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45-533: One of the key aspects of the P910 is its ability to input text via several methods: multi-tap and T9 text input using the numerical keypad, hand-writing recognition with the pre-installed Jot-Pro software and touchscreen, virtual keyboard on screen and the new QWERTY keyboard on the inside of the flip. Other enhancements (compared to the P900) include support for HTML browsing, a new numerical keypad with larger keys and

90-502: A 3×4 numeric keypad ), originally developed by Tegic Communications , now part of Nuance Communications . T9 stands for Text on 9 keys. T9 was used on phones from Verizon , NEC , Nokia , Samsung Electronics , Siemens , Sony Mobile , Sanyo , SAGEM and others, as well as PDAs such as Avigo during the late 1990s. The main competing technologies include iTap created by Motorola , SureType created by RIM , Eatoni's LetterWise and WordWise , and Intelab's Tauto . It still

135-595: A binary search algorithm (with cost ⁠ O ( log ⁡ n ) {\displaystyle O(\log n)} ⁠ ) outperforms a sequential search (cost ⁠ O ( n ) {\displaystyle O(n)} ⁠ ) when used for table lookups on sorted lists or arrays. The analysis, and study of algorithms is a discipline of computer science . Algorithms are often studied abstractly, without referencing any specific programming language or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on

180-468: A flowchart offers a way to describe and document an algorithm (and a computer program corresponding to it). It has four primary symbols: arrows showing program flow, rectangles (SEQUENCE, GOTO), diamonds (IF-THEN-ELSE), and dots (OR-tie). Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. It is often important to know how much time, storage, or other cost an algorithm may require. Methods have been developed for

225-745: A function . Starting from an initial state and initial input (perhaps empty ), the instructions describe a computation that, when executed , proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic ; some algorithms, known as randomized algorithms , incorporate random input. Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic"). In

270-435: A heuristic is an approach to solving problems that do not have well-defined correct or optimal results. For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an effective method , an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating

315-680: A computer-executable form, but are also used to define or document algorithms. There are many possible representations and Turing machine programs can be expressed as a sequence of machine tables (see finite-state machine , state-transition table , and control table for more), as flowcharts and drakon-charts (see state diagram for more), as a form of rudimentary machine code or assembly code called "sets of quadruples", and more. Algorithm representations can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description. A high-level description describes qualities of

360-719: A computing machine or a human who could only carry out specific elementary operations on symbols . Most algorithms are intended to be implemented as computer programs . However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain performing arithmetic or an insect looking for food), in an electrical circuit , or a mechanical device. Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in Babylonian mathematics (around 2500 BC), Egyptian mathematics (around 1550 BC), Indian mathematics (around 800 BC and later),

405-477: A numeric-style 4×3 keyboard. Such T9 formats for text entry therefore remain available in all latest [as of August 2020] iterations of LG keyboards, certain Samsung keyboards, and third party T9 keyboards such as Go keyboard for Androids and Type Nine for iPhones, as shown on this LG V60. Algorithm In mathematics and computer science , an algorithm ( / ˈ æ l ɡ ə r ɪ ð əm / )

450-477: A primary and secondary language and access matches from both languages. This enables users to write messages in their native language, as well as a foreign one. Some implementations learn commonly used word pairs and provide word prediction (e.g. if one often writes "eat food", after entering "eat" the phone will suggest "food", which can be confirmed by pressing Next). T9 can automatically recognize and correct typing/texting errors, by looking at neighboring keys on

495-525: A programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures : SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction . By themselves, algorithms are not usually patentable. In

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540-477: A sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and any prescribed bureaucratic procedure or cook-book recipe . In general, a program is an algorithm only if it stops eventually —even though infinite loops may sometimes prove desirable. Boolos, Jeffrey & 1974, 1999 define an algorithm to be an explicit set of instructions for determining an output, that can be followed by

585-522: A slightly changed outer casing. Its closest competitors are the palmOne Treo 650 , and the Nokia 9500 Communicator . Other competitors include several PDA-phones powered by Windows and manufactured by Taiwan-based HTC. Sony Ericsson released the successor to the P910, the P990 , in 2006. T9 (predictive text) T9 is a predictive text technology for mobile phones (specifically those that contain

630-472: Is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation . Algorithms are used as specifications for performing calculations and data processing . More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making ) and deduce valid inferences (referred to as automated reasoning ). In contrast,

675-416: Is a method or mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as divide-and-conquer or dynamic programming within operation research . Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern. One of

