Approximate Counting of
            k
            -Paths: Simpler, Deterministic, and in Polynomial Space

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

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Subspace Clustering for High-Dimensional Data Using Cluster Structure Similarity

International Journal of Intelligent Information Technologies ◽

10.4018/ijiit.2018070103 ◽

2018 ◽

Vol 14 (3) ◽

pp. 38-55 ◽

Cited By ~ 2

Author(s):

Kavan Fatehi ◽

Mohsen Rezvani ◽

Mansoor Fateh ◽

Mohammad-Reza Pajoohan

Keyword(s):

Similarity Measure ◽

State Of The Art ◽

Clustering Algorithms ◽

Cluster Structure ◽

High Dimensional Data ◽

Subspace Clustering ◽

The State ◽

High Dimensional ◽

Running Time ◽

Structure Similarity

This article describes how recently, because of the curse of dimensionality in high dimensional data, a significant amount of research has been conducted on subspace clustering aiming at discovering clusters embedded in any possible attributes combination. The main goal of subspace clustering algorithms is to find all clusters in all subspaces. Previous studies have mostly been generating redundant subspace clusters, leading to clustering accuracy loss and also increasing the running time of the algorithms. A bottom-up density-based approach is suggested in this article, in which the cluster structure serves as a similarity measure to generate the optimal subspaces which result in raising the accuracy of the subspace clustering. Based on this idea, the algorithm discovers similar subspaces by considering similarity in their cluster structure, then combines them and the data in the new subspaces would be clustered again. Finally, the algorithm determines all the subspaces and also finds all clusters within them. Experiments on various synthetic and real datasets show that the results of the proposed approach are significantly better in quality and runtime than the state-of-the-art on clustering high-dimensional data.

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Enumerating Maximal k-Plexes with Worst-Case Time Guarantee

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i03.5625 ◽

2020 ◽

Vol 34 (03) ◽

pp. 2442-2449

Author(s):

Yi Zhou ◽

Jingwei Xu ◽

Zhenyu Guo ◽

Mingyu Xiao ◽

Yan Jin

Keyword(s):

Community Detection ◽

Real World ◽

State Of The Art ◽

Real Life ◽

The State ◽

Enumeration Algorithm ◽

Worst Case ◽

Running Time ◽

Real World Applications ◽

First Time

The problem of enumerating all maximal cliques in a graph is a key primitive in a variety of real-world applications such as community detection and so on. However, in practice, communities are rarely formed as cliques due to data noise. Hence, k-plex, a subgraph in which any vertex is adjacent to all but at most k vertices, is introduced as a relaxation of clique. In this paper, we investigate the problem of enumerating all maximal k-plexes and present FaPlexen, an enumeration algorithm which integrates the “pivot” heuristic and new branching schemes. To our best knowledge, for the first time, FaPlexen lists all maximal k-plexes with provably worst-case running time O(n2γn) in a graph with n vertices, where γ < 2. Then, we propose another algorithm CommuPlex which non-trivially extends FaPlexen to find all maximal k-plexes of prescribed size for community detection in massive real-life networks. We finally carry out experiments on both real and synthetic graphs and demonstrate that our algorithms run much faster than the state-of-the-art algorithms.

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A Scalable Scheme for Counting Linear Extensions

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2018/710 ◽

2018 ◽

Author(s):

Topi Talvitie ◽

Kustaa Kangas ◽

Teppo Niinimäki ◽

Mikko Koivisto

Keyword(s):

Artificial Intelligence ◽

Partial Order ◽

Polynomial Time ◽

State Of The Art ◽

Hard Problem ◽

Approximate Counting ◽

Linear Extensions ◽

Exponential Time ◽

Running Time ◽

Cnf Formulas

Counting the linear extensions of a given partial order not only has several applications in artificial intelligence but also represents a hard problem that challenges modern paradigms for approximate counting. Recently, Talvitie et al. (AAAI 2018) showed that an exponential time scheme beats the fastest known polynomial time schemes in practice, even if allowing hours of running time. Here, we present a novel scheme, relaxation Tootsie Pop, which in our experiments exhibits polynomial characteristics and significantly outperforms previous schemes. We also instantiate state-of-the-art model counters for CNF formulas; two natural encodings yield schemes that, however, are inferior to the more specialized schemes.

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Practical Picture Processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100051700 ◽

1974 ◽

Vol 32 ◽

pp. 338-339

Author(s):

T. A. Welton

Keyword(s):

Radiation Damage ◽

Coherence Length ◽

Spatial Information ◽

State Of The Art ◽

Coherent Radiation ◽

The State ◽

Energy Spread ◽

Electron Micrograph ◽

Picture Processing ◽

Molecular Skeleton

Various authors have emphasized the spatial information resident in an electron micrograph taken with adequately coherent radiation. In view of the completion of at least one such instrument, this opportunity is taken to summarize the state of the art of processing such micrographs. We use the usual symbols for the aberration coefficients, and supplement these with £ and 6 for the transverse coherence length and the fractional energy spread respectively. He also assume a weak, biologically interesting sample, with principal interest lying in the molecular skeleton remaining after obvious hydrogen loss and other radiation damage has occurred.

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