Algorithmic Complexity
Introduction
Algorithmic complexity, also known as computational complexity, is a field in theoretical computer science that focuses on classifying computational problems based on their inherent difficulty. It is a measure of the amount of resources required to solve a problem, such as time or space, as a function of the size of the input.
Complexity Measures
The two most common types of complexity measures are time complexity and space complexity.
Time Complexity
Time complexity is a function describing the amount of time an algorithm takes in terms of the amount of input to the algorithm. It is usually expressed using big O notation, which describes the upper bound of the time complexity in the worst case scenario. For example, a linear search algorithm has a time complexity of O(n), where n is the size of the input.
Space Complexity
Space complexity, on the other hand, is a function describing the amount of memory (space) an algorithm takes in terms of the amount of input to the algorithm. Like time complexity, it is also expressed using big O notation. For example, an algorithm that sorts an array in-place (i.e., using no additional memory) has a constant space complexity of O(1).
Complexity Classes
Complexity classes are sets of problems grouped together based on their resource usage. The most well-known complexity classes are P, NP, NP-Complete, and NP-Hard.
P
P (Polynomial time) is the complexity class containing decision problems which can be solved by a deterministic Turing machine in polynomial time. In simpler terms, P is the set of problems that can be solved quickly (in polynomial time) by a computer.
NP
NP (Nondeterministic Polynomial time) is the complexity class containing decision problems for which a given solution can be verified as correct or incorrect in polynomial time by a deterministic Turing machine. In other words, if you are given a "yes" answer to a problem, you can check that it is correct quickly (in polynomial time), but it may take a long time to find the solution in the first place.
NP-Complete
NP-Complete problems are the hardest problems in NP, in the sense that they are the most likely not to be in P. If any NP-Complete problem can be solved quickly, then every problem in NP can be solved quickly.
NP-Hard
NP-Hard problems are at least as hard as the hardest problems in NP. If any NP-Hard problem can be solved quickly, then every problem in NP can be solved quickly. However, unlike NP-Complete problems, an NP-Hard problem may not be in NP (i.e., it may not be possible to verify a solution quickly).
P vs NP Problem
The P vs NP problem is one of the most fundamental questions in computer science. It asks whether every problem for which a solution can be verified quickly (in polynomial time) can also be solved quickly. Despite much effort, this question remains open and is one of the seven "Millennium Prize Problems" for which the Clay Mathematics Institute offers a $1,000,000 prize for a correct solution.
Conclusion
Understanding algorithmic complexity is crucial in computer science and programming. It helps developers write more efficient code and provides a theoretical basis for investigating the limits of what computers can and cannot do.