The substring inclusion constraint longest common subsequence problem can be solved in quadratic time

AbstractIn this paper, we study some variants of the Constrained Longest Common Subsequence (CLCS) problem, namely, the substring inclusion CLCS (Substring-IC-CLCS) problem and a generalized version thereof. In the Substring-IC-CLCS problem, we are to find a longest common subsequence (LCS) of two given strings containing a third constraint string (given) as a substring. Previous solution to this problem runs in cubic time, i.e, O(nmk) time, where n,m and k are the length of the 3 input strings. In this paper, we present simple O(nm) time algorithms to solve the Substring-IC-CLCS problem. We also study the Generalized Substring-IC-LCS problem where we are given two strings of length n and m respectively and an ordered list of p strings and the goal is to find an LCS containing each of them as a substring in the order they appear in the list. We present an O(nmp) algorithm for this generalized version of the problem

Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, K\"unnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$. [...]Comment: Presented at SODA'18. Full Version. 66 page

Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, K\"unnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$. [...

A review of EBMT using proportional analogies

Some years ago a number of papers reported an experimental implementation of Example Based Machine Translation (EBMT) using Proportional Analogy. This approach, a type of analogical learning, was attractive because of its simplicity; and the papers reported considerable success with the method. This paper reviews what we believe to be the totality of research reported using this method, as an introduction to our own experiments in this framework, reported in a companion paper. We report first some lack of clarity in the previously published work, and then report our findings that the purity of the proportional analogy approach imposes huge run-time complexity for the EBMT task even when heuristics as hinted at in the original literature are applied to reduce the amount of computation

Festparameter-Algorithmen fuer die Konsens-Analyse Genomischer Daten

Fixed-parameter algorithms offer a constructive and powerful approach to efficiently obtain solutions for NP-hard problems combining two important goals: Fixed-parameter algorithms compute optimal solutions within provable time bounds despite the (almost inevitable) computational intractability of NP-hard problems. The essential idea is to identify one or more aspects of the input to a problem as the parameters, and to confine the combinatorial explosion of computational difficulty to a function of the parameters such that the costs are polynomial in the non-parameterized part of the input. This makes especially sense for parameters which have small values in applications. Fixed-parameter algorithms have become an established algorithmic tool in a variety of application areas, among them computational biology where small values for problem parameters are often observed. A number of design techniques for fixed-parameter algorithms have been proposed and bounded search trees are one of them. In computational biology, however, examples of bounded search tree algorithms have been, so far, rare. This thesis investigates the use of bounded search tree algorithms for consensus problems in the analysis of DNA and RNA data. More precisely, we investigate consensus problems in the contexts of sequence analysis, of quartet methods for phylogenetic reconstruction, of gene order analysis, and of RNA secondary structure comparison. In all cases, we present new efficient algorithms that incorporate the bounded search tree paradigm in novel ways. On our way, we also obtain results of parameterized hardness, showing that the respective problems are unlikely to allow for a fixed-parameter algorithm, and we introduce integer linear programs (ILP's) as a tool for classifying problems as fixed-parameter tractable, i.e., as having fixed-parameter algorithms. Most of our algorithms were implemented and tested on practical data.Festparameter-Algorithmen bieten einen konstruktiven Ansatz zur Loesung von kombinatorisch schwierigen, in der Regel NP-harten Problemen, der zwei Ziele beruecksichtigt: innerhalb von beweisbaren Laufzeitschranken werden optimale Ergebnisse berechnet. Die entscheidende Idee ist dabei, einen oder mehrere Aspekte der Problemeingabe als Parameter der Problems aufzufassen und die kombinatorische Explosion der algorithmischen Schwierigkeit auf diese Parameter zu beschraenken, so dass die Laufzeitkosten polynomiell in Bezug auf den nicht-parametrisierten Teil der Eingabe sind. Gibt es einen Festparameter-Algorithmus fuer ein kombinatorisches Problem, nennt man das Problem festparameter-handhabbar. Die Entwicklung von Festparameter-Algorithmen macht vor allem dann Sinn, wenn die betrachteten Parameter im Anwendungsfall nur kleine Werte annehmen. Festparameter-Algorithmen sind zu einem algorithmischen Standardwerkzeug in vielen Anwendungsbereichen geworden, unter anderem in der algorithmischen Biologie, wo in vielen Anwendungen kleine Parameterwerte beobachtet werden koennen. Zu den bekannten Techniken fuer den Entwurf von Festparameter-Algorithmen gehoeren unter anderem groessenbeschraenkte Suchbaeume. In der algorithmischen Biologie gibt es bislang nur wenige Beispiele fuer die Anwendung von groessenbeschraenkten Suchbaeumen. Diese Arbeit untersucht den Einsatz groessenbeschraenkter Suchbaeume fuer NP-harte Konsens-Probleme in der Analyse von DNS- und RNS-Daten. Wir betrachten Konsens-Probleme in der Analyse von DNS-Sequenzdaten, in der Analyse von sogenannten Quartettdaten zur Erstellung von phylogenetischen Hypothesen, in der Analyse von Daten ueber die Anordnung von Genen und beim Vergleich von RNS-Strukturdaten. In allen Faellen stellen wir neue effiziente Algorithmen vor, in denen das Paradigma der groessenbeschraenkten Suchbaeume auf neuartige Weise realisiert wird. Auf diesem Weg zeigen wir auch Ergebnisse parametrisierter Haerte, die zeigen, dass fuer die dabei betrachteten Probleme ein Festparameter-Algorithmus unwahrscheinlich ist. Ausserdem fuehren wir ganzzahliges lineares Programmieren als eine neue Technik ein, um die Festparameter-Handhabbarkeit eines Problems zu zeigen. Die Mehrzahl der hier vorgestellten Algorithmen wurde implementiert und auf Anwendungsdaten getestet

