Exact Algorithms for Maximum Independent Set

Abstract

Given two strings S D s 1 s 2 : : : s n and R D r 1 r 2 : : : r m (wlog let n m) over an alphabet D f 1 ; ; 2 ; : : : : ` g, the standard edit distance between S and R, denoted ED(S, R) is the minimum number of single character edits, specifically insertions, deletions and replacements, to transform S into R (equivalently R into S). If the input strings S and R are permutations of the alphabet ¢ (so that jS j D jRj D j j) then an analogous permutation edit distance between S and R, denoted PED(S, R) can be defined as the minimum number of single character moves, to transform S into R (or vice versa). A generalization of the standard edit distance is edit distance with moves, which, for input strings S and R is denoted EDM(S, R), and is defined as the minimum number of character edits and substring (block) moves to transform one of the strings into the other. A move of block s[j, k] to position h transforms S D s 1 s 2 If the input strings S and R are permutations of the alphabet ¢ (so that jS j D jRj D j j) then EDM(S, R) is also called as the transposition distance and is denoted TED(S, R) [1]. Perhaps the most general form of the standard edit distance that involves edit operations on blocks/substrings is the block edit distance, denoted BED(S, R). It is defined as the minimum number of single character edits, block moves, as well as block copies and block uncopies to transform one of the strings into the other. Copying of a block s[j, k] to position h transforms S D s 1 s 2 A block uncopy is the inverse of a block copy: it deletes a block s[j, k] provided there exists sOEj 0 ; k 0  D sOEj; k which does not overlap with s[j, k] and transforms S into S 0 D s 1 : : : s j 1 s kC1 : : : s n. Throughout this discussion all edit operations have unit cost and they may overlap; i.e., a character can be edited on multiple times.