Computing longest common substring/subsequence of non-linear texts

A non-linear text is a directed graph where each vertex is labeled with a string. In this paper, we introduce the longest common substring/subsequence problems on non-linear texts. Firstly, we present an algorithm to compute the longest common substring of non-linear texts G ₁ and G ₂ in O(|E ₁||E ₂|) time and O(|V ₁||V ₂|) space, when at least one of G ₁ and G ₂ is acyclic. Here, V _i and E _i are the sets of vertices and arcs of input non-linear text G _i, respectively, for 1≤i≤2. Secondly, we present algorithms to compute the longest common subsequence of G ₁ and G ₂ in O(|E ₁||E ₂|) time and O(|V ₁||V ₂|) space, when both G ₁ and G ₂ are acyclic, and in O(|E ₁||E ₂|+|V ₁||V ₂| log |Σ|) time and O(|V ₁||V ₂|) space if G ₁ and/or G ₂ are cyclic, where, Σ denotes the alphabet.