Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering

Wang, Yiqiu; Yu, Shangdi; Gu, Yan; Shun, Julian

doi:10.1145/3448016.3457296

Cited by 26 publications

(19 citation statements)

References 50 publications

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“…Other practical cases based on MST include COVID-19 pandemic transmission forecasting [141], clustering [142], constructing trees for broadcasting [143], image registration and segmentation [144], circuit design [145] and emotion recognition [146]. There do exist several primary algorithms, namely, classic algorithm and faster algorithm; however, in consideration that such MST-like models will facilitate the development of multifarious industrial areas, more and more individuals have turned to design a variety of intuitive algorithms to figure out such issues, such as reinforcement learning [147], genetic algorithm [148] and fast parallel algorithm [149].…”

Section: Minimum Spanning Treementioning

confidence: 99%

A Review: Machine Learning for Combinatorial Optimization Problems in Energy Areas

et al. 2022

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Combinatorial optimization problems (COPs) are a class of NP-hard problems with great practical significance. Traditional approaches for COPs suffer from high computational time and reliance on expert knowledge, and machine learning (ML) methods, as powerful tools have been used to overcome these problems. In this review, the COPs in energy areas with a series of modern ML approaches, i.e., the interdisciplinary areas of COPs, ML and energy areas, are mainly investigated. Recent works on solving COPs using ML are sorted out firstly by methods which include supervised learning (SL), deep learning (DL), reinforcement learning (RL) and recently proposed game theoretic methods, and then problems where the timeline of the improvements for some fundamental COPs is the layout. Practical applications of ML methods in the energy areas, including the petroleum supply chain, steel-making, electric power system and wind power, are summarized for the first time, and challenges in this field are analyzed.

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Section: Minimum Spanning Treementioning

confidence: 99%

A Review: Machine Learning for Combinatorial Optimization Problems in Energy Areas

et al. 2022

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“…, where > 0 is fixed and MST(R) is the exact minimum spanning tree [3], [87]. The most recent approaches [88] focus on developing parallel algorithms for the construction of MST(R).…”

Section: Minimum Spanning Tree For Any Finite Metric Spacementioning

confidence: 99%

A new compressed cover tree for k-nearest neighbour search and the stable-under-noise mergegram of a point cloud

Yury¹

2022

Preprint

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This thesis consists of two topics related to computational geometry and one topic related to topological data analysis (TDA), which combines fields of computational geometry and algebraic topology for analyzing data.The first part studies the classical problem of finding k nearest neighbors to m query points in a larger set of n reference points in any metric space. The well-known work by Beygelzimer, Kakade, and Langford in ICML 2006 introduced cover trees and claimed to guarantee a nearlinear time complexity in the number n of reference points. Section 5.3 of Curtin's PhD (2015) pointed out that the proof of this result was wrong. The key step of the original proof attempted to show that the number of iterations can be estimated by multiplying the length of the longest rootto-leaf path in a cover tree by a constant factor. However, this estimate can miss many potential nodes in several branches of a cover tree, that should be considered during the search. The same erroneous argument was unfortunately repeated in several subsequent papers. This chapter gives formal counterexamples to the time complexity estimates of the cover tree construction algorithm and the nearest neighbors search in the past work. We prove correct analogs of the past claims with slightly better bounds for any number k ≥ 1 of neighbors. A new compressed cover tree guarantees a parameterized time complexity that is near-linear in the maximum size of both query and reference set.The second part is about the construction of a Minimum Spanning Tree (MST) on any finite metric space. This is an efficient way to visualize any unstructured data given only by distances, for example by any metric graph connecting abstract data points. Though many efficient algorithms for MST were developed, there was only one past attempt to justify a nearlinear time complexity in the size of a given metric space. In 2010 March, Ram, and Gray claimed that MST of any finite metric space can be constructed in a parametrized near-linear time. However, in this work we have discovered a counterexample showing that the provided proof was incorrect. The past obstacles are now resolved by the new concepts of a compressed cover tree and a minimized expansion constant. The main contribution is the first MST algorithm that correctly justifies a parametrized a near-linear time complexity.The third part extends the key concept of persistence within Topological Data Analysis in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case, the distance-based persistence is determined by a i First of all, I would like to thank my primary supervisor, Dr. Vitaliy Kurlin, for giving me the opportunity to pursue a Ph.D. in a field about which I am very passionate and for always supporting me during these years. His incredible experience has been inspiring to me. I am grateful for all our fruitful conversations that helped me progress in my research.I would also like to thank my second supervisor, Dr. Marja K...

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“…Besides the linkage criteria considered in this paper, other popular criteria for HAC include single, centroid, and median linkage. Single linkage with the Eulidean metric is closely related to the Euclidean minimum spanning tree problem, and can be solved efficiently using variants of Boruvka's algorithm for minimum spanning tree [49,67]. Centroid and median linkage do not satisfy the reducibility property and cannot take advantage of the NNC algorithm.…”

Section: Related Workmentioning

confidence: 99%

“…Unfortunately, exact HAC algorithms usually require Ω(𝑛 2 ) work, since the distances between all pairs of points have to be computed. To accelerate exact HAC algorithms due to their significant computational cost, there have been several parallel exact HAC algorithms proposed in the literature [25,33,35,43,44,59,67,70], but most of them maintain a distance matrix, which requires quadratic memory, making them unscalable to large data sets. The only parallel exact algorithm that works for the metrics that we consider and uses subquadratic space is by Zhang et al [70], but it has not been shown to scale to large data sets.…”

Section: Introductionmentioning

confidence: 99%

ParChain: A Framework for Parallel Hierarchical Agglomerative Clustering using Nearest-Neighbor Chain

Wang

et al. 2021

Preprint

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This paper studies the hierarchical clustering problem, where the goal is to produce a dendrogram that represents clusters at varying scales of a data set. We propose the ParChain framework for designing parallel hierarchical agglomerative clustering (HAC) algorithms, and using the framework we obtain novel parallel algorithms for the complete linkage, average linkage, and Ward's linkage criteria. Compared to most previous parallel HAC algorithms, which require quadratic memory, our new algorithms require only linear memory, and are scalable to large data sets. ParChain is based on our parallelization of the nearest-neighbor chain algorithm, and enables multiple clusters to be merged on every round. We introduce two key optimizations that are critical for efficiency: a range query optimization that reduces the number of distance computations required when finding nearest neighbors of clusters, and a caching optimization that stores a subset of previously computed distances, which are likely to be reused.Experimentally, we show that our highly-optimized implementations using 48 cores with two-way hyper-threading achieve 5.8-110.1x speedup over state-of-the-art parallel HAC algorithms and achieve 13.75-54.23x self-relative speedup. Compared to state-ofthe-art algorithms, our algorithms require up to 237.3x less space. Our algorithms are able to scale to data set sizes with tens of millions of points, which existing algorithms are not able to handle.

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Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering

Cited by 26 publications

References 50 publications

A Review: Machine Learning for Combinatorial Optimization Problems in Energy Areas

A Review: Machine Learning for Combinatorial Optimization Problems in Energy Areas

A new compressed cover tree for k-nearest neighbour search and the stable-under-noise mergegram of a point cloud

ParChain: A Framework for Parallel Hierarchical Agglomerative Clustering using Nearest-Neighbor Chain

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