Large Edit Distance with Multiple Block Operations
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We consider the addition of some or all of the operations block move, block delete, block copy, block reversals, and block copy reversals, to the traditional edit distance problem (finding the minimum number of insert-character and delete-character operations to convert one string to another). When all of the above operations are allowed, the problem, called the nearest neighbors problem, is NP hard, and the best known approximation is O(logn log* n), which was achieved by Muthukrishnan and Sahinalp [2000,2002a]. In this paper we show that this problem can be approximated by a constant factor of 3.5 using a simple sliding window method. When eliminating reversals, the same method reduces the best known approximation of 12, achieved by Ergun, Muthukrishnan and Sahinalp , down to a factor of 4. Both constant factors are proved to be tight. Allowing only subsets of these operations does not necessarily make the problem easier. Shapira and Storer  present a logn factor approximation algorithm for edit distance with block moves (which is also an NP-complete problem). Here, we show that edit distance with block deletions can be solved optimally, but edit distance with block moves and block deletions remains NP-complete and can be reduced to the problem of block moves only, keeping the same logn factor approximation.
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