pBWT: Achieving succinct data structures for parameterized pattern matching and related problems

The fields of succinct data structures and compressed text indexing have seen quite a bit of progress over the last two decades. An important achievement, primarily using techniques based on the Burrows-Wheeler Transform (BWT), was obtaining the full functionality of the suffix tree in the optimal number of bits. A crucial property that allows the use of BWT for designing compressed indexes is order-preserving suffix links. Specifically, the relative order between two suffixes in the subtree of an internal node is same as that of the suffixes obtained by truncating the furst character of the two suffixes. Unfortunately, in many variants of the text-indexing problem, for e.g., parameterized pattern matching, 2D pattern matching, and order-isomorphic pattern matching, this property does not hold. Consequently, the compressed indexes based on BWT do not directly apply. Furthermore, a compressed index for any of these variants has been elusive throughout the advancement of the field of succinct data structures. We achieve a positive breakthrough on one such problem, namely the Parameterized Pattern Matching problem. Let T be a text that contains n characters from an alphabet , which is the union of two disjoint sets: containing static characters (s-characters) and containing parameterized characters (p-characters). A pattern P (also over ) matches an equal-length substring S of T i the s-characters match exactly, and there exists a one-to-one function that renames the p-characters in S to that in P. The task is to find the starting positions (occurrences) of all such substrings S. Previous index [Baker, STOC 1993], known as Parameterized Suffix Tree, requires (n log n) bits of space, and can find all occ occurrences in time O(jPj log +occ), where = jj. We introduce an n log +O(n)-bit index with O(jPj log +occlog n log ) query time. At the core, lies a new BWT-like transform, which we call the Parame- terized Burrows-Wheeler Transform (pBWT). The techniques are extended to obtain a succinct index for the Parameterized Dictionary Matching problem of Idury and Schaer [CPM, 1994]

Succinct Data Structures for Parameterized Pattern Matching and Related Problems

Let T be a fixed text-string of length n and P be a varying pattern-string of length |P| \u3c= n. Both T and P contain characters from a totally ordered alphabet Sigma of size sigma \u3c= n. Suffix tree is the ubiquitous data structure for answering a pattern matching query: report all the positions i in T such that T[i + k - 1] = P[k], 1 \u3c= k \u3c= |P|. Compressed data structures support pattern matching queries, using much lesser space than the suffix tree, mainly by relying on a crucial property of the leaves in the tree. Unfortunately, in many suffix tree variants (such as parameterized suffix tree, order-preserving suffix tree, and 2-dimensional suffix tree), this property does not hold. Consequently, compressed representations of these suffix tree variants have been elusive. We present the first compressed data structures for two important variants of the pattern matching problem: (1) Parameterized Matching -- report a position i in T if T[i + k - 1] = f(P[k]), 1 \u3c= k \u3c= |P|, for a one-to-one function f that renames the characters in P to the characters in T[i,i+|P|-1], and (2) Order-preserving Matching -- report a position i in T if T[i + j - 1] and T[i + k -1] have the same relative order as that of P[j] and P[k], 1 \u3c= j \u3c k \u3c= |P|. For each of these two problems, the existing suffix tree variant requires O(n*log n) bits of space and answers a query in O(|P|*log sigma + occ) time, where occ is the number of starting positions where a match exists. We present data structures that require O(n*log sigma) bits of space and answer a query in O((|P|+occ) poly(log n)) time. As a byproduct, we obtain compressed data structures for a few other variants, as well as introduce two new techniques (of independent interest) for designing compressed data structures for pattern matching

