algorithm.design.kleinberg.tardos.solutions.ch8 (23)

Algorithm Design

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Suppose m n ,andlet L denote the maximum length of any string in A B . Suppose there is a string that is a concatenation over both A and B ,andlet u be one of minimum length. We claim that the length of u is at most n 2 L 2 . For suppose not. First, we say that position p in u is of type ( a i ,k ) if in the concatenation over A , it is represented by position k of string a i . We de±ne type ( b i ,k ) analogously. Now, if the length of u is greater than n 2 L 2 , then by the pigeonhole principle, there exist positions
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Unformatted text preview: p and p in u , p < p , so that both are of type ( a i , k ) and ( b j , k ) for some indices i, j, k . But in this case, the string u obtained by deleting positions p, p + 1 , . . ., p − 1 would also be a concatenation over both A and B . As u is shorter than u , this is a contradiction. 1 ex690.144.299 1 Simpo PDF Password Remover Unregistered Version - http://www.simpopdf.com...
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This document was uploaded on 02/12/2012.

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