A Parallel Algorithm for Planar Orthogonal Grid Drawings

2000 ◽  
Vol 10 (01) ◽  
pp. 141-150
Author(s):  
ROBERTO TAMASSIA ◽  
IOANNIS G. TOLLIS ◽  
JEFFREY SCOTT VITTER
Keyword(s):  
Graph Drawing ◽  
Edge Length ◽  
Optimal Time ◽  
Worst Case ◽  
Vlsi Layout ◽  
Orthogonal Grid ◽  
Graph Layouts ◽  
Crew Pram ◽  
Number Of Bends ◽  
Simply Connected

In this paper we consider the problem of constructing planar orthogonal grid drawings (or more simply, layouts) of graphs, with the goal of minimizing the number of bends along the edges. We present optimal parallel algorithms that construct graph layouts with O(n) maximum edge length, O(n2) area, and at most 2n+4 bends (for biconnected graphs) and 2.4n+2 bends (for simply connected graphs). All three of these quality measures for the layouts are optimal in the worst case for biconnected graphs. The algorithm runs on a CREW PRAM in O( log n) time with n/ log n processors, thus achieving optimal time and processor utilization. Applications include VLSI layout, graph drawing, and wireless communication.

LAYOUT OF AN ARBITRARY PERMUTATION IN A MINIMAL RIGHT TRIANGLE AREA

2007 ◽  
Vol 08 (02) ◽  
pp. 101-118
Author(s):  
MARIA ARTISHCHEV-ZAPOLOTSKY ◽  
YEFIM DINITZ ◽  
SHIMON EVEN ◽  
VLADIMIR YANOVSKY
Keyword(s):  
Interconnection Networks ◽  
Minimum Area ◽  
Worst Case ◽  
Perpendicular Line ◽  
Restricted Area ◽  
Vlsi Layout ◽  
Arbitrary Permutation ◽  
Layout Area ◽  
Number Of Bends ◽  
Optimal Area

In VLSI layout of interconnection networks, routing two-point nets in some restricted area is one of the central operations. The main aim is usually minimization of the layout area, while reducing the number of wire bends is also very useful. In this paper, we consider connecting a set of N inputs on a line to a set of N outputs on a perpendicular line inside a right triangle shaped area, where the order of the outputs is a given permutation of the order of the corresponding inputs. Such triangles were used, for example, by Dinitz, Even, and Artishchev-Zapolotsky for an optimal layout of the Butterfly network. That layout was of a particular permutation, while here we solve the problem for an arbitrary permutation case. We show two layouts in an optimal area of ½ N2 + o(N2), with O (N) bends each. We prove that the first layout requires the absolutely minimum area and yields the irreducible number of bends, while containing knock-knees. The second one eliminates knock-knees, still keeping a constant number, 7, of bends per connection. As well, we prove a lower bound of 3N - o(N) for the number of bends in the worst case layout in an optimal area of ½ N2 + o(N2).

On Parallel Methods of Multibody Dynamics

2003 ◽  
Author(s):  
James H. Critchley ◽  
Kurt S. Anderson
Keyword(s):  
Time Complexity ◽  
Multibody System ◽  
Practical Importance ◽  
Divide And Conquer ◽  
Optimal Time ◽  
Worst Case ◽  
Parallel Methods ◽  
Divide And Conquer Algorithm ◽  
Coordinate Reduction ◽  
Parallel Arrays

Optimal time efficient parallel computation methods for large multibody system dynamics are defined and investigated in detail. Comparative observations are made which demonstrate significant deficiencies in operating regions of practical importance and a new parallel algorithm is generated to address them. The new method of Recursive Coordinate Reduction Parallelism (RCRP) outperforms or directly reduces to the fastest general multibody algorithms available for small parallel resources and obtains “O(logk(n))” time complexity in the presence of larger parallel arrays. Performance of this method relative to the Divide and Conquer Algorithm is illustrated with an operations count for the worst case of a multibody chain system.

scholarly journals An Information-Theoretic Framework for Evaluating Edge Bundling Visualization

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 625 ◽  
Author(s):  
Jieting Wu ◽  
Feiyu Zhu ◽  
Xin Liu ◽  
Hongfeng Yu
Keyword(s):  
Information Theory ◽  
Graph Drawing ◽  
Evaluation Framework ◽  
Graph Visualization ◽  
Large Graphs ◽  
Information Theoretic ◽  
Underlying Network ◽  
Graph Layouts ◽  
Edge Bundling ◽  
Visual Results

Edge bundling is a promising graph visualization approach to simplifying the visual result of a graph drawing. Plenty of edge bundling methods have been developed to generate diverse graph layouts. However, it is difficult to defend an edge bundling method with its resulting layout against other edge bundling methods as a clear theoretic evaluation framework is absent in the literature. In this paper, we propose an information-theoretic framework to evaluate the visual results of edge bundling techniques. We first illustrate the advantage of edge bundling visualizations for large graphs, and pinpoint the ambiguity resulting from drawing results. Second, we define and quantify the amount of information delivered by edge bundling visualization from the underlying network using information theory. Third, we propose a new algorithm to evaluate the resulting layouts of edge bundling using the amount of the mutual information between a raw network dataset and its edge bundling visualization. Comparison examples based on the proposed framework between different edge bundling techniques are presented.

scholarly journals Applying Graph Centrality Metrics in Visual Analytics of Scientific Standard Datasets

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 30 ◽  
Author(s):  
Jie Hua ◽  
Mao Lin Huang ◽  
Weidong Huang ◽  
Chenglin Zhao
Keyword(s):  
Visual Analytics ◽  
Graph Drawing ◽  
Graph Centrality ◽  
Standard Data ◽  
Link Diagrams ◽  
Centrality Metrics ◽  
Scientific Standard ◽  
Graph Layouts ◽  
Importance Ranking ◽  
Scientific Standards

