SAMPLING IN DYNAMIC DATA STREAMS AND APPLICATIONS

A dynamic geometric data stream is a sequence of m ADD/REMOVE operations of points from a discrete geometric space {1,…, Δ} d ?. ADD (p) inserts a point p from {1,…, Δ} d into the current point set P , REMOVE(p) deletes p from P . We develop low-storage data structures to (i) maintain ε-nets and ε-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + ε)-approximation of the weight of the Euclidean minimum spanning tree of P . Our data structure for ε-nets uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an e-net for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. Our data structure for ε-approximations uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an ε-approximation for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O ( log (1/δ)( log Δ/ε) O ( d )) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P .

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Board Game Supporting Learning Prim’s Algorithm and Dijkstra’s Algorithm

Methods and Innovations for Multimedia Database Content Management ◽

10.4018/978-1-4666-1791-9.ch015 ◽

2012 ◽

pp. 256-270

Author(s):

Wen-Chih Chang ◽

Te-Hua Wang ◽

Yan-Da Chiu

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Experimental Results ◽

Dijkstra’S Algorithm ◽

Board Game ◽

Tree Algorithm ◽

Dijkstra's Algorithm ◽

Tree Algorithms

The concept of minimum spanning tree algorithms in data structure is difficult for students to learn and to imagine without practice. Usually, learners need to diagram the spanning trees with pen to realize how the minimum spanning tree algorithm works. In this paper, the authors introduce a competitive board game to motivate students to learn the concept of minimum spanning tree algorithms. They discuss the reasons why it is beneficial to combine graph theories and board game for the Dijkstra and Prim minimum spanning tree theories. In the experimental results, this paper demonstrates the board game and examines the learning feedback for the mentioned two graph theories. Advantages summarizing the benefits of combining the graph theories with board game are discussed.

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CLUSTERING WITH MINIMUM SPANNING TREES

International Journal of Artificial Intelligence Tools ◽

10.1142/s0218213011000061 ◽

2011 ◽

Vol 20 (01) ◽

pp. 139-177 ◽

Cited By ~ 8

Author(s):

YAN ZHOU ◽

OLEKSANDR GRYGORASH ◽

THOMAS F. HAIN

Keyword(s):

Spanning Tree ◽

Clustering Algorithm ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Clustering Algorithms ◽

Predefined Criterion ◽

Point Set ◽

Color Clustering ◽

Representative Points ◽

Set Of Points

We propose two Euclidean minimum spanning tree based clustering algorithms — one a k-constrained, and the other an unconstrained algorithm. Our k-constrained clustering algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of a set of representative points and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. Our unconstrained clustering algorithm partitions a point set into a group of clusters by maximally reducing the overall standard deviation of the edges in the Euclidean minimum spanning tree constructed from a given point set, without prescribing the number of clusters. We present our experimental results comparing our proposed algorithms with k-means, X-means, CURE, Chameleon, and the Expectation-Maximization (EM) algorithm on both artificial data and benchmark data from the UCI repository. We also apply our algorithms to image color clustering and compare them with the standard minimum spanning tree clustering algorithm as well as CURE, Chameleon, and X-means.

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THREE TECHNIQUES FOR PARALLEL MAINTENANCE OF A MINIMUM SPANNING TREE UNDER BATCH OF UPDATES

Parallel Processing Letters ◽

10.1142/s0129626496000212 ◽

1996 ◽

Vol 06 (02) ◽

pp. 213-222 ◽

Cited By ~ 3

Author(s):

PAOLO FERRAGINA ◽

FABRIZIO LUCCIO

Keyword(s):

Data Structure ◽

Parallel Algorithms ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Undirected Graph ◽

Batch Size ◽

Insertions And Deletions ◽

Erew Pram ◽

Pram Model ◽

Proper Values

In this paper we provide three simple techniques to maintain in parallel the minimum spanning tree of an undirected graph under single or batch of edge updates (i.e., insertions and deletions). Our results extend the use of the sparsification data structure to the EREW PRAM model. For proper values of the batch size, our algorithms require less time and work than the best known dynamic parallel algorithms.

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Practical innovation about the Minimum Spanning Tree (MST) in data structure teaching

Frontiers in Computer Education ◽

10.1201/b18444-26 ◽

2015 ◽

pp. 113-117

Author(s):

Jing Yi ◽

Sheng'en Li ◽

Xiaobing Tang ◽

Xiaofei Niu

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Minimum Spanning Tree

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Fully Retroactive Minimum Spanning Tree Problem

The Computer Journal ◽

10.1093/comjnl/bxaa135 ◽

2020 ◽

Author(s):

José Wagner de Andrade Júnior ◽

Rodrigo Duarte Seabra

Keyword(s):

Data Structure ◽

Empirical Analysis ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Square Root ◽

Minimum Spanning Tree Problem

Abstract This article describes an algorithm that solves a fully dynamic variant of the minimum spanning tree (MST) problem. The fully retroactive MST allows adding an edge to time $t$, or to obtain the current MST at time $t$. By using the square root technique and a data structure link-cut tree, it was possible to obtain an algorithm that runs each query in $O(\sqrt{m} \lg{|V(G)|})$ amortized, in which $|V(G)|$ is the number of nodes in graph $G$ and $m$ is the size of the timeline. We use a different approach to solve the MST problem instead of the standard algorithms, such as Prim or Kruskal, and this allows using the square root technique to improve the final complexity of the algorithm. Our empirical analysis shows that the proposed algorithm runs faster than re-executing the standard algorithms, and this difference only increases when the number of nodes in these graphs is larger.

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Board Game Supporting Learning Prim’s Algorithm and Dijkstra’s Algorithm

International Journal of Multimedia Data Engineering and Management ◽

10.4018/jmdem.2010100102 ◽

2010 ◽

Vol 1 (4) ◽

pp. 16-30

Author(s):

Wen-Chih Chang ◽

Te-Hua Wang ◽

Yan-Da Chiu

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Experimental Results ◽

Dijkstra’S Algorithm ◽

Board Game ◽

Tree Algorithm ◽

Dijkstra's Algorithm ◽

Tree Algorithms

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AN EREW PARM ALGORITHM FOR UPDATING MINIMUM SPANNING TREES

Parallel Processing Letters ◽

10.1142/s012962649900013x ◽

1999 ◽

Vol 09 (01) ◽

pp. 111-122 ◽

Cited By ~ 2

Author(s):

SAJAL K. DAS ◽

PAOLO FERRAGINA

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Undirected Graph ◽

Substantial Improvement ◽

Minimum Spanning Trees ◽

Single Edge ◽

Sequential Approach ◽

Work Complexity

We propose a parallel algorithm for the EREW PRAW model that maintains a minimum spanning tree (MST) of an undirected graph under single edge insertions and deletions. For a graph of n nodes and m edges, each update requires O( log n) time and O(m 2/3 log n) work. This is a substantial improvement over the known bounds on the work complexity. Our algorithm uses a partition of the MST, similar to the sequential approach due to Frederickson [6], and also employs a novel data structure for efficiently managing edge insertions in parallel.

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