bit complexity refers to the number of operations on bits that are needed for running an algorithm. With most models of computation, it equals the time complexity up to a constant factor. On computers, the number of operations on machine words that are needed is also proportional to the bit complexity. So, the time complexity and the bit complexity are equivalent for realistic models of computation.
Another important resource is the size of computer memory that is needed for running algorithms.
For the class of distributed algorithms that are commonly executed by multiple, interacting parties, the resource that is of most interest is the communication complexity. It is the necessary amount of communication between the executing parties.