Window-accumulated subsequence matching problem is linear

Tree inclusion problems

“…These problem generalize in a natural way the subsequence problems for words: we proved in [BCGM01], that the problem of counting the number of w-windows of a text t containing a pattern p as a subsequence (i.e. letters of p appear in the window in the same order as in p but are not necessarily consecutive and may be interleaved with other letters) can be solved in time O(n) where n is the size of t. The generalization to trees can be stated as follows: P is an embedded subtree of T if P can be obtained by deleting some nodes from T (if a node v is deleted, the ingoing edge to v (if it exists) is also deleted, and outgoing edges are replaced by edges going from the parent of v (if it exists) to the children of v).…”

“…It is easy to see [BCGM01,K92] that a window of height w of T contains P as an embedded subtree iff it contains a minimal subtree of T containing P ; therefore, it is enough to count the number of w-windows of T containing a minimal subtree containing P .…”

Tree inclusions in windows and slices

“…Idea of the algorithm It is easy to see [BCGM01,K92] that a window of height w of T contains P as an embedded subtree iff it contains a minimal subtree of T containing P ; therefore, it is enough to count the number of w-windows of T containing a minimal subtree containing P .…”

“…Given two trees (a target T and a pattern P ) and a natural number w, we address two problems: 1. counting the number of w-windows of T which contain pattern P as an embedded subtree, and 2. counting the number of w-slices of T which contain pattern P as an embedded subtree. These problems generalize in a natural way the subsequence problems for words: we proved in [BCGM01], that the problem of counting the number of w-windows of a text t containing a pattern p as a subsequence (i.e. letters of p appear in the window in the same order as in p but are not necessarily consecutive and may be interleaved with other letters) can be solved in time O(n) where n is the size of t.…”

A Conditional Lower Bound for Episode Matching

“…Das et al [12] gave an upper bound of O(nm/ log n), where n is the length of S and m is the length of P . Even though the problem and its variations have been thoroughly studied [4,8,10,11,23,24] and have numerous applications in data mining [4,5,17,21,25], the original O(nm/ log n) upper bound has never been improved. In this paper we give an argument for why this is the case, by proving a lower bound conditioned on the Strong Exponential Time Hypothesis (SETH) (see Conjecture 1).…”