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Approximate NFA universality and related problems motivated by information theory

THEORETICAL COMPUTER SCIENCE 2023年第1期972卷

作者： Konstantinidis, Stavros Mastnak, Mitja Moreira, Nelma Reis, Rogerio St Marys Univ Math & CS 923 Robie Str Halifax NS B3H 3C3 Canada Univ Porto Fac Ciencias CMUP & DM DCC Rua Campo Alegre P-4169007 Porto Portugal

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is 'approximately' maximal, which then boils down to whether a given NFA is 'approximately' universal. Here we introduce the notion of a (1 - & epsilon;)-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable & epsilon;.& COPY;2023 Elsevier B.V. All rights reserved.

关键词： NFA universality Approximation randomized algorithm PSPACE hard

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Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

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JOURNAL OF APPROXIMATION THEORY 2020年 251卷 105342-105342页

作者： Gnewuch, M. Wnuk, M. Univ Osnabruck Inst Math Albrechtstr 28 A D-49076 Osnabruck Germany

Smolyak's method, also known as sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem. In this paper we study the randomized setting, i.e., we randomize Smolyak's method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst case root mean square error (the typical error criterion for randomized algorithms, sometimes referred to as "randomized error",) and the root mean square worst case error (sometimes referred to as "worst-case error"). randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of Nth minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinite-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases ("spaces of increasing smoothness") and to the problem of L-2-approximation (function recovery). (C) 2019 Elsevier Inc. All rights reserved.

关键词： Tensor product problem Sparse grid randomized algorithm Infinite-dimensional approximation Multivariate decomposition method Changing dimension algorithm

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Time-Optimal Construction of Overlay Networks 21

Time-Optimal Construction of Overlay Networks

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40th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC)

作者： Goette, Thorsten Hinnenthal, Kristian Scheideler, Christian Werthmann, Julian Paderborn Univ Paderborn Germany

ISBN: (纸本)9781450385480

We show how to construct an overlay network of constant degree and diameter O(log n) in time O(log n) starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of and establish new connections by sending node identifiers. If the initial network's graph is weakly connected and has constant degree, then our algorithm constructs the desired topology with each node sending and receiving only O(log n) messages in each round in time O(log n), w.h.p., which beats the currently best O(log(3/2) n) time algorithm of [Gotte et al., SIROCCO'19]. Since the problem cannot be solved faster than by using pointer jumping for O(log n) rounds (which would even require each node to communicate Omega(n) bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of [Kwok and Lau, APPROX' 14]. Additionally, we show how our algorithm can be used to efficiently solve graph problems in hybrid networks [Augustine et al., SODA'20]. Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the initial edges is unrestricted. In contrast, only polylogarithmically many messages can be communicated over edges that have been established throughout an algorithm's execution. For an (undirected) graph G with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in time O (log n), w.h.p. Furthermore, we show how to compute an MIS in time O(log d + log log n), w.h.p., where d is the initial degree of G.

关键词： Distributed Protocol Peer-to-Peer Network randomized algorithm Expander

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Approximating 4-Cliques in Streaming Graphs: The Power of Dual Sampling 13

Approximating 4-Cliques in Streaming Graphs: The Power of Du...

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IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)

作者： Mann, Anmol Srinivasan, Venkatesh Thomo, Alex Univ Victoria Victoria BC Canada

ISBN: (纸本)9781450391283

Clique counting is considered to be a challenging problem in graph mining. The reason is combinatorial explosion;even moderate graphs with a few million edges could have clique counts in the order of many billions. In this paper, we propose a fast and scalable algorithm for approximating 4-clique counts in a single-pass streaming model. By leveraging a combination of sampling approaches, we estimate the 4-clique count with high accuracy. Our algorithm performs well on massive graphs containing several billions of 4-cliques, and terminates within a reasonable amount of time.

关键词： stream data analysis clique counting approximation randomized algorithm

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Brief Announcement: On the Message Complexity of Fault-Tolerant Computation: Leader Election and Agreement 21

Brief Announcement: On the Message Complexity of Fault-Toler...

