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# ps3sol - Introduction to Algorithms Massachusetts Institute...

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Introduction to Algorithms October 10, 2005 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik D. Demaine and Charles E. Leiserson Handout 13 Problem Set 3 Solutions Problem 3-1. Pattern Matching Principal Skinner has a problem: he is absolutely sure that Bart Simpson has plagiarized some text on a recent book report. One of Bart’s sentences sounds oddly familiar, but Skinner can’t quite figure out where it came from. Skinner decides to see if some smart-alec MIT student can help him out. Skinner gives you a DVD containing the full text of the Springfield public library. The data is stored in a binary string T [1] , T [2] , . . . , T [ n ] , which we view as an array T [1 . . n ] , where each T [ i ] is either 0 or 1 . Skinner also gives you the quote from Bart Simpson’s book report, a shorter binary string P [1 . . m ] , again where each P [ i ] is either 0 or 1 , and where m < n . For a binary string A [1 . . k ] and for integers i, j with 1 i j k , we use the notation A [ i . . j ] to refer to the binary string A [ i ] , A [ i + 1] , . . . , A [ j ] , called a substring of A . The goal of this problem is to determine whether P is a substring of T , i.e., whether P = A [ i . . j ] for some i, j with . 1 i j n For the purpose of this problem, assume that you can manipulate O (log n ) -bit integers in constant time. For example, if x n 7 and y n 5 , then you can calculate x + y in constant time. On the other hand, you may not assume that m -bit integers can be manipulated in constant time, because m may be too large. For example, if m = Θ(log 2 n ) and x and y are each m -bit integers, you cannot calculate x + y in constant time. (In general, it is reasonable to assume that you can manipulate integers of length logarithmic in the input size in constant time, but larger integers require proportionally more time.) (a) Assume that you have a hash function h ( x ) that computes a hash value of the m - bit binary string x = A [ i . . ( i + m 1)] , for some binary string A [1 . . k ] and some 1 i k m + 1 . Moreover, assume that the hash function is perfect: if x = y , then h ( x ) = h ( y ) . Assume that you can calculate the hash function in O ( m ) time. Show how to determine whether P is a substring of T in O ( mn ) time. Solution: We compute the hash of the pattern string, and compare it to the hash of all possible length- m substrings of A , i.e., compare h ( P ) to h ( A [ i . . ( i + m 1)]) , for 1 i < n m + 1 . Since the hash function is perfect, h ( P ) = h ( A [ i . . ( i + m 1)]) if and only if P = A [ i, . . ( i + m 1)] . There are O ( n ) hash functions to compute, O ( n ) comparisons of hash values, and each computation and comparison requires O ( m ) time, for a total running time of O ( mn ) . Note that because calculation of the hash function takes O ( m ) time, this algorithm is not asymptotically any better than simply comparing the substrings directly. This part is designed as motivation for the rest of the problem.

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2 Handout 13: Problem Set 3 Solutions (b) Consider the following family of hash functions h p , parameterized by a prime num- 4 ber p in the range [2 , cn ] for some constant c > 0 : h p ( x ) = x (mod p ) .
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