720-581: Is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: One of the simplest algorithms finds the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be described in plain English as: High-level description: (Quasi-)formal description: Written in prose but much closer to

765-424: Is a side effect of the requirements for small database sizes on the lower end embedded devices. On a phone with a numeric keypad, each time a key (1-9) is pressed (when in a text field), the algorithm returns a guess for what letters are most likely for the keys pressed to that point. For example, to enter the word 'the', the user would press 8 then 4 then 3, and the display would display 't' then 'th' then 'the'. If

810-453: Is an example of a letter changing while entering words. Many smart keyboards now exist, such as Swype or Swiftkey , that have taken the idea of T9 and combined it with the advanced touchscreen technology found in Android phones and iPhones . These advances have made T9 obsolete in newer cellphones for many users, since it is predicated on the use of a keypad with nothing besides numbers,

855-431: Is used on niche products as Punkt mp-02. T9 is available on certain phones without a touchscreen, and is available on Apple iPhone (as of iOS 18 ) and Android phones as a custom keyboard. T9's objective is to make it easier to enter text messages . It allows words to be formed by a single keypress for each letter, which is an improvement over the multi-tap approach used in conventional mobile phone text entry at

900-453: Is useful for uncovering unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are non-trivial to perform fairly. To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in

945-1107: The Entscheidungsproblem (decision problem) posed by David Hilbert . Later formalizations were framed as attempts to define " effective calculability " or "effective method". Those formalizations included the Gödel – Herbrand – Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church 's lambda calculus of 1936, Emil Post 's Formulation 1 of 1936, and Alan Turing 's Turing machines of 1936–37 and 1939. Algorithms can be expressed in many kinds of notation, including natural languages , pseudocode , flowcharts , drakon-charts , programming languages or control tables (processed by interpreters ). Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts, and control tables are structured expressions of algorithms that avoid common ambiguities of natural language. Programming languages are primarily for expressing algorithms in

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990-629: The Jacquard loom , a precursor to Hollerith cards (punch cards), and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the telegraph , the precursor of the telephone, was in use throughout the world. By the late 19th century, the ticker tape ( c.  1870s ) was in use, as were Hollerith cards (c. 1890). Then came the teleprinter ( c.  1910 ) with its punched-paper use of Baudot code on tape. Telephone-switching networks of electromechanical relays were invented in 1835. These led to

1035-792: The Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC), and Arabic mathematics (around 800 AD). The earliest evidence of algorithms is found in ancient Mesopotamian mathematics. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to c.  2500 BC describes the earliest division algorithm . During the Hammurabi dynasty c.  1800  – c.  1600 BC , Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy . Babylonian clay tablets describe and employ algorithmic procedures to compute

1080-596: The United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), so algorithms are not patentable (as in Gottschalk v. Benson ). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr , the application of a simple feedback algorithm to aid in the curing of synthetic rubber

1125-454: The algorithm itself, ignoring how it is implemented on the Turing machine. An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states. In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine. The graphical aid called

1170-588: The algorithm's properties, not implementation. Pseudocode is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their algorithmic efficiency is tested using real code. The efficiency of a particular algorithm may be insignificant for many "one-off" problems but it may be critical for algorithms designed for fast interactive, commercial or long life scientific usage. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing

1215-403: The analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the elements of a list of n numbers would have a time requirement of ⁠ O ( n ) {\displaystyle O(n)} ⁠ , using big O notation . The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in

1260-518: The asterisk and the pound key (known as the hash key in Commonwealth countries). Many features, such as predictive text, have been adopted by and improved by future generations of keyboard software. However, T9 remains viable. For example, those with larger fingertips still use the T9 based keyboard on smartphones for text entry, because key press accuracy increases with the larger screen area per key on

1305-521: The earliest codebreaking algorithm. Bolter credits the invention of the weight-driven clock as "the key invention [of Europe in the Middle Ages ]," specifically the verge escapement mechanism producing the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata " in the 13th century and "computational machines"—the difference and analytical engines of Charles Babbage and Ada Lovelace in

1350-523: The early 12th century, Latin translations of said al-Khwarizmi texts involving the Hindu–Arabic numeral system and arithmetic appeared, for example Liber Alghoarismi de practica arismetrice , attributed to John of Seville , and Liber Algorismi de numero Indorum , attributed to Adelard of Bath . Hereby, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with

1395-427: The field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power. Algorithm design

