Parameterized Strings: Algorithms and Data Structures

A parameterized string (p-string) T = T[1] T[2]...T[n] is a sophisticated string of length n composed of symbols from a constant alphabet Sigma and a parameter alphabet pi. Given a pair of p-strings S and T, the parameterized pattern matching (p-match) problem is to verify whether the individual constant symbols match and whether there exists a bijection between the parameter symbols of S and T. If the two conditions are met, S is said to be a p-match of T. A significant breakthrough in the p-match area is the prev encoding, which is proven to identify a p-match between S and T if and only if prev(S) == prev(T). In order to utilize suffix data structures in terms of p-matching, we must account for the dynamic nature of the parameterized suffixes (p-suffixes) of T, namely prev(T[ i...n]) ∀ i, 1 ≤ i ≤ n.;In this work, we propose transformative approaches to the direct parameterized suffix sorting (p-suffix sorting) problem by generating and sorting lexicographically numeric fingerprints and arithmetic codes that correspond to individual p-suffixes. Our algorithm to p-suffix sort via fingerprints is the first theoretical linear time algorithm for p-suffix sorting for non-binary parameter alphabets, which assumes that each code is represented by a practical integer. We eliminate the key problems of fingerprints by introducing an algorithm that exploits the ordering of arithmetic codes to sort p-suffixes in linear time on average.;The longest previous factor (LPF) problem is defined for traditional strings exclusively from the constant alphabet Sigma. We generalize the LPF problem to the parameterized longest previous factor (pLPF) problem defined for p-strings. Subsequently, we present a linear time solution to construct the pLPF array. Given our pLPF algorithm, we show how to construct the pLCP (parameterized longest common prefix) array in linear time. Our algorithm is further exploited to construct the standard LPF and LCP arrays all in linear time.;We then study the structural string (s-string), a variant of the p-string that extends the p-string alphabets to include complementary parameters that correspond to one another. The s-string problem involves the new encoding schemes sencode and compl in order to identify a structural match (s-match). Current s-match solutions use a structural suffix tree (s-suffix tree) to study structural matches in RNA sequences. We introduce the suffix array, LCP, and LPF data structures for the s-string encoding schemes. Using our new data structures, we identify the first suffix array solution to the s-match problem. Our algorithms and data structures are shown to apply to s-strings and also p-strings and traditional strings

Linear Time Runs Over General Ordered Alphabets

A run in a string is a maximal periodic substring. For example, the string $\texttt{bananatree}$ contains the runs $\texttt{anana} = (\texttt{an})^{3/2}$ and $\texttt{ee} = \texttt{e}^2$. There are less than $n$ runs in any length-$n$ string, and computing all runs for a string over a linearly-sortable alphabet takes $\mathcal{O}(n)$ time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an $\mathcal{O}(n\alpha(n))$ time algorithm (where $\alpha(n)$ is the extremely slowly growing inverse Ackermann function). We show how to achieve $\mathcal{O}(n)$ time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov's conjecture.Comment: This work has been submitted to ICALP 202

An Elegant Algorithm for the Construction of Suffix Arrays

The suffix array is a data structure that finds numerous applications in string processing problems for both linguistic texts and biological data. It has been introduced as a memory efficient alternative for suffix trees. The suffix array consists of the sorted suffixes of a string. There are several linear time suffix array construction algorithms (SACAs) known in the literature. However, one of the fastest algorithms in practice has a worst case run time of $O(n^2)$. The problem of designing practically and theoretically efficient techniques remains open. In this paper we present an elegant algorithm for suffix array construction which takes linear time with high probability; the probability is on the space of all possible inputs. Our algorithm is one of the simplest of the known SACAs and it opens up a new dimension of suffix array construction that has not been explored until now. Our algorithm is easily parallelizable. We offer parallel implementations on various parallel models of computing. We prove a lemma on the $\ell$-mers of a random string which might find independent applications. We also present another algorithm that utilizes the above algorithm. This algorithm is called RadixSA and has a worst case run time of $O(n\log{n})$. RadixSA introduces an idea that may find independent applications as a speedup technique for other SACAs. An empirical comparison of RadixSA with other algorithms on various datasets reveals that our algorithm is one of the fastest algorithms to date. The C++ source code is freely available at http://www.engr.uconn.edu/~man09004/radixSA.zi

Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing

This paper presents a general technique for optimally transforming any dynamic data structure that operates on atomic and indivisible keys by constant-time comparisons, into a data structure that handles unbounded-length keys whose comparison cost is not a constant. Examples of these keys are strings, multi-dimensional points, multiple-precision numbers, multi-key data (e.g.~records), XML paths, URL addresses, etc. The technique is more general than what has been done in previous work as no particular exploitation of the underlying structure of is required. The only requirement is that the insertion of a key must identify its predecessor or its successor. Using the proposed technique, online suffix tree can be constructed in worst case time $O(\log n)$ per input symbol (as opposed to amortized $O(\log n)$ time per symbol, achieved by previously known algorithms). To our knowledge, our algorithm is the first that achieves $O(\log n)$ worst case time per input symbol. Searching for a pattern of length $m$ in the resulting suffix tree takes $O(\min(m\log |\Sigma|, m + \log n) + tocc)$ time, where $tocc$ is the number of occurrences of the pattern. The paper also describes more applications and show how to obtain alternative methods for dealing with suffix sorting, dynamic lowest common ancestors and order maintenance

Compressed Suffix Arrays and Suffix Trees with Applications to Text Indexing and String Matching

AMS subject classifications. 68W05, 68Q25, 68P05, 68P10, 68P30 DOI. 10.1137/S0097539702402354The proliferation of online text, such as found on the World Wide Web and in online databases, motivates the need for space-efficient text indexing methods that support fast string searching. We model this scenario as follows: Consider a text T consisting of n symbols drawn from a fixed alphabet Σ. The text T can be represented in n lg |Σ| bits by encoding each symbol with lg |Σ| bits. The goal is to support fast online queries for searching any string pattern P of m symbols, with T being fully scanned only once, namely, when the index is created at preprocessing time. The text indexing schemes published in the literature are greedy in terms of space usage: they require Ω(n lg n) additional bits of space in the worst case. For example, in the standard unit cost RAM, suffix trees and suffix arrays need Ω(n) memory words, each of Ω(lg n) bits. These indexes are larger than the text itself by a multiplicative factor of Ω(lg|Σ| n), which is significant when Σ is of constant size, such as in ascii or unicode. On the other hand, these indexes support fast searching, either in O(mlg |Σ|) time or in O(m+lg n) time, plus an output-sensitive cost O(occ) for listing the occ pattern occurrences. We present a new text index that is based upon compressed representations of suffix arrays and suffix trees. It achieves a fast O(m/lg|Σ| n + lg | Σ| n) search time in the worst case, for any constant 0 < ≤ 1, using at most −1 + O(1) n lg |Σ| bits of storage. Our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case, and for many scenarios, the space is actually sublinear in practice. As a concrete example, the compressed suffix array for a typical 100 MB ascii file can require 30–40 MB or less, while the raw suffix array requires 500 MB. Our theoretical bounds improve both time and space of previous indexing schemes. Listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output-sensitive cost, giving O(occ lg | Σ| n) time as a result. When the patterns are sufficiently long, we can use auxiliary data structures in O(n lg |Σ|) bits to obtain a total search bound of O(m/lg|Σ| n + occ) time, which is optimal

Faster algorithms for 1-mappability of a sequence

In the k-mappability problem, we are given a string x of length n and integers m and k, and we are asked to count, for each length-m factor y of x, the number of other factors of length m of x that are at Hamming distance at most k from y. We focus here on the version of the problem where k = 1. The fastest known algorithm for k = 1 requires time O(mn log n/ log log n) and space O(n). We present two algorithms that require worst-case time O(mn) and O(n log^2 n), respectively, and space O(n), thus greatly improving the state of the art. Moreover, we present an algorithm that requires average-case time and space O(n) for integer alphabets if m = {\Omega}(log n/ log {\sigma}), where {\sigma} is the alphabet size

Computing All Distinct Squares in Linear Time for Integer Alphabets

Given a string on an integer alphabet, we present an algorithm that computes the set of all distinct squares belonging to this string in time linear to the string length. As an application, we show how to compute the tree topology of the minimal augmented suffix tree in linear time. Asides from that, we elaborate an algorithm computing the longest previous table in a succinct representation using compressed working space