This document provides the foundations behind the functionality provided by the @math G library (this https URL), focusing on the basic operations the library provides: subsumption, refinement of directed labeled graphs, and distance similarity assessment between directed labeled graphs. @math G development was initially supported by the National Science Foundation, by the EAGER grant IIS-1551338.
Similarity measures for specific application domains, such as molecular structures in domains of biology or chemistry have also been studied @cite_30 , and they are typically grouped into three classes @cite_43 : sequence similarities (e.g., for DNA fragments), fingerprint-based (transform each molecule into a sequence of binary features representing whether a particular molecule exhibits particular properties or contains certain substructures) and graph-based (based on maximum common sub-graphs). The latter is a computationally expensive process, and thus there are a number of strategies to simplify the computations (e.g., @cite_17 ).
@cite_30 @cite_43 @cite_17
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Measures of sequence similarity and distance based more or less explicitly on subword composition are attracting an increasing interest driven by intensive applications such as massive document classification and genome-wide molecular taxonomy. A uniform character of such measures is in some underlying notion of relative compressibility, whereby two similar sequences are expected to share a larger number of common substrings than two distant ones. This paper reviews some of the approaches to sequence comparison based on subword composition and suggests that their common denominator may ultimately reside in special classes of subwords, the nature of which resonates in interesting ways with the structure of popular subword trees and graphs. Preface Introduction Molecular Biology Mathematics, Statistics, and Computer Science Some Molecular Biology DNA and Proteins The Central Dogma The Genetic Code Transfer RNA and Protein Sequences Genes Are Not Simple Biological Chemistry Restriction Maps Introduction Graphs Interval Graphs Measuring Fragment Sizes Multiple Maps Double Digest Problem Classifying Multiple Solutions Algorithms for DDP Algorithms and Complexity DDP is N P-Complete Approaches to DDP Simulated Annealing: TSP and DDP Mapping with Real Data Cloning and Clone Libraries A Finite Number of Random Clones Libraries by Complete Digestion Libraries by Partial Digestion Genomes per Microgram Physical Genome Maps: Oceans, Islands, and Anchors Mapping by Fingerprinting Mapping by Anchoring An Overview of Clone Overlap Putting It Together Sequence Assembly Shotgun Sequencing Sequencing by Hybridization Shotgun Sequencing Revisited Databases and Rapid Sequence Analysis DNA and Protein Sequence Databases A Tree Representation of a Sequence Hashing a Sequence Repeats in a Sequence Sequence Comparison by Hashing Sequence Comparison with at most l Mismatches Sequence Comparison by Statistical Content Dynamic Programming Alignment of Two Sequences The Number of Alignments Shortest and Longest Paths in a Network Global Distance Alignment Global Similarity Alignment Fitting One Sequence into Another Local Alignment and Clumps Linear Space Algorithms Tracebacks Inversions Map Alignment Parametric Sequence Comparisons Multiple Sequence Alignment The Cystic Fibrosis Gene Dynamic Programming in r-Dimensions Weighted-Average Sequences Profile Analysis Alignment by Hidden Markov Models Consensus Word Analysis Probability and Statistics for Sequence Alignment Global Alignment Local Alignment Extreme Value Distributions The Chein-Stein Method Poisson Approximation and Long Matches Sequence Alignment with Scores Probability and Statistics for Sequence Patterns A Central Limit Theorem Nonoverlapping Pattern Counts Poisson Approximation Site Distributions RNA Secondary Structure Combinatorics Minimum Free-energy Structures Consensus folding Trees and Sequences Trees Distance Parsimony Maximum Likelihood Trees Sources and Perspectives Molecular Biology Physical Maps and Clone Libraries Sequence Assembly Sequence Comparisons Probability and Statistics RNA Secondary Structure Trees and Sequences References Problem Solutions and Hints Mathematical Notation Algorithm Index Author Index Subject Index This paper reviews the use of similarity searching in chemical databases. It begins by introducing the concept of similarity searching, differentiating it from the more common substructure searching, and then discusses the current generation of fragment-based measures that are used for searching chemical structure databases. The next sections focus upon two of the principal characteristics of a similarity measure: the coefficient that is used to quantify the degree of structural resemblance between pairs of molecules and the structural representations that are used to characterize molecules that are being compared in a similarity calculation. New types of similarity measure are then compared with current approaches, and examples are given of several applications that are related to similarity searching. A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance," based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class and it minorizes every computable distance in the class (that is, it is universal in that it discovers all computable similarities). We demonstrate that it is a metric and call it the similarity metric . This theory forms the foundation for a new practical tool. To evidence generality and robustness, we give two distinctive applications in widely divergent areas using standard compression programs like gzip and GenCompress. First, we compare whole mitochondrial genomes and infer their evolutionary history. This results in a first completely automatic computed whole mitochondrial phylogeny tree. Secondly, we fully automatically compute the language tree of 52 different languages.