final-s00-prac - CS 70 Spring 2000 Discrete Mathematics for...

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CS 70 Discrete Mathematics for CS Spring 2000 Wagner Sample Final You will have three hours for the final itself. This sample final is similar in style and scope to the actual final; however, it is rather longer than the actual final. The exam is open-book, open-notes. We have not assigned points to the questions (but we will do this for the final itself). 1. (?? pts.) The longest common subsequence problem In this question, we will need an induction principle for pairs of strings over an alphabet Σ . We say that a pair of strings ( a 0 , b 0 ) is “less than” another pair ( a , b ) , written ( a 0 , b 0 ) ( a , b ) , if | a 0 | + | b 0 | < | a | + | b | , where | a | denotes the length of string a Σ * . Our induction principle is the following, where P ( · , · ) is a proposition that applies to pairs of strings: • if 2 a , b Σ * [ 2 a 0 , b 0 Σ * , ( a 0 , b 0 ) ( a , b ) P ( a 0 , b 0 )] P ( a , b ) then 2 a , b Σ * P ( a , b ) . Now consider the following problem. A subsequence of a string a is any sequence that can be obtained by deleting one or more (not necessarily consecutive) letters from a , leaving the order of the remaining letters unchanged. A longest common subsequence (written lcs ) of two strings a and b is a sequence of maximum length that is a subsequence of both a and b . Thus for example a lcs of a = GACTTACCCAGT and b = GTTATGTACA is GATTACA ; a lcs of a = ATT and b = GCG is the empty string λ . Suppose we want to compute a lcs of two given strings, a and b . [This problem arises, for example, in DNA analysis and in file comparison utilities.] To make the notation simpler, we’ll actually just compute the length of a lcs (which is essentially the same problem). Here is a suggested algorithm: algorithm lcs length(a,b) if a = or b = then return(0) else if head ( a ) = head ( b ) then return( max { 1 + lcs length ( tail ( a ) , tail ( b )) , lcs length
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This note was uploaded on 09/03/2011 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at University of California, Berkeley.

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final-s00-prac - CS 70 Spring 2000 Discrete Mathematics for...

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