Misplaced Pages

#669330

59-490: (Redirected from Critical Path ) Critical path may refer to: The longest series of sequential operations in a parallel computation; see analysis of parallel algorithms Critical path method , an algorithm for scheduling a set of project activities Critical Path (book) , by Buckminster Fuller The Critical Path: An Essay on the Social Context of Literary Criticism ,

118-480: A computational hardness assumption to prove the difficulty of several other problems in computational game theory , property testing , and machine learning . The complexity class QP consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of DTIME as follows. In complexity theory, the unsolved P versus NP problem asks if all problems in NP have polynomial-time algorithms. All

177-529: A constant multiplier , and such a multiplier is irrelevant to big O classification, the standard usage for logarithmic-time algorithms is O ( log ⁡ n ) {\displaystyle O(\log n)} regardless of the base of the logarithm appearing in the expression of T . Algorithms taking logarithmic time are commonly found in operations on binary trees or when using binary search . An O ( log ⁡ n ) {\displaystyle O(\log n)} algorithm

236-554: A 1971 book by Northrop Frye The Critical Path , a podcast by Horace Dediu Critical Path (video game) , an interactive movie computer game Critical Path, Inc. , a provider of messaging services Critical Path Institute , an organization for improvement of the drug development process Critical Path Project , a video archive Critical Path Project, early source of HIV/AIDS information founded by Kiyoshi Kuromiya See also [ edit ] All pages with titles containing Critical path Topics referred to by

295-422: A constant. Linear time is the best possible time complexity in situations where the algorithm has to sequentially read its entire input. Therefore, much research has been invested into discovering algorithms exhibiting linear time or, at least, nearly linear time. This research includes both software and hardware methods. There are several hardware technologies which exploit parallelism to provide this. An example

354-792: A deterministic Turing machine form the complexity class known as EXP . Sometimes, exponential time is used to refer to algorithms that have T ( n ) = 2 , where the exponent is at most a linear function of n . This gives rise to the complexity class E . An algorithm is said to be factorial time if T(n) is upper bounded by the factorial function n! . Factorial time is a subset of exponential time (EXP) because n ! ≤ n n = 2 n log ⁡ n = O ( 2 n 1 + ϵ ) {\displaystyle n!\leq n^{n}=2^{n\log n}=O\left(2^{n^{1+\epsilon }}\right)} for all ϵ > 0 {\displaystyle \epsilon >0} . However, it

413-440: A deterministic machine which is robust in terms of machine model changes. (For example, a change from a single-tape Turing machine to a multi-tape machine can lead to a quadratic speedup, but any algorithm that runs in polynomial time under one model also does so on the other.) Any given abstract machine will have a complexity class corresponding to the problems which can be solved in polynomial time on that machine. An algorithm

472-422: A given size. Less common, and usually specified explicitly, is the average-case complexity , which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed as a function of the size of the input. Since this function is generally difficult to compute exactly, and

531-466: A paper dictionary. As a result, the search space within the dictionary decreases as the algorithm gets closer to the target word. An algorithm is said to run in polylogarithmic time if its time T ( n ) {\displaystyle T(n)} is O ( ( log ⁡ n ) k ) {\displaystyle O{\bigl (}(\log n)^{k}{\bigr )}} for some constant k . Another way to write this

590-488: A term n c {\displaystyle n^{c}} for any c > 1 {\displaystyle c>1} . Algorithms which run in quasilinear time include: In many cases, the O ( n log ⁡ n ) {\displaystyle O(n\log n)} running time is simply the result of performing a Θ ( log ⁡ n ) {\displaystyle \Theta (\log n)} operation n times (for

649-457: Is O ( log k ⁡ n ) {\displaystyle O(\log ^{k}n)} . For example, matrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine , and a graph can be determined to be planar in a fully dynamic way in O ( log 3 ⁡ n ) {\displaystyle O(\log ^{3}n)} time per insert/delete operation. An algorithm

