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136-397: In computing , a hash table is a data structure that implements an associative array , also called a dictionary or simply map ; an associative array is an abstract data type that maps keys to values . A hash table uses a hash function to compute an index , also called a hash code , into an array of buckets or slots , from which the desired value can be found. During lookup,

272-399: A L o o k u p ( k e y , command ) {\displaystyle \mathrm {Lookup} (\mathrm {key} ,{\text{command}})} wrapper such that each element in the bucket gets rehashed and its procedure involve as follows: Linear hashing is an implementation of the hash table which enables dynamic growths or shrinks of the table one bucket at

408-480: 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

544-529: 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

680-444: A dynamic array found to be more cache-friendly is used in the place where a linked list or self-balancing binary search trees is usually deployed, since the contiguous allocation pattern of the array could be exploited by hardware-cache prefetchers —such as translation lookaside buffer —resulting in reduced access time and memory consumption. Open addressing is another collision resolution technique in which every entry record

816-755: A function of the size of the input. Since this function is generally difficult to compute exactly, and 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

952-484: A broad array of electronic, wireless, and optical networking technologies. The Internet carries an extensive range of information resources and services, such as the inter-linked hypertext documents of the World Wide Web and the infrastructure to support email. Computer programming is the process of writing, testing, debugging, and maintaining the source code and documentation of computer programs. This source code

1088-497: A computer's capabilities, but typically do not directly apply them in the performance of tasks that benefit the user, unlike application software. Application software, also known as an application or an app , is computer software designed to help the user perform specific tasks. Examples include enterprise software , accounting software , office suites , graphics software , and media players . Many application programs deal principally with documents . Apps may be bundled with

1224-422: 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

1360-792: 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

1496-440: 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

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1632-432: A hash function works, one can then focus on finding the fastest possible such hash function. A search algorithm that uses hashing consists of two parts. The first part is computing a hash function which transforms the search key into an array index . The ideal case is such that no two search keys hashes to the same array index. However, this is not always the case and is impossible to guarantee for unseen given data. Hence

1768-466: 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

1904-422: A property that, the cost of finding the desired item from any given buckets within the neighbourhood is very close to the cost of finding it in the bucket itself; the algorithm attempts to be an item into its neighbourhood—with a possible cost involved in displacing other items. Each bucket within the hash table includes an additional "hop-information"—an H -bit bit array for indicating the relative distance of

2040-458: A set of instructions called a computer program . The program has an executable form that the computer can use directly to execute the instructions. The same program in its human-readable source code form, enables a programmer to study and develop a sequence of steps known as an algorithm . Because the instructions can be carried out in different types of computers, a single set of source instructions converts to machine instructions according to

2176-406: A solution is to perform the resizing gradually to avoid storage blip—typically at 50% of new table's size—during rehashing and to avoid memory fragmentation that triggers heap compaction due to deallocation of large memory blocks caused by the old hash table. In such case, the rehashing operation is done incrementally through extending prior memory block allocated for the old hash table such that

2312-435: A superposition, i.e. in both states of one and zero, simultaneously. Thus, the value of the qubit is not between 1 and 0, but changes depending on when it is measured. This trait of qubits is known as quantum entanglement , and is the core idea of quantum computing that allows quantum computers to do large scale computations. Quantum computing is often used for scientific research in cases where traditional computers do not have

2448-488: 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

2584-559: A time. Computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery . It includes the study and experimentation of algorithmic processes, and the development of both hardware and software. Computing has scientific, engineering, mathematical, technological, and social aspects. Major computing disciplines include computer engineering , computer science , cybersecurity , data science , information systems , information technology , and software engineering . The term computing

2720-457: 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

2856-517: 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

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2992-422: 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

3128-421: Is spintronics . Spintronics can provide computing power and storage, without heat buildup. Some research is being done on hybrid chips, which combine photonics and spintronics. There is also research ongoing on combining plasmonics , photonics, and electronics. Cloud computing is a model that allows for the use of computing resources, such as servers or applications, without the need for interaction between