Parameterized Strings: Algorithms and Applications

The parameterized string (p-string), a generalization of the traditional string, is composed of constant and parameter symbols. A parameterized match (p-match) exists between two p-strings if the constants match exactly and there exists a bijection between the parameter symbols. Historically, p-strings have been employed in source code cloning, plagiarism detection, and structural similarity between biological sequences. By handling the intricacies of the parameterized suffix, we can efficiently address complex applications with data structures also reusable in traditional matching scenarios. In this dissertation, we extend data structures for p-strings (and variants) to address sophisticated string computations.;We introduce a taxonomy of classes for longest factor problems. Using this taxonomy, we show an interesting connection between the parameterized longest previous factor (pLPF) and familiar data structures in string theory, including the border array, prefix array, longest common prefix array, and analogous p-string data structures. Exploiting this connection, we construct a multitude of data structures using the same general pLPF framework.;Before this dissertation, the p-match was defined predominately by the matching between uncompressed p-strings. Here, we introduce the compressed parameterized pattern match to find all p-matches between a pattern and a text, using only the pattern and a compressed form of the text. We present parameterized compression (p-compression) as a new way to losslessly compress data to support p-matching. Experimentally, it is shown that p-compression is competitive with standard compression schemes. Using p-compression, we address the compressed p-match independent of the underlying compression routine.;Currently, p-string theory lacks the capability to support indeterminate symbols, a staple essential for applications involving inexact matching such as in music analysis. In this work, we propose and efficiently address two new types of p-matching with indeterminate symbols. (1) We introduce the indeterminate parameterized match (ip-match) to permit matching with indeterminate holes in a p-string. We support the ip-match by introducing data structures that extend the prefix array. (2) From a different perspective, the equivalence parameterized match (e-match) evolves the p-match to consider intra-alphabet symbol classes as equivalence classes. We propose a method to perform the e-match using the p-string suffix array framework, i.e. the parameterized suffix array (pSA) and parameterized longest common prefix array (pLCP). Historically, direct constructions of the pSA and pLCP have suffered from quadratic time bounds in the worst-case. Here, we introduce new p-string theory to efficiently construct the pSA/pLCP and break the theoretical worst-case time barrier.;Biological applications have become a classical use of p-string theory. Here, we introduce the structural border array to provide a lightweight solution to the biologically-oriented variant of the p-match, i.e. the structural match (s-match) on structural strings (s-strings). Following the s-match, we show how to use s-string suffix structures to support various pattern matching problems involving RNA secondary structures. Finally, we propose/construct the forward stem matrix (FSM), a data structure to access RNA stem structures, and we apply the FSM to the detection of hairpins and pseudoknots in an RNA sequence.;This dissertation advances the state-of-the-art in p-string theory by developing data structures for p-strings/s-strings and using p-string/s-string theory in new and old contexts to address various applications. Due to the flexibility of the p-string/s-string, the data structures and algorithms in this work are also applicable to the myriad of problems in the string community that involve traditional strings

Space-Efficient Dictionaries for Parameterized and Order-Preserving Pattern Matching

Let S and S\u27 be two strings of the same length.We consider the following two variants of string matching. * Parameterized Matching: The characters of S and S\u27 are partitioned into static characters and parameterized characters. The strings are parameterized match iff the static characters match exactly and there exists a one-to-one function which renames the parameterized characters in S to those in S\u27. * Order-Preserving Matching: The strings are order-preserving match iff for any two integers i,j in [1,|S|], S[i] <= S[j] iff S\u27[i] <= S\u27[j]. Let P be a collection of d patterns {P_1, P_2, ..., P_d} of total length n characters, which are chosen from an alphabet Sigma. Given a text T, also over Sigma, we consider the dictionary indexing problem under the above definitions of string matching. Specifically, the task is to index P, such that we can report all positions j where at least one of the patterns P_i in P is a parameterized-match (resp. order-preserving match) with the same-length substring of $T$ starting at j. Previous best-known indexes occupy O(n * log(n)) bits and can report all occ positions in O(|T| * log(|Sigma|) + occ) time. We present space-efficient indexes that occupy O(n * log(|Sigma|+d) * log(n)) bits and reports all occ positions in O(|T| * (log(|Sigma|) + log_{|Sigma|}(n)) + occ) time for parameterized matching and in O(|T| * log(n) + occ) time for order-preserving matching

Answering Regular Path Queries on Workflow Provenance

This paper proposes a novel approach for efficiently evaluating regular path queries over provenance graphs of workflows that may include recursion. The approach assumes that an execution g of a workflow G is labeled with query-agnostic reachability labels using an existing technique. At query time, given g, G and a regular path query R, the approach decomposes R into a set of subqueries R1, ..., Rk that are safe for G. For each safe subquery Ri, G is rewritten so that, using the reachability labels of nodes in g, whether or not there is a path which matches Ri between two nodes can be decided in constant time. The results of each safe subquery are then composed, possibly with some small unsafe remainder, to produce an answer to R. The approach results in an algorithm that significantly reduces the number of subqueries k over existing techniques by increasing their size and complexity, and that evaluates each subquery in time bounded by its input and output size. Experimental results demonstrate the benefit of this approach

New Algorithms for Position Heaps

We present several results about position heaps, a relatively new alternative to suffix trees and suffix arrays. First, we show that, if we limit the maximum length of patterns to be sought, then we can also limit the height of the heap and reduce the worst-case cost of insertions and deletions. Second, we show how to build a position heap in linear time independent of the size of the alphabet. Third, we show how to augment a position heap such that it supports access to the corresponding suffix array, and vice versa. Fourth, we introduce a variant of a position heap that can be simulated efficiently by a compressed suffix array with a linear number of extra bits

Super-resolution Line Spectrum Estimation with Block Priors

We address the problem of super-resolution line spectrum estimation of an undersampled signal with block prior information. The component frequencies of the signal are assumed to take arbitrary continuous values in known frequency blocks. We formulate a general semidefinite program to recover these continuous-valued frequencies using theories of positive trigonometric polynomials. The proposed semidefinite program achieves super-resolution frequency recovery by taking advantage of known structures of frequency blocks. Numerical experiments show great performance enhancements using our method.Comment: 7 pages, double colum