Algorithms for the Longest Common Subsequence Problem

TLDR: A lgor i thm is appl icable in the genera l case and requi res O ( p n + n log n) t ime for any input strings o f lengths m and n even though the lower bound on T ime of O ( m n ) need not apply to all inputs.

Abstract: We start by def ining conven t ions and t e rmino logy that will be used th roughou t this paper . String C = clc~ ... cp is a subsequence of string A = aja2 "'" am if there is a mapp ing F : {1, 2 . . . . , p} ~ {1, 2, ... , m} such that F(i) = k only if c~ = ak and F is a m o n o t o n e strictly increasing funct ion (i .e. F(i) = u, F(]) = v, and i < j imply that u < v). C can be fo rmed by delet ing m p (not necessari ly ad jacen t ) symbols f rom A . F o r example , " c o u r s e " is a subsequence of " c o m p u t e r sc ience . " Str ing C is a c o m m o n s ubs equence of strings A and B if C is a s u b s e q u e n c e of A and also a subsequence of B. String C is a longest c o m m o n subsequence (abbrev ia ted LCS) of string A and B if C is a c o m m o n subsequence of A and B of maximal length , i .e. there is no c o m m o n subsequence of A and B that has grea te r length. Th roughou t this paper , we assume that A and B are strings of lengths m and n , m _< n , that have an LCS C of (unknown) length p . We assume that the symbols that may appea r in these strings c o m e f rom some a lphabet of size t . A symbol can be s tored in m e m o r y by using log t bits, which we assume will fit in one word of memory . Symbols can be c o m p a r e d (a -< b?) in one t ime unit . The n u m b e r of di f ferent symbols that actual ly appear in string B is def ined to be s (which must be less than n and t). The longest c o m m o n s u b s e q u e n c e prob lem has been solved by using a recurs ion re la t ionship on the length of the solut ion [7, 12, 16, 21]. These are general ly appl icable a lgor i thms that take O ( m n ) t ime for any input strings o f lengths m and n even though the lower bound on t ime of O ( m n ) need not apply to all inputs [2]. We present a lgor i thms that , depend ing on the na ture of the Input, may not requ i re quadra t ic t ime to r ecove r an LCS. The first a lgor i thm is appl icable in the genera l case and requi res O ( p n + n log n) t ime. T h e second a lgor i thm requi res t ime b o u n d e d by O((m + 1 p )p log n). In the c o m m o n special case where p is close to m , this a lgor i thm takes t ime