3264-460: Is a discipline that integrates several fields of electrical engineering and computer science required to develop computer hardware and software. Computer engineers usually have training in electronic engineering (or electrical engineering ), software design , and hardware-software integration, rather than just software engineering or electronic engineering. Computer engineers are involved in many hardware and software aspects of computing, from

3400-461: 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

3536-429: Is a common method of implementation of hash tables. Let T {\displaystyle T} and x {\displaystyle x} be the hash table and the node respectively, the operation involves as follows: If the element is comparable either numerically or lexically , and inserted into the list by maintaining the total order , it results in faster termination of the unsuccessful searches. If

3672-407: Is a hash function, and h ( x ) < m {\displaystyle h(x)<m} . Under reasonable assumptions, hash tables have better time complexity bounds on search, delete, and insert operations in comparison to self-balancing binary search trees . Hash tables are also commonly used to implement sets, by omitting the stored value for each key and merely tracking whether

3808-405: Is a non-integer real-valued constant and m {\displaystyle m} is the size of the table. An advantage of the hashing by multiplication is that the m {\displaystyle m} is not critical. Although any value A {\displaystyle A} produces a hash function, Donald Knuth suggests using the golden ratio . Uniform distribution of

3944-414: Is a person who writes computer software. The term computer programmer can refer to a specialist in one area of computer programming or to a generalist who writes code for many kinds of software. One who practices or professes a formal approach to programming may also be known as a programmer analyst. A programmer's primary computer language ( C , C++ , Java , Lisp , Python , etc.) is often prefixed to

4080-427: 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

4216-409: Is also synonymous with counting and calculating . In earlier times, it was used in reference to the action performed by mechanical computing machines , and before that, to human computers . The history of computing is longer than the history of computing hardware and includes the history of methods intended for pen and paper (or for chalk and slate) with or without the aid of tables. Computing

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4352-414: Is an area of research that brings together the disciplines of computer science, information theory, and quantum physics. While the idea of information as part of physics is relatively new, there appears to be a strong tie between information theory and quantum mechanics. Whereas traditional computing operates on a binary system of ones and zeros, quantum computing uses qubits . Qubits are capable of being in

4488-458: Is an open addressing based collision resolution algorithm; the collisions are resolved through favouring the displacement of the element that is farthest—or longest probe sequence length (PSL)—from its "home location" i.e. the bucket to which the item was hashed into. Although Robin Hood hashing does not change the theoretical search cost , it significantly affects the variance of the distribution of

4624-430: 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

4760-640: Is available, values can be stored without regard for their keys, and a binary search or linear search can be used to retrieve the element. In many situations, hash tables turn out to be on average more efficient than search trees or any other table lookup structure. For this reason, they are widely used in many kinds of computer software , particularly for associative arrays , database indexing , caches , and sets . The idea of hashing arose independently in different places. In January 1953, Hans Peter Luhn wrote an internal IBM memorandum that used hashing with chaining. The first example of open addressing

4896-489: 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

5032-423: Is computer software designed to operate and control computer hardware, and to provide a platform for running application software. System software includes operating systems , utility software , device drivers , window systems , and firmware . Frequently used development tools such as compilers , linkers , and debuggers are classified as system software. System software and middleware manage and integrate

5168-591: 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

5304-436: 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

5440-476: Is denoted CMOS-integrated nanophotonics (CINP). One benefit of optical interconnects is that motherboards, which formerly required a certain kind of system on a chip (SoC), can now move formerly dedicated memory and network controllers off the motherboards, spreading the controllers out onto the rack. This allows standardization of backplane interconnects and motherboards for multiple types of SoCs, which allows more timely upgrades of CPUs. Another field of research