SECTION 10

#1733093750670

708-517: Is content-addressable memory . This concept of linear time is used in string matching algorithms such as the Boyer–Moore string-search algorithm and Ukkonen's algorithm . An algorithm is said to run in quasilinear time (also referred to as log-linear time ) if T ( n ) = O ( n log k ⁡ n ) {\displaystyle T(n)=O(n\log ^{k}n)} for some positive constant k ; linearithmic time

767-422: Is n . Another example was the graph isomorphism problem , which the best known algorithm from 1982 to 2016 solved in 2 O ( n log ⁡ n ) {\displaystyle 2^{O\left({\sqrt {n\log n}}\right)}} . However, at STOC 2016 a quasi-polynomial time algorithm was presented. It makes a difference whether the algorithm is allowed to be sub-exponential in

826-461: Is a polynomial time algorithm . The following table summarizes some classes of commonly encountered time complexities. In the table, poly( x ) = x , i.e., polynomial in  x . formerly-best algorithm for graph isomorphism An algorithm is said to be constant time (also written as O ( 1 ) {\textstyle O(1)} time) if the value of T ( n ) {\textstyle T(n)} (the complexity of

885-427: Is a synonym for "tractable", "feasible", "efficient", or "fast". Some examples of polynomial-time algorithms: These two concepts are only relevant if the inputs to the algorithms consist of integers. The concept of polynomial time leads to several complexity classes in computational complexity theory. Some important classes defined using polynomial time are the following. P is the smallest time-complexity class on

944-430: Is an open problem. Other computational problems with quasi-polynomial time solutions but no known polynomial time solution include the planted clique problem in which the goal is to find a large clique in the union of a clique and a random graph . Although quasi-polynomially solvable, it has been conjectured that the planted clique problem has no polynomial time solution; this planted clique conjecture has been used as

1003-489: Is clearly superpolynomial, but some algorithms are only very weakly superpolynomial. For example, the Adleman–Pomerance–Rumely primality test runs for n time on n -bit inputs; this grows faster than any polynomial for large enough n , but the input size must become impractically large before it cannot be dominated by a polynomial with small degree. An algorithm that requires superpolynomial time lies outside

1062-591: Is considered highly efficient, as the ratio of the number of operations to the size of the input decreases and tends to zero when n increases. An algorithm that must access all elements of its input cannot take logarithmic time, as the time taken for reading an input of size n is of the order of n . An example of logarithmic time is given by dictionary search. Consider a dictionary D which contains n entries, sorted in alphabetical order . We suppose that, for 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} , one may access

1121-436: Is defined to take superpolynomial time if T ( n ) is not bounded above by any polynomial. Using little omega notation , it is ω ( n ) time for all constants c , where n is the input parameter, typically the number of bits in the input. For example, an algorithm that runs for 2 steps on an input of size n requires superpolynomial time (more specifically, exponential time). An algorithm that uses exponential resources

1180-483: Is different from Wikidata All article disambiguation pages All disambiguation pages Analysis of parallel algorithms A so-called work-time (WT) (sometimes called work-depth, or work-span) framework was originally introduced by Shiloach and Vishkin for conceptualizing and describing parallel algorithms. In the WT framework, a parallel algorithm is first described in terms of parallel rounds. For each round,

1239-423: Is known. Such problems arise in approximation algorithms; a famous example is the directed Steiner tree problem , for which there is a quasi-polynomial time approximation algorithm achieving an approximation factor of O ( log 3 ⁡ n ) {\displaystyle O(\log ^{3}n)} ( n being the number of vertices), but showing the existence of such a polynomial time algorithm

SECTION 20

#1733093750670

1298-438: Is not a subset of E. An example of an algorithm that runs in factorial time is bogosort , a notoriously inefficient sorting algorithm based on trial and error . Bogosort sorts a list of n items by repeatedly shuffling the list until it is found to be sorted. In the average case, each pass through the bogosort algorithm will examine one of the n ! orderings of the n items. If the items are distinct, only one such ordering

1357-401: Is not generally agreed upon, however the two most widely used are below. A problem is said to be sub-exponential time solvable if it can be solved in running times whose logarithms grow smaller than any given polynomial. More precisely, a problem is in sub-exponential time if for every ε > 0 there exists an algorithm which solves the problem in time O (2 ). The set of all such problems

