IA Scholar Query: Loopless Generation of k-Ary Tree Sequences.
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Internet Archive Scholar query results feedeninfo@archive.orgTue, 26 Jul 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Combinatorial Gray codes-an updated survey
https://scholar.archive.org/work/zryp7sxkrbczrguasg4ugmgfee
A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a 'small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.Torsten Mützework_zryp7sxkrbczrguasg4ugmgfeeTue, 26 Jul 2022 00:00:00 GMTThe Smallest Hard Trees
https://scholar.archive.org/work/mn6ifeo7qvcnjokezzvwzwjave
We find an orientation of a tree with 20 vertices such that the corresponding fixed-template constraint satisfaction problem (CSP) is NP-complete, and prove that for every orientation of a tree with fewer vertices the corresponding CSP can be solved in polynomial time. We also compute the smallest tree that is NL-hard (assuming L is not NL), the smallest tree that cannot be solved by arc consistency, and the smallest tree that cannot be solved by Datalog. Our experimental results also support a conjecture of Bulin concerning a question of Hell, Nesetril and Zhu, namely that "easy trees lack the ability to count". Most proofs are computer-based and make use of the most recent universal-algebraic theory about the complexity of finite-domain CSPs. However, further ideas are required because of the huge number of orientations of trees. In particular, we use the well-known fact that it suffices to study orientations of trees that are cores and show how to efficiently decide whether a given orientation of a tree is a core using the arc-consistency procedure. Moreover, we present a method to generate orientations of trees that are cores that works well in practice. In this way we found interesting examples for the open research problem to classify finite-domain CSPs in NL.Manuel Bodirsky, Jakub Bulin, Florian Starke, Michael Wernthalerwork_mn6ifeo7qvcnjokezzvwzwjaveMon, 16 May 2022 00:00:00 GMTHomomorphism Tensors and Linear Equations
https://scholar.archive.org/work/nscyvqvqkvc5jnsnf7ki5pgmb4
Lovász (1967) showed that two graphs G and H are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph F, the number of homomorphisms from F to G equals the number of homomorphisms from F to H. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).Martin Grohe, Gaurav Rattan, Tim Seppeltwork_nscyvqvqkvc5jnsnf7ki5pgmb4Mon, 25 Apr 2022 00:00:00 GMTMoser-Tardos Algorithm with small number of random bits
https://scholar.archive.org/work/jxk5cvngfzay5ebq6p2qqp47sa
We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have some fixed subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two colloraries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which in any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.Endre Csóka, Łukasz Grabowski, András Máthé, Oleg Pikhurko, Konstantinos Tyroswork_jxk5cvngfzay5ebq6p2qqp47saFri, 11 Mar 2022 00:00:00 GMTCycle structure and colorings of directed graphs
https://scholar.archive.org/work/4oljppo7pnh3vlrxkkkho6zd4m
This thesis deals with problems from the theory of finite directed graphs. A directed graph (digraph for short) is a binary relation whose domain has finite size. With that digraphs can be seen as a very general way of representing (possibly asymmetric) relations between pairs from a finite set of objects. Undoubtedly, such a generality allows to encode many structures by digraphs. This works particularly well if important properties of the structure at hand can be expressed as relations or connections between objects. To give some selected examples, let us mention road networks, electricity networks, radio networks, the world wide web, circuits in electronic devices, or neural networks. A main focus of the thesis at hand is the investigation of properties of one of the most fundamental objects all over graph theory, the so-called cycle (sometimes also called circuit). A cycle in a graph is determined by a closed alternating sequence of cyclically connected vertices and edges. In a graph of finite size one will typically see loads of distinct cycles of various types. Therefore cycles constitute an important and recurring motive in almost all branches of graph theory, for instance, they play important roles in structural graph theory, in the theory of flows on directed networks, in theoretical characterizations of graph classes, as well as in the theory of graph colorings. Additionally, cycles play a decisive role in numerous algorithmic problems and their solutions, such as in the Traveling Salesman Problem, algorithms for finding a largest matching in a given graph, in the max-flow problem, and also in subprocedures such as Kruskal's algorithm for finding a minimum weight spanning tree. For those reasons, a substantial amount of research in graph theory has specialised on the structure of cycles in graphs. In the first major part of this thesis we deal with cycles which occur in directed graphs, and prove several necessary and sufficient theoretical conditions for the existence of cycles of certain types. Additi [...]Raphael Mario Steiner, Technische Universität Berlin, Stefan Felsnerwork_4oljppo7pnh3vlrxkkkho6zd4mThu, 30 Dec 2021 00:00:00 GMTOn the scalability, resilience, and privacy of decentralized blockchain networks