5576-462: Is empty, the element is inserted, and the leftmost bit of bitmap is set to 1; if not empty, linear probing is used for finding an empty slot in the table, the bitmap of the bucket gets updated followed by the insertion; if the empty slot is not within the range of the neighbourhood, i.e. H -1, subsequent swap and hop-info bit array manipulation of each bucket is performed in accordance with its neighbourhood invariant properties . Robin Hood hashing

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5712-460: Is found, which indicates an unsuccessful search. Well-known probe sequences include: The performance of open addressing may be slower compared to separate chaining since the probe sequence increases when the load factor α {\displaystyle \alpha } approaches 1. The probing results in an infinite loop if the load factor reaches 1, in the case of a completely filled table. The average cost of linear probing depends on

5848-415: Is independent of the number of elements stored in the table. Many hash table designs also allow arbitrary insertions and deletions of key–value pairs , at amortized constant average cost per operation. Hashing is an example of a space-time tradeoff . If memory is infinite, the entire key can be used directly as an index to locate its value with a single memory access. On the other hand, if infinite time

5984-485: Is intimately tied to the representation of numbers, though mathematical concepts necessary for computing existed before numeral systems . The earliest known tool for use in computation is the abacus , and it is thought to have been invented in Babylon circa between 2700 and 2300 BC. Abaci, of a more modern design, are still used as calculation tools today. The first recorded proposal for using digital electronics in computing

6120-710: Is its potential to support energy efficiency. Allowing thousands of instances of computation to occur on one single machine instead of thousands of individual machines could help save energy. It could also ease the transition to renewable energy source, since it would suffice to power one server farm with renewable energy, rather than millions of homes and offices. However, this centralized computing model poses several challenges, especially in security and privacy. Current legislation does not sufficiently protect users from companies mishandling their data on company servers. This suggests potential for further legislative regulations on cloud computing and tech companies. Quantum computing

6256-423: 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

6392-438: 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

6528-401: 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

6664-485: 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

6800-482: Is resolved through maintaining two hash tables, each having its own hashing function, and collided slot gets replaced with the given item, and the preoccupied element of the slot gets displaced into the other hash table. The process continues until every key has its own spot in the empty buckets of the tables; if the procedure enters into infinite loop —which is identified through maintaining a threshold loop counter—both hash tables get rehashed with newer hash functions and

6936-478: Is said to be perfect for a given set S {\displaystyle S} if it is injective on S {\displaystyle S} , that is, if each element x ∈ S {\displaystyle x\in S} maps to a different value in 0 , . . . , m − 1 {\displaystyle {0,...,m-1}} . A perfect hash function can be created if all

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7072-425: 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

7208-449: 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

7344-620: 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

7480-559: Is sometimes considered a sub-discipline of electrical engineering , telecommunications, computer science , information technology, or computer engineering , since it relies upon the theoretical and practical application of these disciplines. The Internet is a global system of interconnected computer networks that use the standard Internet Protocol Suite (TCP/IP) to serve billions of users. This includes millions of private, public, academic, business, and government networks, ranging in scope from local to global. These networks are linked by

7616-406: Is stored in the bucket array itself, and the hash resolution is performed through probing . When a new entry has to be inserted, the buckets are examined, starting with the hashed-to slot and proceeding in some probe sequence , until an unoccupied slot is found. When searching for an entry, the buckets are scanned in the same sequence, until either the target record is found, or an unused array slot

7752-410: 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

7888-687: Is the application of computers and telecommunications equipment to store, retrieve, transmit, and manipulate data, often in the context of a business or other enterprise. The term is commonly used as a synonym for computers and computer networks, but also encompasses other information distribution technologies such as television and telephones. Several industries are associated with information technology, including computer hardware, software, electronics , semiconductors , internet, telecom equipment , e-commerce , and computer services . DNA-based computing and quantum computing are areas of active research for both computing hardware and software, such as

8024-549: 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

8160-588: 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 )