1416-485: Is of great practical importance. An algorithm is said to be of polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm, that is, T ( n ) = O ( n ) for some positive constant k . Problems for which a deterministic polynomial-time algorithm exists belong to the complexity class P , which is central in the field of computational complexity theory . Cobham's thesis states that polynomial time

1475-425: Is said to be subquadratic time if T ( n ) = o ( n 2 ) {\displaystyle T(n)=o(n^{2})} . For example, simple, comparison-based sorting algorithms are quadratic (e.g. insertion sort ), but more advanced algorithms can be found that are subquadratic (e.g. shell sort ). No general-purpose sorts run in linear time, but the change from quadratic to sub-quadratic

1534-449: Is said to run in sub-linear time (often spelled sublinear time ) if T ( n ) = o ( n ) {\displaystyle T(n)=o(n)} . In particular this includes algorithms with the time complexities defined above. The specific term sublinear time algorithm commonly refers to randomized algorithms that sample a small fraction of their inputs and process them efficiently to approximately infer properties of

1593-620: Is significantly faster than exponential time . The worst case running time of a quasi-polynomial time algorithm is 2 O ( log c ⁡ n ) {\displaystyle 2^{O(\log ^{c}n)}} for some fixed c > 0 {\displaystyle c>0} . When c = 1 {\displaystyle c=1} this gives polynomial time, and for c < 1 {\displaystyle c<1} it gives sub-linear time. There are some problems for which we know quasi-polynomial time algorithms, but no polynomial time algorithm

1652-410: Is that 3SAT , the satisfiability problem of Boolean formulas in conjunctive normal form with at most three literals per clause and with n variables, cannot be solved in time 2 . More precisely, the hypothesis is that there is some absolute constant c > 0 such that 3SAT cannot be decided in time 2 by any deterministic Turing machine. With m denoting the number of clauses, ETH is equivalent to

1711-549: Is the case k = 1 {\displaystyle k=1} . Using soft O notation these algorithms are O ~ ( n ) {\displaystyle {\tilde {O}}(n)} . Quasilinear time algorithms are also O ( n 1 + ε ) {\displaystyle O(n^{1+\varepsilon })} for every constant ε > 0 {\displaystyle \varepsilon >0} and thus run faster than any polynomial time algorithm whose time bound includes

1770-588: Is the class of all parameterized problems ( L , k ) {\displaystyle (L,k)} for which there is a computable function f : N → N {\displaystyle f:\mathbb {N} \to \mathbb {N} } with f ∈ o ( k ) {\displaystyle f\in o(k)} and an algorithm that decides L in time 2 f ( k ) ⋅ poly ( n ) {\displaystyle 2^{f(k)}\cdot {\text{poly}}(n)} . The exponential time hypothesis ( ETH )

1829-452: Is the complexity class SUBEXP which can be defined in terms of DTIME as follows. This notion of sub-exponential is non-uniform in terms of ε in the sense that ε is not part of the input and each ε may have its own algorithm for the problem. Some authors define sub-exponential time as running times in 2 o ( n ) {\displaystyle 2^{o(n)}} . This definition allows larger running times than

Critical path - Misplaced Pages Continue

1888-531: Is the size in units of bits needed to represent the input. Algorithmic complexities are classified according to the type of function appearing in the big O notation. For example, an algorithm with time complexity O ( n ) {\displaystyle O(n)} is a linear time algorithm and an algorithm with time complexity O ( n α ) {\displaystyle O(n^{\alpha })} for some constant α > 0 {\displaystyle \alpha >0}

1947-421: Is unrealistic, but not a problem, since any computation that can run in parallel on N processors can be executed on p < N processors by letting each processor execute multiple units of work. A result called Brent's law states that one can perform such a "simulation" in time T p , bounded by or, less precisely, An alternative statement of the law bounds T p above and below by showing that

