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t computers can and cannot do the p versus np problem one of the seven millennium prize problems 1 is part of the field of computational complexity closely related fields in theoretical computer science are analysis of algorithms and computability theory a key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem more precisely computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources in turn imposing restrictions on the available resources is what distinguishes computational complexity from computability theory the latter theory asks what kinds of problems can in principle be solved algorithmically computational problems edit a traveling salesman tour through 14 german cities problem instances edit a computational problem can be viewed as an infinite collection of instances together with a set possibly empty of solutions for every instance the input string for a computational problem is referred to as a problem instance and should not be confused with the problem itself in computational complexity theory a problem refers to the abstract question to be solved in contrast an instance of this problem is a rather concrete utterance which can serve as the input for a decision problem for example consider the problem of primality testing the instance is a number e g 15 and the solution is yes if the number is prime and no otherwise in this case 15 is not prime and the answer is no stated another way the instance is a particular input to the problem and the solution is the output corresponding to the given input to further highlight the difference between a problem and an instance consider the following instance of the decision version of the travelling salesman problem is there a route of at most 2000 kilometres passing through all of germany s 14 largest cities the quantitative answer to this particular problem instance is of little use for solving other instances of the problem such as asking for a round trip through 14 sites in milan whose total length is at most 10 km for this reason complexity theory addresses computational problems and not particular problem instances representing problem instances edit when considering computational problems a problem instance is a string over an alphabet usually the alphabet is taken to be the binary alphabet i e the set 0 1 and thus the strings are bitstrings as in a real world computer mathematical objects other than bitstrings must be suitably encoded for example integers can be represented in binary notation and graphs can be encoded directly via their adjacency matrices or by encoding their adjacency lists in binary even though some proofs of complexity theoretic theorems regularly assume some concrete choice of input encoding one tries to keep the discussion abstract enough to be independent of the precise choice of encoding this can be achieved by ensuring that different representations can be transformed into each other efficiently decision problems as formal languages edit a decision problem has only two possible outputs yes or no or alternately 1 or 0 on any input decision problems are one of the central objects of study in computational complexity theory a decision problem is a type of computational problem where the answer is either yes or no alternatively 1 or 0 a decision problem can be viewed as a formal language where the members of the language are instances whose output is yes and the non members are those instances whose output is no the objective is to decide with the aid of an algorithm whether a given input string is a member of the formal language under consideration if the algorithm deciding this problem returns the answer yes the algorithm is said to accept the input string otherwise it is said to reject the input an example of a decision problem is the following the input is an arbitrary graph the problem consists in deciding whether the given graph is connected or not the formal language associated with this decision problem is then the set of all connected graphs to obtain a precise definition of this language one has to decide how graphs are encoded as binary strings function problems edit a function problem is a computational problem where a single output of a total function is expected for every input but the output can be more complex than that of a decision problem that is the output is not just yes or no notable examples include the traveling salesman problem and the integer factorization problem it is tempting to think that the notion of function problems is much richer than the notion of decision problems however this is not really the case since function problems can be recast as decision problems for example the multiplication of two integers can be expressed as the set of triples a b c displaystyle a b c such that the relation a b c displaystyle a times b c holds deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers measuring the size of an instance edit to measure the difficulty of solving a computational problem one may wish to see how much time the best algorithm requires to solve the problem however the running time may in general depend on the instance in particular larger instances will require more time to solve thus the time required to solve a problem or the space required or any measure of complexity is calculated as a function of the size of the instance the input size is typically measured in bits complexity theory studies how algorithms scale as input size increases for instance in the problem of finding whether a graph is connected how much more time does it take to solve a problem for a graph with 2 n displaystyle 2n vertices compared to the time taken for a graph with n displaystyle n vertices if the input size is n displaystyle n the time taken can be expressed as a function of n displaystyle n since the time taken on different inputs of the same size can be different the worst case time complexity t n displaystyle t n is defined to be the maximum time taken over all inputs of size n displaystyle n if t n displaystyle t n is a polynomial in n displaystyle n then the algorithm is said to be a polynomial time algorithm cobham s thesis argues that a problem can be solved with a feasible amount of resources if and only if it admits a polynomial time algorithm machine models and complexity measures edit turing machine edit main article turing machine an illustration of a turing machine a turing machine is a mathematical model of a general computing machine it is a theoretical device that manipulates symbols contained on a strip of tape turing machines are not intended as a practical computing technology but rather as a general model of a computing machine anything from an advanced supercomputer to a mathematician with a pencil and paper it is believed that if a problem can be solved by an algorithm there exists a turing machine that solves the problem indeed this is the statement of the church turing thesis furthermore it is known that everything that can be computed on other models of computation known to us today such as a ram machine conway s game of life cellular automata lambda calculus or any programming language can be computed on a turing machine since turing machines are easy to analyze