8296-452: 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

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8432-482: Is the hash value of x ∈ S {\displaystyle x\in S} and m {\displaystyle m} is the size of the table. The scheme in hashing by multiplication is as follows: h ( x ) = ⌊ m ( ( x A ) mod 1 ) ⌋ {\displaystyle h(x)=\lfloor m{\bigl (}(xA){\bmod {1}}{\bigr )}\rfloor } Where A {\displaystyle A}

8568-531: 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}

8704-509: Is the study of complementary networks of hardware and software (see information technology) that people and organizations use to collect, filter, process, create, and distribute data . The ACM 's Computing Careers describes IS as: "A majority of IS [degree] programs are located in business schools; however, they may have different names such as management information systems, computer information systems, or business information systems. All IS degrees combine business and computing topics, but

8840-449: Is thus often developed by a team of domain experts, each a specialist in some area of development. However, the term programmer may apply to a range of program quality, from hacker to open source contributor to professional. It is also possible for a single programmer to do most or all of the computer programming needed to generate the proof of concept to launch a new killer application . A programmer, computer programmer, or coder

8976-429: Is written in a programming language , which is an artificial language that is often more restrictive than natural languages , but easily translated by the computer. Programming is used to invoke some desired behavior (customization) from the machine. Writing high-quality source code requires knowledge of both the computer science domain and the domain in which the application will be used. The highest-quality software

9112-554: The CPU type. The execution process carries out the instructions in a computer program. Instructions express the computations performed by the computer. They trigger sequences of simple actions on the executing machine. Those actions produce effects according to the semantics of the instructions. Computer hardware includes the physical parts of a computer, including the central processing unit , memory , and input/output . Computational logic and computer architecture are key topics in

9248-417: 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

9384-485: 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

9520-444: The function of the program it implements, either by directly providing instructions to the computer hardware or by serving as input to another piece of software. The term was coined to contrast with the old term hardware (meaning physical devices). In contrast to hardware, software is intangible. Software is also sometimes used in a more narrow sense, meaning application software only. System software, or systems software,

9656-409: The integer universe assumption that all elements of the table stem from the universe U = { 0 , . . . , u − 1 } {\displaystyle U=\{0,...,u-1\}} , where the bit length of u {\displaystyle u} is confined within the word size of a computer architecture . A hash function h {\displaystyle h}

9792-524: 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

9928-407: The worst-case time complexity , which is the maximum amount of time required for inputs of 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

10064-880: The above titles, and those who work in a web environment often prefix their titles with Web . The term programmer can be used to refer to a software developer , software engineer, computer scientist , or software analyst . However, members of these professions typically possess other software engineering skills, beyond programming. The computer industry is made up of businesses involved in developing computer software, designing computer hardware and computer networking infrastructures, manufacturing computer components, and providing information technology services, including system administration and maintenance. The software industry includes businesses engaged in development , maintenance , and publication of software. The industry also includes software services , such as training , documentation , and consulting. Computer engineering

10200-454: 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

10336-442: The application. In particular, if one uses dynamic resizing with exact doubling and halving of the table size, then the hash function needs to be uniform only when the size is a power of two . Here the index can be computed as some range of bits of the hash function. On the other hand, some hashing algorithms prefer to have the size be a prime number . For open addressing schemes, the hash function should also avoid clustering ,

10472-437: 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

10608-437: 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

10744-748: The bucket array holds exactly one item. Therefore an open-addressed hash table cannot have a load factor greater than 1. The performance of open addressing becomes very bad when the load factor approaches 1. Therefore a hash table that uses open addressing must be resized or rehashed if the load factor α {\displaystyle \alpha } approaches 1. With open addressing, acceptable figures of max load factor α max {\displaystyle \alpha _{\max }} should range around 0.6 to 0.75. A hash function h : U → { 0 , . . . , m − 1 } {\displaystyle h:U\rightarrow \{0,...,m-1\}} maps