2006-417: The complexity class P . Cobham's thesis posits that these algorithms are impractical, and in many cases they are. Since the P versus NP problem is unresolved, it is unknown whether NP-complete problems require superpolynomial time. Quasi-polynomial time algorithms are algorithms whose running time exhibits quasi-polynomial growth , a type of behavior that may be slower than polynomial time but yet

2065-485: The floor function . If w = D ( ⌊ n 2 ⌋ ) {\displaystyle w=D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} --that is to say, the word w is exactly in the middle of the dictionary--then we are done. Else, if w < D ( ⌊ n 2 ⌋ ) {\displaystyle w<D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} --i.e., if

2124-524: The k th entry of the dictionary in a constant time. Let D ( k ) {\displaystyle D(k)} denote this k th entry. Under these hypotheses, the test to see if a word w is in the dictionary may be done in logarithmic time: consider D ( ⌊ n 2 ⌋ ) {\displaystyle D\left(\left\lfloor {\frac {n}{2}}\right\rfloor \right)} , where ⌊ ⌋ {\displaystyle \lfloor \;\rfloor } denotes

2183-408: The parallel random-access machine PRAM model) and, as well as in the class notes . The overview below explains how the WT framework can be used for analyzing more general parallel algorithms, even when their description is not available within the WT framework. Suppose computations are executed on a machine that has p processors. Let T p denote the time that expires between the start of

2242-454: The algorithm) is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an array takes constant time as only one operation has to be performed to locate it. In a similar manner, finding the minimal value in an array sorted in ascending order; it is the first element. However, finding the minimal value in an unordered array is not a constant time operation as scanning over each element in

2301-444: The algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor . Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity , which is the maximum amount of time required for inputs of

2360-437: The array is needed in order to determine the minimal value. Hence it is a linear time operation, taking O ( n ) {\textstyle O(n)} time. If the number of elements is known in advance and does not change, however, such an algorithm can still be said to run in constant time. Despite the name "constant time", the running time does not have to be independent of the problem size, but an upper bound for

2419-437: The best-known algorithms for NP-complete problems like 3SAT etc. take exponential time. Indeed, it is conjectured for many natural NP-complete problems that they do not have sub-exponential time algorithms. Here "sub-exponential time" is taken to mean the second definition presented below. (On the other hand, many graph problems represented in the natural way by adjacency matrices are solvable in subexponential time simply because

Critical path - Misplaced Pages Continue

2478-408: The computation and its end. Analysis of the computation's running time focuses on the following notions: Several useful results follow from the definitions of work, span and cost: Using these definitions and laws, the following measures of performance can be given: Analysis of parallel algorithms is usually carried out under the assumption that an unbounded number of processors is available. This

2537-425: The entire instance. This type of sublinear time algorithm is closely related to property testing and statistics . Other settings where algorithms can run in sublinear time include: An algorithm is said to take linear time , or O ( n ) {\displaystyle O(n)} time, if its time complexity is O ( n ) {\displaystyle O(n)} . Informally, this means that

2596-403: The first definition of sub-exponential time. An example of such a sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the general number field sieve , which runs in time about 2 O ~ ( n 1 / 3 ) {\displaystyle 2^{{\tilde {O}}(n^{1/3})}} , where the length of the input

2655-418: The hypothesis that k SAT cannot be solved in time 2 for any integer k ≥ 3 . The exponential time hypothesis implies P ≠ NP . An algorithm is said to be exponential time , if T ( n ) is upper bounded by 2 , where poly( n ) is some polynomial in n . More formally, an algorithm is exponential time if T ( n ) is bounded by O (2 ) for some constant k . Problems which admit exponential time algorithms on

2714-439: The known inapproximability results for the set cover problem. The term sub-exponential time is used to express that the running time of some algorithm may grow faster than any polynomial but is still significantly smaller than an exponential. In this sense, problems that have sub-exponential time algorithms are somewhat more tractable than those that only have exponential algorithms. The precise definition of "sub-exponential"