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40th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC)

作者： Kumar, Manish Molla, Anisur Rahaman Indian Stat Inst Kolkata India

ISBN: (纸本)9781450385480

This paper investigates on the message complexity of the two fundamental problems, namely, leader election and agreement in the crash-fault synchronous and fully-connected distributed network. We present randomized algorithms for both the problems and also develop non-trivial lower bounds on the message complexity. In the so-called implicit version of the two problems, our algorithms achieve sublinear message complexity while tolerating more than a constant fraction of faulty nodes. The algorithms work in anonymous networks, where nodes do not know each other. Specifically, our main results are: (1) A randomized leader election algorithm which elects a leader with high probability in a complete network with n nodes, in which at least [alpha n] nodes are non-faulty and the remaining can be faulty, where alpha >= (log(2) n)In. The time complexity of the algorithm is O(log n/alpha) rounds and message complexity is O((n(0.5) log(2.5) n)/alpha(2.5)) with high probability (i.e., with probability >= 1 - 1/n). A non-trivial lower bound of Omega (n(0.)(5)/alpha(1.5)) messages for any leader election algorithm that tolerates at most (1- alpha)-fraction faulty nodes and succeeds with high probability. (2) A randomized algorithm, tolerating at most left perpendicular(1 - alpha)nright perpendicular faulty nodes, solves agreement in O (log n/alpha) rounds and with high probability uses only O ((n(0.5) log(1.5)n)/alpha(1.)(5)) messages. A match- ing lower bound (up to a polylog n factor) of Omega(n(0.)(5)/alpha(1.5)) messages for any agreement algorithm that tolerates at most (1-alpha)-fraction faulty nodes and succeeds with high probability. A full version of the paper is available at [17].

关键词： Distributed algorithm randomized algorithm fault-tolerant algorithm crash-fault leader election agreement message complexity

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Fruit-fly Inspired Neighborhood Encoding for Classification 21

Fruit-fly Inspired Neighborhood Encoding for Classification

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27th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD)

作者： Sinha, Kaushik Ram, Parikshit Wichita State Univ Wichita KS 67260 USA IBM Res AI New York NY USA

ISBN: (纸本)9781450383325

Inspired by the fruit-fly olfactory circuit, the Fly Bloom Filter [4] is able to efficiently summarize the data with a single pass and has been used for novelty detection. We propose a new classifier that effectively encodes the different local neighborhoods for each class with a per-class Fly Bloom Filter. The inference on test data requires an efficient FlyHash [6] operation followed by a high-dimensional, but very sparse, dot product with the per-class Bloom Filters. On the theoretical side, we establish conditions under which the predictions of our proposed classifier agrees with the predictions of the nearest neighbor classifier. We extensively evaluate our proposed scheme with 71 data sets of varied data dimensionality to demonstrate that the predictive performance of our proposed neuroscience inspired classifier is competitive to the nearest-neighbor classifiers and other single-pass classifiers.

关键词： nearest-neighbor classification randomized algorithm bio-inspired

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A simple randomized algorithm for two-dimensional strip packing

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COMPUTERS & OPERATIONS RESEARCH 2013年第1期40卷 1-8页

作者： Yang, Shuangyuan Han, Shuihua Ye, Weiguo Xiamen Univ Software Sch Xiamen 361005 Peoples R China Xiamen Univ Dept Management Sci Xiamen 361005 Peoples R China

Two-dimensional strip packing problem is to pack given rectangular pieces on a strip of stock sheet having fixed width and infinite height. Its aim is to minimize the height of the strip such that non-guillotinable and fix orientation constraints are meet. In this paper, an improved scoring rule is developed and the least waste priority strategy is introduced, and a randomized algorithm is presented for solving this problem. This algorithm is very simple and does not need to set any parameters. Computational results on a wide range of benchmark problem instances show that the proposed algorithm obtains a better or matching performance as compared to the most of the previously published meta-heuristics. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Packing problem Heuristic randomized algorithm

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A fast sketching-based algorithm for rank-(L, L, 1) block term decomposition

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APPLIED MATHEMATICS LETTERS 2022年第0期133卷

作者： Zhang, Hao Huang, Ting-Zhu Zhao, Xi-Le Univ Elect Sci & Technol China Res Ctr Image & Vis Comp Sch Math Sci Chengdu Sichuan Peoples R China

