An Encoding for Order-Preserving Matching

Encoding data structures store enough information to answer the queries they are meant to support but not enough to recover their underlying datasets. In this paper we give the first encoding data structure for the challenging problem of order-preserving pattern matching. This problem was introduced only a few years ago but has already attracted significant attention because of its applications in data analysis. Two strings are said to be an order-preserving match if the relative order of their characters is the same: e.g., (4, 1, 3, 2) and (10, 3, 7, 5) are an order-preserving match. We show how, given a string S[1..n] over an arbitrary alphabet of size sigma and a constant c >=1, we can build an O(n log log n)-bit encoding such that later, given a pattern P[1..m] with m >= log^c n, we can return the number of order-preserving occurrences of P in S in O(m) time. Within the same time bound we can also return the starting position of some order-preserving match for P in S (if such a match exists). We prove that our space bound is within a constant factor of optimal if log(sigma) = Omega(log log n); our query time is optimal if log(sigma) = Omega(log n). Our space bound contrasts with the Omega(n log n) bits needed in the worst case to store S itself, an index for order-preserving pattern matching with no restrictions on the pattern length, or an index for standard pattern matching even with restrictions on the pattern length. Moreover, we can build our encoding knowing only how each character compares to O(log^c n) neighbouring characters

A Compact Index for Order-Preserving Pattern Matching

Order-preserving pattern matching was introduced recently but it has already attracted much attention. Given a reference sequence and a pattern, we want to locate all substrings of the reference sequence whose elements have the same relative order as the pattern elements. For this problem we consider the offline version in which we build an index for the reference sequence so that subsequent searches can be completed very efficiently. We propose a space-efficient index that works well in practice despite its lack of good worst-case time bounds. Our solution is based on the new approach of decomposing the indexed sequence into an order component, containing ordering information, and a delta component, containing information on the absolute values. Experiments show that this approach is viable, faster than the available alternatives, and it is the first one offering simultaneously small space usage and fast retrieval.Comment: 16 pages. A preliminary version appeared in the Proc. IEEE Data Compression Conference, DCC 2017, Snowbird, UT, USA, 201

A compact index for order-preserving pattern matching

Order-preserving pattern matching has been introduced recently, but it has already attracted much attention. Given a reference sequence and a pattern, we want to locate all substrings of the reference sequence whose elements have the same relative order as the pattern elements. For this problem, we consider the offline version in which we build an index for the reference sequence so that subsequent searches can be completed very efficiently. We propose a space-efficient index that works well in practice despite its lack of good worst-case time bounds. Our solution is based on the new approach of decomposing the indexed sequence into an order component, containing ordering information, and a \u3b4 component, containing information on the absolute values. Experiments show that this approach is viable, is faster than the available alternatives, and is the first one offering simultaneously small space usage and fast retrieval

Wheeler graphs: A framework for BWT-based data structures

The famous Burrows\u2013Wheeler Transform (BWT) was originally defined for a single string but variations have been developed for sets of strings, labeled trees, de Bruijn graphs, etc. In this paper we propose a framework that includes many of these variations and that we hope will simplify the search for more. We first define Wheeler graphs and show they have a property we call path coherence. We show that if the state diagram of a finite-state automaton is a Wheeler graph then, by its path coherence, we can order the nodes such that, for any string, the nodes reachable from the initial state or states by processing that string are consecutive. This means that even if the automaton is non-deterministic, we can still store it compactly and process strings with it quickly. We then rederive several variations of the BWT by designing straightforward finite-state automata for the relevant problems and showing that their state diagrams are Wheeler graphs