2773-655: The notation, see Big O notation § Family of Bachmann–Landau notations ). For example, binary tree sort creates a binary tree by inserting each element of the n -sized array one by one. Since the insert operation on a self-balancing binary search tree takes O ( log ⁡ n ) {\displaystyle O(\log n)} time, the entire algorithm takes O ( n log ⁡ n ) {\displaystyle O(n\log n)} time. Comparison sorts require at least Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} comparisons in

2832-399: The operations to be performed are characterized, but several issues can be suppressed. For example, the number of operations at each round need not be clear, processors need not be mentioned and any information that may help with the assignment of processors to jobs need not be accounted for. Second, the suppressed information is provided. The inclusion of the suppressed information is guided by

2891-408: The proof of a scheduling theorem due to Brent, which is explained later in this article. The WT framework is useful since while it can greatly simplify the initial description of a parallel algorithm, inserting the details suppressed by that initial description is often not very difficult. For example, the WT framework was adopted as the basic presentation framework in the parallel algorithms books (for

2950-647: The running time for small inputs is usually not consequential, one commonly focuses on the behavior of the complexity when the input size increases—that is, the asymptotic behavior of the complexity. Therefore, the time complexity is commonly expressed using big O notation , typically O ( n ) {\displaystyle O(n)} , O ( n log ⁡ n ) {\displaystyle O(n\log n)} , O ( n α ) {\displaystyle O(n^{\alpha })} , O ( 2 n ) {\displaystyle O(2^{n})} , etc., where n

3009-418: The running time has to be independent of the problem size. For example, the task "exchange the values of a and b if necessary so that a ≤ b {\textstyle a\leq b} " is called constant time even though the time may depend on whether or not it is already true that a ≤ b {\textstyle a\leq b} . However, there is some constant t such that

SECTION 50

#1733093750670

3068-418: The running time increases at most linearly with the size of the input. More precisely, this means that there is a constant c such that the running time is at most c n {\displaystyle cn} for every input of size n . For example, a procedure that adds up all elements of a list requires time proportional to the length of the list, if the adding time is constant, or, at least, bounded by

3127-423: The same term [REDACTED] This disambiguation page lists articles associated with the title Critical path . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Critical_path&oldid=1207046845 " Category : Disambiguation pages Hidden categories: Short description

3186-430: The size of the input is the square of the number of vertices.) This conjecture (for the k-SAT problem) is known as the exponential time hypothesis . Since it is conjectured that NP-complete problems do not have quasi-polynomial time algorithms, some inapproximability results in the field of approximation algorithms make the assumption that NP-complete problems do not have quasi-polynomial time algorithms. For example, see

3245-427: The size of the instance, the number of vertices, or the number of edges. In parameterized complexity , this difference is made explicit by considering pairs ( L , k ) {\displaystyle (L,k)} of decision problems and parameters k . SUBEPT is the class of all parameterized problems that run in time sub-exponential in k and polynomial in the input size n : More precisely, SUBEPT

3304-404: The span (depth) T ∞ and the work T 1 together provide reasonable bounds on the computation time. Time complexity In theoretical computer science , the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm . Time complexity is commonly estimated by counting the number of elementary operations performed by

3363-419: The time required is always at most t . An algorithm is said to take logarithmic time when T ( n ) = O ( log ⁡ n ) {\displaystyle T(n)=O(\log n)} . Since log a ⁡ n {\displaystyle \log _{a}n} and log b ⁡ n {\displaystyle \log _{b}n} are related by

3422-418: The word w comes earlier in alphabetical order than the middle word of the whole dictionary--we continue the search in the same way in the left (i.e. earlier) half of the dictionary, and then again repeatedly until the correct word is found. Otherwise, if it comes after the middle word, continue similarly with the right half of the dictionary. This algorithm is similar to the method often used to find an entry in

3481-460: The worst case because log ⁡ ( n ! ) = Θ ( n log ⁡ n ) {\displaystyle \log(n!)=\Theta (n\log n)} , by Stirling's approximation . They also frequently arise from the recurrence relation T ( n ) = 2 T ( n 2 ) + O ( n ) {\textstyle T(n)=2T\left({\frac {n}{2}}\right)+O(n)} . An algorithm

#669330