Quantum Graph Parameters

This dissertation considers some of the advantages, and limits, of applying quantum computing to solve two important graph problems. The first is estimating a graph\u27s quantum chromatic number. The quantum chromatic number is the minimum number of colors necessary in a two-player game where the players cannot communicate but share an entangled state and must convince a referee with probability one that they have a proper vertex coloring. We establish several spectral lower bounds for the quantum chromatic number. These lower bounds extend the well-known Hoffman lower bound for the classical chromatic number. The second is the Pattern Matching on Labeled Graphs Problem (PMLG). Here the objective is to match a string (called a pattern) P to a walk in an edge labeled graph G = (V, E). In addition to providing a new quantum algorithm for PMLG, this work establishes conditional lower bounds on the time complexity of any quantum algorithm for PMLG. These include a conditional lower bound based on the recently proposed NC-QSETH and a reduction from the Longest Common Subsequence Problem (LCS). For PMLG where substitutions are allowed to the pattern, our results demonstrate that (i) a quantum algorithm running in time O(|E|m1-ε + |E|1-εm) for any constant ε \u3e 0 would provide an algorithm for LCS on two strings X and Y running in time Õ(|X||Y|1-ε + |X|1-ε|Y|), which is better than any known quantum algorithm for LCS, and (ii) a quantum algorithm running in time O(|E|m½-ε + |E|½-εm) would violate NC-QSETH. Results (i) and (ii) hold even when restricted to binary alphabets for P and the edge labels in G. Our quantum algorithm is for all versions of PMLG (exact, only substitutions, and substitutions/insertions/deletions) and runs in time Õ(√|V||E|· m), making it an improvement over the classical O(|E|m) time algorithm when the graph is non-sparse

