Efficient Reconstruction of Sequences from Their Subsequences or Supersequences

In the paper two combinatorial problems for the set Fnq of sequences of length n over the alphabet Fq={0, 1, ?, q?1} are considered. The maximum size N?q(n, t) of the set of common subsequences of length n?t and the maximum size N+q(n, t) of the set of common supersequences of length n+t of two different sequences of Fnq are found for any nonnegative integers n and t. The number N?q(n, t)+1 (respectively, N+q(n, t)+1) is equal to the minimum number N of different subsequences of length n?t (supersequences of length n+t) of an unknown sequence X?Fnq which are sufficient for its reconstruction. Simple algorithms to recover X?Fnq from N?q(n, t)+1 of its subsequences of length n?t and from N+q(n, t)+1 of its supersequences of length n+t are given.