https://scholar.archive.org/work/xje2kuh225aq3axtdx47xk65oa
Cryptocurrencies such as Bitcoin or Ethereum promise to establish themselves as decentralized alternatives to financial infrastructures that so far have been reliant on centralized trust models. These decentralized blockchain networks are built around the foundational principles introduced with Bitcoin's consensus protocol, in which a peer-to-peer network manages a globally distributed ledger of transactions—the blockchain. While this new and rather unorthodox approach to achieve Byzantine agreement in open and decentralized networks offers a number of promising properties and features, it suffers from very limited throughput scalability. As distributed systems under the pressure to scale are often at risk of neglecting other essential qualities, improving scalability while considering resilience and decentralization poses a fundamental challenge for the research on open blockchain networks today. This thesis is therefore dedicated to the study of blockchain scalability from a computer networking perspective. In this regard, we focus on the two main approaches towards blockchain scalability—on-chain and off-chain scaling—, study the currently deployed state-of-the-art protocols and architectures, and propose improvements that consider decentralization, security, and privacy first-class design goals. As on-chain scalability has been previously shown to be highly dependent on the reliability and performance of the underlying networking layer, the first part of this thesis studies the peer-to-peer networks utilized for block and transaction propagation. To this end, we research the Bitcoin and Zcash networks through longitudinal measurement studies and enable their model-based evaluation through the introduction of a network-centric simulation framework. Furthermore, we present Kadcast, a new transport protocol based on a structured overlay network. We show that Kadcast enables faster and more efficient block and transaction propagation while maintaining decentralization. In the second part, we direct our attention [...]Elias Rohrer, Technische Universität Berlin, Florian Tschorschwork_xje2kuh225aq3axtdx47xk65oaTue, 21 Dec 2021 00:00:00 GMTThe Internet of Federated Things (IoFT): A Vision for the Future and In-depth Survey of Data-driven Approaches for Federated Learning
https://scholar.archive.org/work/vrnmp6taazg4hdyjkoypvtujv4
The Internet of Things (IoT) is on the verge of a major paradigm shift. In the IoT system of the future, IoFT, the cloud will be substituted by the crowd where model training is brought to the edge, allowing IoT devices to collaboratively extract knowledge and build smart analytics/models while keeping their personal data stored locally. This paradigm shift was set into motion by the tremendous increase in computational power on IoT devices and the recent advances in decentralized and privacy-preserving model training, coined as federated learning (FL). This article provides a vision for IoFT and a systematic overview of current efforts towards realizing this vision. Specifically, we first introduce the defining characteristics of IoFT and discuss FL data-driven approaches, opportunities, and challenges that allow decentralized inference within three dimensions: (i) a global model that maximizes utility across all IoT devices, (ii) a personalized model that borrows strengths across all devices yet retains its own model, (iii) a meta-learning model that quickly adapts to new devices or learning tasks. We end by describing the vision and challenges of IoFT in reshaping different industries through the lens of domain experts. Those industries include manufacturing, transportation, energy, healthcare, quality & reliability, business, and computing.Raed Kontar, Naichen Shi, Xubo Yue, Seokhyun Chung, Eunshin Byon, Mosharaf Chowdhury, Judy Jin, Wissam Kontar, Neda Masoud, Maher Noueihed, Chinedum E. Okwudire, Garvesh Raskutti, Romesh Saigal, Karandeep Singh, Zhisheng Yework_vrnmp6taazg4hdyjkoypvtujv4Tue, 09 Nov 2021 00:00:00 GMTCombinatorial generation via permutation languages. I. Fundamentals
https://scholar.archive.org/work/zseyxpcbu5f2tg7xnpioqr5p64
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group S_n. Recently, Pilaud and Santos realized all those lattice congruences as (n-1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, Aaron Williamswork_zseyxpcbu5f2tg7xnpioqr5p64Wed, 03 Nov 2021 00:00:00 GMT