VLSI algorithms for solving recurrence equations and applications

Optimal linear-time algorithms for solving recurrence equations on simple systolic arrays are presented. The systolic arrays use only one-way communication between processors and communicate with the external environment through only one I/O port. Because of their architectural simplicity, the arrays are well suited for direct VLSI implementation. Applications to some pattern recognition and sequence comparison problems are given. For example, it is shown that the set of (k + 2)-tuples of strings (x
1
, . . . , x
k+1
, Y) such that y is a shuffle of x
1
,. . . , x
k+1
can be recognized by a one-way k-dimensional systolic array in (k + 1)n - k time. The longest common subsequence (LCS) problem and the string-to-string correction problem are also considered: the length of an LCS of k + 1 sequences can be computed by a one-way k-dimensional systolic array in (k + 1) n - k time; the edit distance between two strings can be computed by a one-way dimensional systolic array in 2n - 1 time. Applications to other related problems, e.g., dynamic time warping and optimum generalized alignment, as well as optimal-time simulations of multihead acceptors and multitape transducers are also given.