10880-438: The bucket array stores a pointer to a list or array of data. Separate chaining hash tables suffer gradually declining performance as the load factor grows, and no fixed point beyond which resizing is absolutely needed. With separate chaining, the value of α max {\displaystyle \alpha _{\max }} that gives best performance is typically between 1 and 3. With open addressing, each slot of

11016-454: The buckets of the hash table remain unaltered. A common approach for amortized rehashing involves maintaining two hash functions h old {\displaystyle h_{\text{old}}} and h new {\displaystyle h_{\text{new}}} . The process of rehashing a bucket's items in accordance with the new hash function is termed as cleaning , which is implemented through command pattern by encapsulating

11152-615: The buckets or nodes link within the table. The algorithm is ideally suited for fixed memory allocation . The collision in coalesced hashing is resolved by identifying the largest-indexed empty slot on the hash table, then the colliding value is inserted into that slot. The bucket is also linked to the inserted node's slot which contains its colliding hash address. Cuckoo hashing is a form of open addressing collision resolution technique which guarantees O ( 1 ) {\displaystyle O(1)} worst-case lookup complexity and constant amortized time for insertions. The collision

11288-457: The challenges in implementing computations. For example, programming language theory studies approaches to the description of computations, while the study of computer programming investigates the use of programming languages and complex systems . The field of human–computer interaction focuses on the challenges in making computers and computations useful, usable, and universally accessible to humans. The field of cybersecurity pertains to

11424-560: The computer and its system software, or may be published separately. Some users are satisfied with the bundled apps and need never install additional applications. The system software manages the hardware and serves the application, which in turn serves the user. Application software applies the power of a particular computing platform or system software to a particular purpose. Some apps, such as Microsoft Office , are developed in multiple versions for several different platforms; others have narrower requirements and are generally referred to by

11560-414: The computing power to do the necessary calculations, such in molecular modeling . Large molecules and their reactions are far too complex for traditional computers to calculate, but the computational power of quantum computers could provide a tool to perform such calculations. Time complexity Since an algorithm's running time may vary among different inputs of the same size, one commonly considers

11696-571: The constraint of unique keys . In the hash table implementation of associative arrays, an array A {\displaystyle A} of length m {\displaystyle m} is partially filled with n {\displaystyle n} elements, where m ≥ n {\displaystyle m\geq n} . A value x {\displaystyle x} gets stored at an index location A [ h ( x ) ] {\displaystyle A[h(x)]} , where h {\displaystyle h}

11832-448: The design of individual microprocessors , personal computers, and supercomputers , to circuit design . This field of engineering includes not only the design of hardware within its own domain, but also the interactions between hardware and the context in which it operates. Software engineering is the application of a systematic, disciplined, and quantifiable approach to the design, development, operation, and maintenance of software, and

11968-414: The development of quantum algorithms . Potential infrastructure for future technologies includes DNA origami on photolithography and quantum antennae for transferring information between ion traps. By 2011, researchers had entangled 14 qubits . Fast digital circuits , including those based on Josephson junctions and rapid single flux quantum technology, are becoming more nearly realizable with

12104-437: The discovery of nanoscale superconductors . Fiber-optic and photonic (optical) devices, which already have been used to transport data over long distances, are starting to be used by data centers, along with CPU and semiconductor memory components. This allows the separation of RAM from CPU by optical interconnects. IBM has created an integrated circuit with both electronic and optical information processing in one chip. This

12240-686: The emphasis between technical and organizational issues varies among programs. For example, programs differ substantially in the amount of programming required." The study of IS bridges business and computer science , using the theoretical foundations of information and computation to study various business models and related algorithmic processes within a computer science discipline. The field of Computer Information Systems (CIS) studies computers and algorithmic processes, including their principles, their software and hardware designs, their applications, and their impact on society while IS emphasizes functionality over design. Information technology (IT)