The rank-(L, L, 1) block term decomposition (BTD) of the tensor has recently received increasing attention in diverse high-order data processing, e.g., hyperspectral image restoration and blind source separation. However, the standard alternating least squares algorithm for the rank-(L, L, 1) BTD (BTD-ALS) is computationally expensive, which hinders its application on real large-scale data. In this paper, we propose a fast sketching-based algorithm for the rank-(L, L, 1) BTD (FastBTD) to address the computational burden of BTD-ALS. Since the dominant cost of BTD-ALS at each iteration is the sequence of large-scale least squares subproblems, we first project these original subproblems into the low-dimensional subspace by the tensor sketching operator. Then, we can fastly obtain the approximate solutions of the original subproblems by solving these sketched small-scale subproblems at each iteration of FastBTD. The computational complexity of FastBTD (O(2JILR + JIR)) is significantly lower than that of BTD-ALS (O(2I(3)LR + (IR)-R-3)) for solving the least squares subproblems at each iteration, when J << I-2. Moreover, we provide the theoretical error bound for FastBTD. Extensive experiments on both synthetic and real data demonstrate that FastBTD achieves substantial speedup while maintaining accuracy compared with BTD-ALS. (C) 2022 Elsevier Ltd. All rights reserved.

关键词： Block term decomposition randomized algorithm Tensor sketching High-order data processing

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odeN: Simultaneous Approximation of Multiple Motif Counts in Large Temporal Networks 21

odeN: Simultaneous Approximation of Multiple Motif Counts in...

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30th ACM International Conference on Information and Knowledge Management (CIKM)

作者： Sarpe, The Vandin, Fabio Univ Padua Dept Informat Engn Padua Italy

ISBN: (纸本)9781450384469

Counting the number of occurrences of small connected subgraphs, called temporal motifs, has become a fundamental primitive for the analysis of temporal networks, whose edges are annotated with the time of the event they represent. One of the main complications in studying temporal motifs is the large number of motifs that can be built even with a limited number of vertices or edges. As a consequence, since in many applications motifs are employed for exploratory analyses, the user needs to iteratively select and analyze several motifs that represent different aspects of the network, resulting in an inefficient, time-consuming process. This problem is exacerbated in large networks, where the analysis of even a single motif is computationally demanding. As a solution, in this work we propose and study the problem of simultaneously counting the number of occurrences of multiple temporal motifs, all corresponding to the same (static) topology (e.g., a triangle). Given that for large temporal networks computing the exact counts is unfeasible, we propose odeN, a sampling-based algorithm that provides an accurate approximation of all the counts of the motifs. We provide analytical bounds on the number of samples required by odeN to compute rigorous, probabilistic, relative approximations. Our extensive experimental evaluation shows that odeN enables the approximation of the counts of motifs in temporal networks in a fraction of the time needed by state-of-the-art methods, and that it also reports more accurate approximations than such methods.

关键词： temporal motifs sampling algorithm temporal networks randomized algorithm

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Stable Resolving - A randomized Local Search Heuristic for MaxSAT 43rd

Stable Resolving - A Randomized Local Search Heuristic for M...

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43rd German Conference on Artificial Intelligence (KI)

作者： Reisch, Julian Grossmann, Peter Kliewer, Natalia Synopt GmbH Chemnitzer Str 48b D-01187 Dresden Germany Free Univ Berlin Garystr 21 D-14195 Berlin Germany

ISBN: (纸本)9783030582845;9783030582852

Many problems from industrial applications and AI can be encoded as Maximum Satisfiability (MaxSAT). Often, it is more desirable to produce practicable results in very short time compared to optimal solutions after an arbitrary long computation time. In this paper, we propose Stable Resolving (SR), a novel randomized local search heuristic for MaxSAT with that aim. SR works for both weighted and unweighted instances. Starting from a feasible initial solution, the algorithm repeatedly performs the three steps of perturbation, improvements and solution checking. In the perturbation, the search space is explored at the cost of possibly worsening the current solution. The local improvements work by repeatedly flipping signs of variables in over-satisfied clauses. Finally, the algorithm performs a solution checking in a simulated annealing fashion. We compare our approach to state-of-the-art MaxSAT solvers and show by numerical experiments on benchmark instances from the annual MaxSAT competition that SR performs comparable on average and is even the best solver for particular problem instances.

关键词： Maximum Satisfiability MaxSAT Incomplete solving randomized algorithm Local search algorithm Simulated annealing

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