assign4 - CS648 Randomized Algorithms Semester II, 2007-08....

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Unformatted text preview: CS648 Randomized Algorithms Semester II, 2007-08. Assignment 4 Due on : 1st April Note : Give complete details of the analysis of your solution. Be very rigorous in providing any mathematical detail in support of your arguments. Also mention the Lemma/Theorem you use. 1. (15) (a) In this problem we will use a different finger printing technique to solve the pattern matching problem. The idea is to map any bit string s into a 2 2 matrix M ( s ) , as follows. For the empty string , M ( ) = bracketleftBigg 1 1 bracketrightBigg . M (0) = bracketleftBigg 1 1 1 bracketrightBigg . M [1] = bracketleftBigg 1 1 1 bracketrightBigg . For non-empty strings x and y , M ( xy ) = M ( x ) M ( y ) . Work out the proof of the following assertions only to convince yourself (no need to submit the proofs). i. M ( x ) is well-defined for all x { , 1 } * . ii. M ( x ) = M ( y ) x = y . iii. For x { , 1 } n , the entries in M ( x ) are bounded by Fibonacci number F n . Can you con- struct such a sequence of n entries where entries are of the order of n th Fibonacci number ? By considering the matrices M ( x ) modulo a suitable prime p , show how you would perform efficient randomized pattern matching. (b) Consider two dimensional version of the pattern matching problem. The text is an n n matrix X , and the pattern is an m m matrix Y . A pattern matches occurs if Y appears as a contiguous sub-matrix of X . To apply the randomized algorithm described above, we convert the matrix....
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assign4 - CS648 Randomized Algorithms Semester II, 2007-08....

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