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1440-450: The input list. If the space required to store the input numbers is not counted, it has a space requirement of ⁠ O ( 1 ) {\displaystyle O(1)} ⁠ , otherwise ⁠ O ( n ) {\displaystyle O(n)} ⁠ is required. Different algorithms may complete the same task with a different set of instructions in less or more time, space, or ' effort ' than others. For example,

1485-490: The invention of the digital adding device by George Stibitz in 1937. While working in Bell Laboratories, he observed the "burdensome" use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device". In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve

1530-604: The keypad to determine an incorrect keypress. For example, the word "testing" is entered with the key combination "8378464". Entering the same number but with two incorrect keypresses of neighboring keys, e.g. "8278494", results in T9 suggesting the words "tasting" (8278464), "testing" (8378464), and "tapping" (8277464). In order to achieve compression ratios of close to 1 byte per word, T9 uses an optimized algorithm that maintains word order and partial words (also known as stems); however, because of this compression, it over-generates words that are sometimes visible as "junk words". This

1575-450: The less-common word 'fore' is intended (3673) the predictive algorithm may select 'Ford'. Pressing the 'next' key (typically the '*' key) might bring up 'dose', and finally 'fore'. If 'fore' is selected, then the next time the user presses the sequence 3673, fore will be more likely to be the first word displayed. If the word "Felix" is intended, however, when entering 33549, the display shows ' E ', then 'De', 'Del', 'Deli', and ' F elix.' This

1620-429: The mid-19th century. Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real Turing-complete computer instead of just a calculator . Although a full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer". Bell and Newell (1971) write that

1665-627: The most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g., an algorithm's run-time growth as the size of its input increases. Per the Church–Turing thesis , any algorithm can be computed by any Turing complete model. Turing completeness only requires four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in " spaghetti code ",

1710-564: The phrase Dixit Algorismi , or "Thus spoke Al-Khwarizmi". Around 1230, the English word algorism is attested and then by Chaucer in 1391, English adopted the French term. In the 15th century, under the influence of the Greek word ἀριθμός ( arithmos , "number"; cf. "arithmetic"), the Latin word was altered to algorithmus . One informal definition is "a set of rules that precisely defines

1755-400: The possessive 's ). Depending on the language, T9 also supports word breaking after punctuation to support clitics such as l' and n' in French and 's in English. For words entered by the user, word completion can be enabled. When the user enters matching keypresses, in addition to words and stems, the system also provides completions. In later versions of T9, the user can select

1800-675: The time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics , dating back to the Rhind Mathematical Papyrus c.  1550 BC . Algorithms were later used in ancient Hellenistic mathematics . Two examples are the Sieve of Eratosthenes , which was described in the Introduction to Arithmetic by Nicomachus , and the Euclidean algorithm , which

1845-403: The time, in which several letters are associated with each key, and selecting one letter often requires multiple keypresses. T9 combines the groups of letters on each phone key with a fast-access dictionary of words. It will then look up in the dictionary all words corresponding to the sequence of keypresses and order them by frequency of use. As T9 "gains familiarity" with the words and phrases

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1890-430: The user commonly uses, it speeds up the process by offering the most frequently used words first and then letting the user access other choices with one or more presses of a predefined "Next" key. The dictionary is expandable. After introducing a new word, the next time the user tries to produce that word, T9 adds it to the predictive dictionary. The user database (UDB) can be expanded via multi-tap . The implementation of

1935-554: The user database is dependent on the version of T9 and how T9 is actually integrated on the device. Some phone manufacturers supply a permanent user database, while others do so for the duration of the session. Some T9 implementations feature smart punctuation . This feature allows the user to insert sentence and word punctuation using the '1'-key. Depending on the context, smart punctuation inserts sentence punctuation (period or 'full stop') or embedded punctuation (period or hyphen) or word punctuation (apostrophe in can't, won't, isn't, and

1980-449: Was deemed patentable. The patenting of software is controversial, and there are criticized patents involving algorithms, especially data compression algorithms, such as Unisys 's LZW patent . Additionally, some cryptographic algorithms have export restrictions (see export of cryptography ). Another way of classifying algorithms is by their design methodology or paradigm . Some common paradigms are: For optimization problems there

2025-692: Was first described in Euclid's Elements ( c.  300 BC ). Examples of ancient Indian mathematics included the Shulba Sutras , the Kerala School , and the Brāhmasphuṭasiddhānta . The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi , a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages . He gave the first description of cryptanalysis by frequency analysis ,

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