Time complexity in the context of Algorithmic mechanism design

In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called satisfiable, else unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.

SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no known algorithm that efficiently solves each SAT problem (where "efficiently" means "deterministically in polynomial time"). Although such an algorithm is generally believed not to exist, this belief has not been proven or disproven mathematically. Resolving the question of whether SAT has a polynomial-time algorithm would settle the P versus NP problem - one of the most important open problems in the theory of computing.

In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process.

For maximum efficiency it is desirable to minimize resource usage. However, different resources such as time and space complexity cannot be compared directly, so which of two algorithms is considered to be more efficient often depends on which measure of efficiency is considered most important.

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines.

The design and analysis of approximation algorithms crucially involves a mathematical proof certifying the quality of the returned solutions in the worst case. This distinguishes them from heuristics such as annealing or genetic algorithms, which find reasonably good solutions on some inputs, but provide no clear indication at the outset on when they may succeed or fail.

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.

In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same size may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm.

The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem. These estimates provide an insight into reasonable directions of search for efficient algorithms.

In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as n in asymptotic notation). It gives an upper bound on the resources required by the algorithm.

In the case of running time, the worst-case time complexity indicates the longest running time performed by an algorithm given any input of size n, and thus guarantees that the algorithm will finish in the indicated period of time. The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms.

CPU time (or process time) is the amount of time that a central processing unit (CPU) was used for processing a specific set of instructions of a computer program or operating system, as opposed to total time from start to finish, where the CPU might have spent time on other processes. See the figure. CPU time is measured in clock ticks or seconds. Sometimes it is useful to convert CPU time into a percentage of the CPU capacity, giving the CPU usage.

Measuring CPU time for two functionally identical programs that process identical inputs can indicate which program is faster, but it is a common misunderstanding that CPU time can be used to compare algorithms. Comparing programs by their CPU time compares specific implementations of algorithms. (It is possible to have both efficient and inefficient implementations of the same algorithm.) Algorithms are more commonly compared using measures of time complexity and space complexity.

The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space.

Similar to time complexity, space complexity is often expressed asymptotically in big O notation, such as ${\displaystyle O(n),}$${\displaystyle O(n\log n),}$ ${\displaystyle O(n^{\alpha }),}$ ${\displaystyle O(2^{n}),}$ etc., where n is a characteristic of the input influencing space complexity.

A software engine is a core component of a complex software system. The word "engine" is a metaphor of a car's engine. Thus a software engine is a complex subsystem; not unlike how a car engine functions. Software engines work in conjunction with other components of a process or system. They typically have an input and an output, and the productivity is usually linear to running speed.

There is no formal guideline for what should be called an engine, but the term has become widespread in the software industry.

In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.

Binary search runs in logarithmic time in the worst case, making ${\displaystyle O(\log n)}$ comparisons, where ${\displaystyle n}$ is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. However, binary search can be used to solve a wider range of problems, such as finding the next-smallest or next-largest element in the array relative to the target even if it is absent from the array.

In computer science, linear search or sequential search is a method for finding an element within a list. It sequentially checks each element of the list until a match is found or the whole list has been searched.

A linear search runs in linear time in the worst case, and makes at most n comparisons, where n is the length of the list. If each element is equally likely to be searched, then linear search has an average case of ⁠n+1/2⁠ comparisons, but the average case can be affected if the search probabilities for each element vary. Linear search is rarely practical because other search algorithms and schemes, such as the binary search algorithm and hash tables, allow significantly faster searching for all but short lists.

In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource.Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements.

In real-time computing, the worst-case execution time is often of particular concern since it is important to know how much time might be needed in the worst case to guarantee that the algorithm will always finish on time.

In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop is fixed before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions.

The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not particularly easy to devise a computable function that is not primitive recursive; some examples are shown in section § Limitations below.