Faster Approximate Pattern Matching: {A} Unified Approach

Approximate pattern matching is a natural and well-studied problem on strings: Given a text $T$, a pattern $P$, and a threshold $k$, find (the starting positions of) all substrings of $T$ that are at distance at most $k$ from $P$. We consider the two most fundamental string metrics: the Hamming distance and the edit distance. Under the Hamming distance, we search for substrings of $T$ that have at most $k$ mismatches with $P$, while under the edit distance, we search for substrings of $T$ that can be transformed to $P$ with at most $k$ edits. Exact occurrences of $P$ in $T$ have a very simple structure: If we assume for simplicity that $|T| \le 3|P|/2$ and trim $T$ so that $P$ occurs both as a prefix and as a suffix of $T$, then both $P$ and $T$ are periodic with a common period. However, an analogous characterization for the structure of occurrences with up to $k$ mismatches was proved only recently by Bringmann et al. [SODA'19]: Either there are $O(k^2)$ $k$-mismatch occurrences of $P$ in $T$, or both $P$ and $T$ are at Hamming distance $O(k)$ from strings with a common period $O(m/k)$. We tighten this characterization by showing that there are $O(k)$ $k$-mismatch occurrences in the case when the pattern is not (approximately) periodic, and we lift it to the edit distance setting, where we tightly bound the number of $k$-edit occurrences by $O(k^2)$ in the non-periodic case. Our proofs are constructive and let us obtain a unified framework for approximate pattern matching for both considered distances. We showcase the generality of our framework with results for the fully-compressed setting (where $T$ and $P$ are given as a straight-line program) and for the dynamic setting (where we extend a data structure of Gawrychowski et al. [SODA'18])

Counting patterns in strings and graphs

We study problems related to finding and counting patterns in strings and graphs. In the string-regime, we are interested in counting how many substring of a text are at Hamming (or edit) distance at most to a pattern . Among others, we are interested in the fully-compressed setting, where both and are given in a compressed representation. For both distance measures, we give the first algorithm that runs in (almost) linear time in the size of the compressed representations. We obtain the algorithms by new and tight structural insights into the solution structure of the problems. In the graph-regime, we study problems related to counting homomorphisms between graphs. In particular, we study the parameterized complexity of the problem #IndSub(), where we are to count all -vertex induced subgraphs of a graph that satisfy the property . Based on a theory of Lovász, Curticapean et al., we express #IndSub() as a linear combination of graph homomorphism numbers to obtain #W[1]-hardness and almost tight conditional lower bounds for properties that are monotone or that depend only on the number of edges of a graph. Thereby, we prove a conjecture by Jerrum and Meeks. In addition, we investigate the parameterized complexity of the problem #Hom(ℋ → ) for graph classes ℋ and . In particular, we show that for any problem in the class #W[1], there are classes ℋ_ and _ such that is equivalent to #Hom(ℋ_ → _ ).Wir untersuchen Probleme im Zusammenhang mit dem Finden und Zählen von Mustern in Strings und Graphen. Im Stringbereich ist die Aufgabe, alle Teilstrings eines Strings zu bestimmen, die eine Hamming- (oder Editier-)Distanz von höchstens zu einem Pattern haben. Unter anderem sind wir am voll-komprimierten Setting interessiert, in dem sowohl , als auch in komprimierter Form gegeben sind. Für beide Abstandsbegriffe entwickeln wir die ersten Algorithmen mit einer (fast) linearen Laufzeit in der Größe der komprimierten Darstellungen. Die Algorithmen nutzen neue strukturelle Einsichten in die Lösungsstruktur der Probleme. Im Graphenbereich betrachten wir Probleme im Zusammenhang mit dem Zählen von Homomorphismen zwischen Graphen. Im Besonderen betrachten wir das Problem #IndSub(), bei dem alle induzierten Subgraphen mit Knoten zu zählen sind, die die Eigenschaft haben. Basierend auf einer Theorie von Lovász, Curticapean, Dell, and Marx drücken wir #IndSub() als Linearkombination von Homomorphismen-Zahlen aus um #W[1]-Härte und fast scharfe konditionale untere Laufzeitschranken zu erhalten für , die monoton sind oder nur auf der Kantenanzahl der Graphen basieren. Somit beweisen wir eine Vermutung von Jerrum and Meeks. Weiterhin beschäftigen wir uns mit der Komplexität des Problems #Hom(ℋ → ) für Graphklassen ℋ und . Im Besonderen zeigen wir, dass es für jedes Problem in #W[1] Graphklassen ℋ_ und _ gibt, sodass äquivalent zu #Hom(ℋ_ → _ ) ist

Using Partitions and Superstrings for Lossless Compression of Pattern Databases

We present an algorithm for compressing pattern databases (PDBs) and a method for fast random access of these com-pressed PDBs. We demonstrate the effectiveness of our technique by compressing two 6-tile sliding-tile PDBs by a factor of 12 and a 7-tile sliding-tile PDB by a factor of 24

Review of Particle Physics

The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,143 new measurements from 709 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Among the 120 reviews are many that are new or heavily revised, including a new review on Machine Learning, and one on Spectroscopy of Light Meson Resonances. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings

Review of Particle Physics

The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,143 new measurements from 709 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Among the 120 reviews are many that are new or heavily revised, including a new review on Machine Learning, and one on Spectroscopy of Light Meson Resonances. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings. The complete Review (both volumes) is published online on the website of the Particle Data Group (pdg.lbl.gov) and in a journal. Volume 1 is available in print as the PDG Book. A Particle Physics Booklet with the Summary Tables and essential tables, figures, and equations from selected review articles is available in print, as a web version optimized for use on phones, and as an Android app.United States Department of Energy (DOE) DE-AC02-05CH11231government of Japan (Ministry of Education, Culture, Sports, Science and Technology)Istituto Nazionale di Fisica Nucleare (INFN)Physical Society of Japan (JPS)European Laboratory for Particle Physics (CERN)United States Department of Energy (DOE

Review of Particle Physics

The Review summarizes much of particle physics and cosmology. Using data from previous editions, plus 2,143 new measurements from 709 papers, we list, evaluate, and average measured properties of gauge bosons and the recently discovered Higgs boson, leptons, quarks, mesons, and baryons. We summarize searches for hypothetical particles such as supersymmetric particles, heavy bosons, axions, dark photons, etc. Particle properties and search limits are listed in Summary Tables. We give numerous tables, figures, formulae, and reviews of topics such as Higgs Boson Physics, Supersymmetry, Grand Unified Theories, Neutrino Mixing, Dark Energy, Dark Matter, Cosmology, Particle Detectors, Colliders, Probability and Statistics. Among the 120 reviews are many that are new or heavily revised, including a new review on Machine Learning, and one on Spectroscopy of Light Meson Resonances. The Review is divided into two volumes. Volume 1 includes the Summary Tables and 97 review articles. Volume 2 consists of the Particle Listings and contains also 23 reviews that address specific aspects of the data presented in the Listings. The complete Review (both volumes) is published online on the website of the Particle Data Group (pdg.lbl.gov) and in a journal. Volume 1 is available in print as the PDG Book. A Particle Physics Booklet with the Summary Tables and essential tables, figures, and equations from selected review articles is available in print, as a web version optimized for use on phones, and as an Android app