Aspects of Metric Spaces in Computation

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate

CAD Tools for DNA Micro-Array Design, Manufacture and Application

Motivation: As the human genome project progresses and some microbial and eukaryotic genomes are recognized, numerous biotechnological processes have attracted increasing number of biologists, bioengineers and computer scientists recently. Biotechnological processes profoundly involve production and analysis of highthroughput experimental data. Numerous sequence libraries of DNA and protein structures of a large number of micro-organisms and a variety of other databases related to biology and chemistry are available. For example, microarray technology, a novel biotechnology, promises to monitor the whole genome at once, so that researchers can study the whole genome on the global level and have a better picture of the expressions among millions of genes simultaneously. Today, it is widely used in many fields- disease diagnosis, gene classification, gene regulatory network, and drug discovery. For example, designing organism specific microarray and analysis of experimental data require combining heterogeneous computational tools that usually differ in the data format; such as, GeneMark for ORF extraction, Promide for DNA probe selection, Chip for probe placement on microarray chip, BLAST to compare sequences, MEGA for phylogenetic analysis, and ClustalX for multiple alignments. Solution: Surprisingly enough, despite huge research efforts invested in DNA array applications, very few works are devoted to computer-aided optimization of DNA array design and manufacturing. Current design practices are dominated by ad-hoc heuristics incorporated in proprietary tools with unknown suboptimality. This will soon become a bottleneck for the new generation of high-density arrays, such as the ones currently being designed at Perlegen [109]. The goal of the already accomplished research was to develop highly scalable tools, with predictable runtime and quality, for cost-effective, computer-aided design and manufacturing of DNA probe arrays. We illustrate the utility of our approach by taking a concrete example of combining the design tools of microarray technology for Harpes B virus DNA data

2-Dimensional String Problems: Data Structures and Quantum Algorithms

The field of stringology studies algorithms and data structures used for processing strings efficiently. The goal of this thesis is to investigate 2-dimensional (2D) variants of some fundamental string problems, including \textit{Exact Pattern Matching} and \textit{Longest Common Substring}. In the 2D pattern matching problem, we are given a matrix \M[1\dd n,1\dd n] that consists of $N = n \times n$ symbols drawn from an alphabet $\Sigma$ of size $\sigma$. The query consists of a $m \times m$ square matrix \PP[1\dd m, 1\dd m] drawn from the same alphabet, and the task is to find all the locations of \PP in \M. For such square patterns, data structures such as suffix trees and suffix arrays exist for the task of efficient pattern matching. However, a suffix tree occupies $O(N \log N)$ bits, which is significantly more than that of the original text\u27s size of $N\log \sigma$ bits. Therefore, the design of compressed data structures, that supports pattern matching queries efficiently and occupies space close to the original text\u27s size, is imperative. In this thesis, we show an interesting result by designing a compact text index of size $O(N \log\log N + N \log\sigma)$ bits that at least supports efficient inverse suffix array queries. Although, the question of designing a compressed text index that would lead to efficient pattern matching is still evasive, this index gives a hope on the existence of a full 2D compressed text index with all functionalities similar to that of 1D case. On the other hand, the Longest Common 2D substring problem consists of two 2D strings (matrices), and the task is to report the size of the longest common 2D substring (submatrix) of these 2D strings. It is interesting to know if there exists a sublinear-time algorithm for solving this task. We answer this question positively by presenting a sublinear-time \textit{quantum} algorithm. In addition to this, we prove that any quantum algorithm requires at least $\tilde{\Omega}(N^{2/3})$ time to solve this problem

Subsequences and Supersequences of Strings

Stringology - the study of strings - is a branch of algorithmics which been the sub-ject of mounting interest in recent years. Very recently, two books [M. Crochemore and W. Rytter, Text Algorithms, Oxford University Press, 1995] and [G. Stephen, String Searching Algorithms, World Scientific, 1994] have been published on the subject and at least two others are known to be in preparation. Problems on strings arise in information retrieval, version control, automatic spelling correction, and many other domains. However the greatest motivation for recent work in stringology has come from the field of molecular biology. String problems occur, for example, in genetic sequence construction, genetic sequence comparison, and phylogenetic tree construction. In this thesis we study a variety of string problems from a theoretical perspective. In particular, we focus on problems involving subsequences and supersequences of strings