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Unformatted text preview: Algorithms Lecture 0: Introduction We should explain, before proceeding, that it is not our object to consider this program with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be perfomed during its complete solution. Ada Augusta Byron King, Countess of Lovelace, translators notes for Luigi F. Menabrea, Sketch of the Analytical Engine invented by Charles Babbage, Esq. (1843) You are right to demand that an artist engage his work consciously, but you confuse two different things: solving the problem and correctly posing the question. Anton Chekhov, in a letter to A. S. Suvorin (October 27, 1888) The more we reduce ourselves to machines in the lower things, the more force we shall set free to use in the higher. Anna C. Brackett, The Technique of Rest (1892) The moment a man begins to talk about technique thats proof that he is fresh out of ideas. Raymond Chandler 0 Introduction 0.1 What is an algorithm? An algorithm is an explicit, precise, unambiguous, mechanically-executable sequence of elementary instructions. For example, here is an algorithm for singing that annoying song 99 Bottles of Beer on the Wall, for arbitrary values of 99: B OTTLES O F B EER ( n ) : For i n down to 1 Sing i bottles of beer on the wall, i bottles of beer, Sing Take one down, pass it around, i- 1 bottles of beer on the wall. Sing No bottles of beer on the wall, no bottles of beer, Sing Go to the store, buy some more, n bottles of beer on the wall. The word algorithm does not derive, as algorithmophobic classicists might guess, from the Greek root algos ( o ), meaning pain. Rather, it is a corruption of the name of the 9th century Persian mathematician Abu Abd Allah Muh . ammad ibn Musa al-Khwarizm . 1 Al-Khwarizm is perhaps best known as the writer of the treatise Al-Kitab al-mukhtas . ar f h sab al-gabr wal-muqabala 2 , from which the modern word algebra derives. The word algorithm is a corruption of the older word algorism (by false connection to the Greek arithmos ( o ), meaning number, and perhaps the aforementioned o ), used to describe the modern decimal system for writing and manipulating numbersin particular, the use of a small circle or sifr to represent a missing quantitywhich al-Khwarizm brought into Persia from India. Thanks to the efforts of the medieval Italian mathematician Leonardo of Pisa, better known as Fibonacci, algorism began to replace the abacus as the preferred system of commercial calculation 3 in Europe in the late 12th century, although it was several more centuries before cyphers became truly ubiquitous. (Counting boards were used by the English and Scottish royal exchequers well into the 1600s.) Thus, until very recently, the word algorithm referred exclusively to pencil-and-paper...
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- Spring '09