Edit Distance with Move Operations

The traditional edit-distance problem is to find the minimum number of insert-character and delete-character (and sometimes change character) operations required to transform one string into another. Here we consider the more general problem of a string represented by a singly linked list (one character per node) and being able to apply these operations to the pointer associated with a vertex as well as the character associated with the vertex. That is, in O(1) time, not only can characters be inserted or deleted, but substrings can be moved or deleted. We limit our attention to the ability to move substrings and leave substring deletions for future research. Note that O(1) time substring move operation implies O(1) substring exchange operation as well, a form of transformation that has been of interest in molecular biology. We show that this problem is NP-complete, and that a “recursive” sequence of moves can be simulated with at most a constant factor increase by a non-recursive sequence. Although a greedy algorithm is known to have poor (a polynomial factor) worst case performance, we present a polynomial time greedy algorithm for non-recursive moves which on a sub class of instances of a problem of size n achieves an approximation factor to optimal of at most O(log n). The development of this greedy algorithm shows how to reduce moves of substrings to moves of characters, and how to convert moves of characters to only inserts and deletes of characters.