Let T be a text of length n, and let P be a pattern of length m. Describe an O(n+m)time method…

Let T be a text of length n, and let P be a pattern of length m. Describe an O(n+m)time method for finding the longest prefix of P that is a substring ofT.

Say that a pattern P of length m is a circular substring of a text T of length n if there is an index 0 ≤ i

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The KMP pattern matching algorithm can be modified to run faster on binary strings by redefining the failure function as

, where denotes the complement of the kth bit of P. Describe how to modify the KMP algorithm to be able to take advantage of this new failure function and also give a method for computing this failure function. Show that this method makes at most n comparisons between the text and the pattern (as opposed to the 2n comparisons needed by the standard KMP algorithm given in Section 12.2.3).

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