12376-644: The engineering paradigm. The generally accepted concepts of Software Engineering as an engineering discipline have been specified in the Guide to the Software Engineering Body of Knowledge (SWEBOK). The SWEBOK has become an internationally accepted standard in ISO/IEC TR 19759:2015. Computer science or computing science (abbreviated CS or Comp Sci) is the scientific and practical approach to computation and its applications. A computer scientist specializes in

12512-425: 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

12648-434: The field of computer hardware. Computer software, or just software , is a collection of computer programs and related data, which provides instructions to a computer. Software refers to one or more computer programs and data held in the storage of the computer. It is a set of programs, procedures, algorithms, as well as its documentation concerned with the operation of a data processing system. Program software performs

12784-403: 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

12920-442: The first silicon dioxide field effect transistors at Bell Labs, the first transistors in which drain and source were adjacent at the surface. Subsequently, a team demonstrated a working MOSFET at Bell Labs 1960. The MOSFET made it possible to build high-density integrated circuits , leading to what is known as the computer revolution or microcomputer revolution . A computer is a machine that manipulates data according to

13056-511: The first working transistor , the point-contact transistor , in 1947. In 1953, the University of Manchester built the first transistorized computer , the Manchester Baby . However, early junction transistors were relatively bulky devices that were difficult to mass-produce, which limited them to a number of specialised applications. In 1957, Frosch and Derick were able to manufacture

13192-457: The hash function's ability to distribute the elements uniformly throughout the table to avoid clustering , since formation of clusters would result in increased search time. Since the slots are located in successive locations, linear probing could lead to better utilization of CPU cache due to locality of references resulting in reduced memory latency . Coalesced hashing is a hybrid of both separate chaining and open addressing in which

13328-420: The hash table and j {\displaystyle j} be the index, the insertion procedure is as follows: Repeated insertions cause the number of entries in a hash table to grow, which consequently increases the load factor; to maintain the amortized O ( 1 ) {\displaystyle O(1)} performance of the lookup and insertion operations, a hash table is dynamically resized and

13464-413: The hash table whenever the load factor α {\displaystyle \alpha } reaches α max {\displaystyle \alpha _{\max }} . Similarly the table may also be resized if the load factor drops below α max / 4 {\displaystyle \alpha _{\max }/4} . With separate chaining hash tables, each slot of

13600-418: The hash values is a fundamental requirement of a hash function. A non-uniform distribution increases the number of collisions and the cost of resolving them. Uniformity is sometimes difficult to ensure by design, but may be evaluated empirically using statistical tests, e.g., a Pearson's chi-squared test for discrete uniform distributions. The distribution needs to be uniform only for table sizes that occur in

13736-418: 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

13872-444: The item which was originally hashed into the current virtual bucket within H -1 entries. Let k {\displaystyle k} and B k {\displaystyle Bk} be the key to be inserted and bucket to which the key is hashed into respectively; several cases are involved in the insertion procedure such that the neighbourhood property of the algorithm is vowed: if B k {\displaystyle Bk}

14008-451: The items of the tables are rehashed into the buckets of the new hash table, since the items cannot be copied over as varying table sizes results in different hash value due to modulo operation . If a hash table becomes "too empty" after deleting some elements, resizing may be performed to avoid excessive memory usage . Generally, a new hash table with a size double that of the original hash table gets allocated privately and every item in

14144-470: The items on the buckets, i.e. dealing with cluster formation in the hash table. Each node within the hash table that uses Robin Hood hashing should be augmented to store an extra PSL value. Let x {\displaystyle x} be the key to be inserted, x . p s l {\displaystyle x.psl} be the (incremental) PSL length of x {\displaystyle x} , T {\displaystyle T} be

14280-437: The key is hashed and the resulting hash indicates where the corresponding value is stored. A map implemented by a hash table is called a hash map . Most hash table designs employ an imperfect hash function . Hash collisions , where the hash function generates the same index for more than one key, therefore typically must be accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup

14416-401: The key is present. A load factor α {\displaystyle \alpha } is a critical statistic of a hash table, and is defined as follows: load factor   ( α ) = n m , {\displaystyle {\text{load factor}}\ (\alpha )={\frac {n}{m}},} where The performance of the hash table deteriorates in relation to

14552-419: The keys are ordered , it could be efficient to use " self-organizing " concepts such as using a self-balancing binary search tree , through which the theoretical worst case could be brought down to O ( log ⁡ n ) {\displaystyle O(\log {n})} , although it introduces additional complexities. In dynamic perfect hashing , two-level hash tables are used to reduce

14688-539: The keys are known ahead of time. The schemes of hashing used in integer universe assumption include hashing by division, hashing by multiplication, universal hashing , dynamic perfect hashing , and static perfect hashing . However, hashing by division is the commonly used scheme. The scheme in hashing by division is as follows: h ( x )   =   x mod m {\displaystyle h(x)\ =\ x\,{\bmod {\,}}m} where h ( x ) {\displaystyle h(x)}

14824-439: 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"

14960-400: The load factor α {\displaystyle \alpha } . The software typically ensures that the load factor α {\displaystyle \alpha } remains below a certain constant, α max {\displaystyle \alpha _{\max }} . This helps maintain good performance. Therefore, a common approach is to resize or "rehash"

15096-493: The look-up complexity to be a guaranteed O ( 1 ) {\displaystyle O(1)} in the worst case. In this technique, the buckets of k {\displaystyle k} entries are organized as perfect hash tables with k 2 {\displaystyle k^{2}} slots providing constant worst-case lookup time, and low amortized time for insertion. A study shows array-based separate chaining to be 97% more performant when compared to

15232-542: The mapping of two or more keys to consecutive slots. Such clustering may cause the lookup cost to skyrocket, even if the load factor is low and collisions are infrequent. The popular multiplicative hash is claimed to have particularly poor clustering behavior. K-independent hashing offers a way to prove a certain hash function does not have bad keysets for a given type of hashtable. A number of K-independence results are known for collision resolution schemes such as linear probing and cuckoo hashing. Since K-independence can prove

15368-655: 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

15504-407: The operations such as A d d ( k e y ) {\displaystyle \mathrm {Add} (\mathrm {key} )} , G e t ( k e y ) {\displaystyle \mathrm {Get} (\mathrm {key} )} and D e l e t e ( k e y ) {\displaystyle \mathrm {Delete} (\mathrm {key} )} through

15640-410: The original hash table gets moved to the newly allocated one by computing the hash values of the items followed by the insertion operation. Rehashing is simple, but computationally expensive. Some hash table implementations, notably in real-time systems , cannot pay the price of enlarging the hash table all at once, because it may interrupt time-critical operations. If one cannot avoid dynamic resizing,

15776-460: The owner of these resources and the end user. It is typically offered as a service, making it an example of Software as a Service , Platforms as a Service , and Infrastructure as a Service , depending on the functionality offered. Key characteristics include on-demand access, broad network access, and the capability of rapid scaling. It allows individual users or small business to benefit from economies of scale . One area of interest in this field

15912-480: The platform they run on. For example, a geography application for Windows or an Android application for education or Linux gaming . Applications that run only on one platform and increase the desirability of that platform due to the popularity of the application, known as killer applications . A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow

16048-480: The problem of search in large files. The first published work on hashing with chaining is credited to Arnold Dumey , who discussed the idea of using remainder modulo a prime as a hash function. The word "hashing" was first published in an article by Robert Morris. A theoretical analysis of linear probing was submitted originally by Konheim and Weiss. An associative array stores a set of (key, value) pairs and allows insertion, deletion, and lookup (search), with

16184-609: The procedure continues. Hopscotch hashing is an open addressing based algorithm which combines the elements of cuckoo hashing , linear probing and chaining through the notion of a neighbourhood of buckets—the subsequent buckets around any given occupied bucket, also called a "virtual" bucket. The algorithm is designed to deliver better performance when the load factor of the hash table grows beyond 90%; it also provides high throughput in concurrent settings , thus well suited for implementing resizable concurrent hash table . The neighbourhood characteristic of hopscotch hashing guarantees

