Input-driven pushdown automata for edit distance neighborhood

Abstract

Edit distance ℓ-neighborhood of a language L is the set of all strings that can be obtained by at most ℓ elementary edit operations—deleting or inserting one symbol in the string—from some string in L. We shall show that if L is recognized by a nondeterministic input-driven pushdown automaton (pda) with |Γ| pushdown symbols and |Q| states, then its edit distance ℓ-neighborhood can be recognized by a nondeterministic input-driven pda with 2⋅|Γ|+1 pushdown symbols and O(|Q|⋅|Γ|ℓ) states. We shall also prove the corresponding lower bound: if |Q|>|Γ|≥2, there exists a language L that is recognized by a deterministic input-driven pda using |Γ| pushdown symbols and |Q| states but any nondeterministic input-driven pda for its edit distance ℓ-neighborhood must use at least (|Q|−1)⋅|Γ|ℓ states. This improves the known upper bound from literature and gives the bound that is asymptotically the best possible. If, beside deleting or inserting one symbol in the string, the measure of edit distance includes the operation of rewriting one symbol with another, the edit distance ℓ-neighborhood can be recognized with 2⋅|Γ|+1 pushdown symbols and O(|Q|⋅|Γ|2⋅ℓ) states.