mathematically and are believed to be as powerful as any other model of computation the turing machine is the most commonly used model in complexity theory many types of turing machines are used to define complexity classes such as deterministic turing machines probabilistic turing machines non deterministic turing machines quantum turing machines symmetric turing machines and alternating turing machines they are all equally powerful in principle but when resources such as time or space are bounded some of these may be more powerful than others a deterministic turing machine is the most basic turing machine which uses a fixed set of rules to determine its future actions a probabilistic turing machine is a deterministic turing machine with an extra supply of random bits the ability to make probabilistic decisions often helps algorithms solve problems more efficiently algorithms that use random bits are called randomized algorithms a non deterministic turing machine is a deterministic turing machine with an added feature of non determinism which allows a turing machine to have multiple possible future actions from a given state one way to view non determinism is that the turing machine branches into many possible computational paths at each step and if it solves the problem in any of these branches it is said to have solved the problem clearly this model is not meant to be a physically realizable model it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes for examples see non deterministic algorithm other machine models edit many machine models different from the standard multi tape turing machines have been proposed in the literature for example random access machines perhaps surprisingly each of these models can be converted to another without providing any extra computational power the time and memory consumption of these alternate models may vary 2 what all these models have in common is that the machines operate deterministically however some computational problems are easier to analyze in terms of more unusual resources for example a non deterministic turing machine is a computational model that is allowed to branch out to check many different possibilities at once the non deterministic turing machine has very little to do with how we physically want to compute algorithms but its branching exactly captures many of the mathematical models we want to analyze so that non deterministic time is a very important resource in analyzing computational problems complexity measures edit for a precise definition of what it means to solve a problem using a given amount of time and space a computational model such as the deterministic turing machine is used the time required by a deterministic turing machine m displaystyle m on input x displaystyle x is the total number of state transitions or steps the machine makes before it halts and outputs the answer yes or no a turing machine m displaystyle m is said to operate within time f n displaystyle f n if the time required by m displaystyle m on each input of length n displaystyle n is at most f n displaystyle f n a decision problem a displaystyle a can be solved in time f n displaystyle f n if there exists a turing machine operating in time f n displaystyle f n that solves the problem since complexity theory is interested in classifying problems based on their difficulty one defines sets of problems based on some criteria for instance the set of problems solvable within time f n displaystyle f n on a deterministic turing machine is then denoted by dtime f n displaystyle f n analogous definitions can be made for space requirements although time and space are the most well known complexity resources any complexity measure can be viewed as a computational resource complexity measures are very generally defined by the blum complexity axioms other complexity measures used in complexity theory include communication complexity circuit complexity and decision tree complexity the complexity of an algorithm is often expressed using big o notation best worst and average case complexity edit visualization of the quicksort algorithm which has average case performance o n log n displaystyle mathcal o n log n the best worst and average case complexity refer to three different ways of measuring the time complexity or any other complexity measure of different inputs of the same size since some inputs of size n displaystyle n may be faster to solve than others we define the following complexities best case complexity this is the complexity of solving the problem for the best input of size n displaystyle n average case complexity this is the complexity of solving the problem on average for inputs of size n this complexity is only defined with respect to a probability distribution over the inputs for instance if all inputs of the same size are assumed to be equally likely to appear the average case complexity can be defined with respect to the uniform distribution over all inputs of size n displaystyle n amortized analysis amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm worst case complexity this is the complexity of solving the problem for the worst input of size n displaystyle n the order from cheap to costly is best average of discrete uniform distribution amortized worst for example the deterministic sorting algorithm quicksort addresses the problem of sorting a list of integers the worst case is when the pivot is always the largest or smallest value in the list so the list is never divided in this case the algorithm takes time o n 2 displaystyle n 2 if we assume that all possible permutations of the input list are equally likely the average time taken for sorting is o n log n displaystyle o n log n the best case occurs when each pivoting divides the list in half also needing o n log n displaystyle o n log n time upper and lower bounds on the complexity of problems edit to classify the computation time or similar resources such as space consumption it is helpful to demonstrate upper and lower bounds on the maximum amount of time required by the most efficient algorithm to solve a given problem the complexity of an algorithm is usually taken to be its worst case complexity unless specified otherwise analyzing a particular algorithm falls under the field of analysis of algorithms to show an upper bound t n displaystyle t n on the time complexity of a problem one needs to show only that there is a particular algorithm with running time at most t n displaystyle t n however proving lower bounds is much more difficult since lower bounds make a statement about all possible algorithms that solve a given problem the phrase all possible algorithms includes not just the algorithms known today but any algorithm that might be discovered in the future to show a lower bound of t n displaystyle t n for a problem requires showing that no algorithm can have time complexity lower than t n displaystyle t n upper and lower bounds are usually stated using the big o notation which hides constant factors and smaller terms this makes the bounds independent of the specific details of the computational model used for instance if t n 7 n 2 15 n 40 displaystyle t n 7n 2 15n 40 in big o notation one ...
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