16320-522: The protection of computer systems and networks. This includes information and data privacy , preventing disruption of IT services and prevention of theft of and damage to hardware, software, and data. Data science is a field that uses scientific and computing tools to extract information and insights from data, driven by the increasing volume and availability of data. Data mining , big data , statistics, machine learning and deep learning are all interwoven with data science. Information systems (IS)

16456-596: The rules and data formats for exchanging information in a computer network, and provide the basis for network programming . One well-known communications protocol is Ethernet , a hardware and link layer standard that is ubiquitous in local area networks . Another common protocol is the Internet Protocol Suite , which defines a set of protocols for internetworking, i.e. for data communication between multiple networks, host-to-host data transfer, and application-specific data transmission formats. Computer networking

16592-418: 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

16728-418: 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

16864-462: The second part of the algorithm is collision resolution. The two common methods for collision resolution are separate chaining and open addressing. In separate chaining, the process involves building a linked list with key–value pair for each search array index. The collided items are chained together through a single linked list, which can be traversed to access the item with a unique search key. Collision resolution through chaining with linked list

17000-454: The sharing of resources and information. When at least one process in one device is able to send or receive data to or from at least one process residing in a remote device, the two devices are said to be in a network. Networks may be classified according to a wide variety of characteristics such as the medium used to transport the data, communications protocol used, scale, topology , and organizational scope. Communications protocols define

17136-430: 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

17272-427: 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

17408-545: The standard linked list method under heavy load. Techniques such as using fusion tree for each buckets also result in constant time for all operations with high probability. The linked list of separate chaining implementation may not be cache-conscious due to spatial locality — locality of reference —when the nodes of the linked list are scattered across memory, thus the list traversal during insert and search may entail CPU cache inefficiencies. In cache-conscious variants of collision resolution through separate chaining,

17544-442: The study of these approaches. That is, the application of engineering to software. It is the act of using insights to conceive, model and scale a solution to a problem. The first reference to the term is the 1968 NATO Software Engineering Conference , and was intended to provoke thought regarding the perceived software crisis at the time. Software development , a widely used and more generic term, does not necessarily subsume

17680-446: The theory of computation and the design of computational systems. Its subfields can be divided into practical techniques for its implementation and application in computer systems , and purely theoretical areas. Some, such as computational complexity theory , which studies fundamental properties of computational problems , are highly abstract, while others, such as computer graphics , emphasize real-world applications. Others focus on

17816-419: 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

17952-452: The universe U {\displaystyle U} of keys to indices or slots within the table, that is, h ( x ) ∈ { 0 , . . . , m − 1 } {\displaystyle h(x)\in \{0,...,m-1\}} for x ∈ U {\displaystyle x\in U} . The conventional implementations of hash functions are based on

18088-418: 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

18224-460: 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

18360-482: Was proposed by A. D. Linh, building on Luhn's memorandum. Around the same time, Gene Amdahl , Elaine M. McGraw , Nathaniel Rochester , and Arthur Samuel of IBM Research implemented hashing for the IBM 701 assembler . Open addressing with linear probing is credited to Amdahl, although Andrey Ershov independently had the same idea. The term "open addressing" was coined by W. Wesley Peterson in his article which discusses

18496-488: Was the 1931 paper "The Use of Thyratrons for High Speed Automatic Counting of Physical Phenomena" by C. E. Wynn-Williams . Claude Shannon 's 1938 paper " A Symbolic Analysis of Relay and Switching Circuits " then introduced the idea of using electronics for Boolean algebraic operations. The concept of a field-effect transistor was proposed by Julius Edgar Lilienfeld in 1925. John Bardeen and Walter Brattain , while working under William Shockley at Bell Labs , built

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