Graphs are often used to model data with a relational structure and graphs are usually visualised into node-link diagrams for a better understanding of the underlying data. Node-link diagrams represent not only data entries in a graph, but also the relations among the data entries. Further, many graph drawing algorithms and graph centrality metrics have been successfully applied in visual analytics of various graph datasets, yet little attention has been paid to analytics of scientific standard data. This study attempts to adopt graph drawing methods (force-directed algorithms) to visualise scientific standard data and provide information with importance ‘ranking’ based on graph centrality metrics such as Weighted Degree, PageRank, Eigenvector, Betweenness and Closeness factors. The outcomes show that our method can produce clear graph layouts of scientific standard for visual analytics, along with the importance ‘ranking’ factors (represent via node colour, size etc.). Our method may assist users with tracking various relationships while understanding scientific standards with fewer relation issues (missing/wrong connection etc.) through focusing on higher priority standards.

scholarly journals Fixed edge-length graph drawing is NP-hard

1990 ◽  
Vol 28 (2) ◽  
pp. 111-134 ◽  
Author(s):  
Peter Eades ◽  
Nicholas C. Wormald
Keyword(s):  
Graph Drawing ◽  
Edge Length ◽  
Np Hard

Machine Learning Prediction and Reliability Analysis Applied to Subsea Spool and Jumper Design

2021 ◽  
Author(s):  
Mengdi Song ◽  
Massyl Gheroufella ◽  
Paul Chartier
Keyword(s):  
Machine Learning ◽  
Reliability Analysis ◽  
Design Process ◽  
Predictive Modelling ◽  
Machine Learning Algorithms ◽  
Case Scenario ◽  
Worst Case ◽  
Deterministic Solution ◽  
Number Of Bends ◽  
Subsea Pipelines

Abstract In subsea pipelines projects, the design of rigid spool and jumper can be a challenging and time-consuming task. The selected spool layout for connecting the pipelines to the subsea structures, including the number of bends and leg lengths, must offer the flexibility to accommodate the pipeline thermal expansion, the pipe-lay target box and misalignments associated with the post-lay survey metrology and spool fabrication. The analysis results are considerably affected by many uncertainties involved. Consequently, a very large amount of calculations is required to assess the full combination of uncertainties and to capture the worst-case scenario. Rather than applying the deterministic solution, this paper uses machine learning prediction to significantly improve the efficiency of the design process. In addition, thanks to the fast predictive model using machine learning algorithms, the uncertainty quantification and propagation analysis using probabilistic statistical method becomes feasible in terms of CPU time and can be incorporated into the design process to evaluate the reliability of the outputs. The latter allows us to perform a systematic probabilistic design by considering a certain level of acceptance on the probability of failure, for example as per DNVGL design code. The machine learning predictive modelling and the reliability analysis based upon the probability distribution of the uncertainties are introduced and explained in this paper. Some project examples are shown to highlight the method’s comprehensive nature and efficient characteristics.

EFFICIENT CONSTRUCTION OF LOW WEIGHTED BOUNDED DEGREE PLANAR SPANNER

2004 ◽  
Vol 14 (01n02) ◽  
pp. 69-84 ◽  
Author(s):  
XIANG-YANG LI ◽  
YU WANG
Keyword(s):  
Minimum Spanning Tree ◽  
Edge Length ◽  
Adjustable Parameter ◽  
Positive Real ◽  
Worst Case ◽  
Bounded Degree ◽  
Localized Algorithm ◽  
Efficient Construction ◽  
Centralized Algorithm ◽  
Degree Bound

Given a set V of n points in a two-dimensional plane, we give an O(n log n)-time centralized algorithm that constructs a planar t-spanner for V, for [Formula: see text] such that the degree of each node is bounded from above by [Formula: see text], where 0<α<π/2 is an adjustable parameter. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most [Formula: see text]. We also show, by applying the greedy method in Ref. [14], how to construct a low weighted bounded degree planar spanner with spanning ratio ρ(α)2(1+∊) and the same degree bound, where ∊ is any positive real constant. Here, a structure is called low weighted if its total edge length is proportional to the total edge length of the Euclidean minimum spanning tree of V. Moreover, we show that our method can be extended to construct a planar bounded degree spanner for unit disk graphs with the adjustable parameter α satisfying 0<α<π/3. Previously, only centralized method6 of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t≃10.02. The distributed implementation of this centralized method takes O(n2) communications in the worst case. Our method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n).

scholarly journals Rotation Algorithms for Balancing Search Trees

1970 ◽  
Vol 10 (1) ◽  
pp. 26-32
Author(s):  
Chumphol Bunkhumpornpat ◽  
Varin Chouvatut ◽  
Saowaluk Rattanaudomsawat
Keyword(s):  
Data Structure ◽  
Hierarchical Structure ◽  
Time Complexity ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Search Tree ◽  
Optimal Time ◽  
Worst Case ◽  
Cluster Membership ◽  
The Hierarchical Structure

A search tree is a data structure constructed from a minimum spanning tree. This data structure is used for determining the cluster membership of a query instance clustered by a similarity-guaranteed clustering algorithm. According to the line topology of a search tree in the worst case, the time complexity of tree traversing is O(n) where n is the number of nodes of the tree. Unfortunately, the AVL tree algorithm cannot solve this problem because the algorithm is unable to maintain the hierarchical structure of a search tree. From the definition of balance factor, our proposed algorithm is designed to rotate nodes until the tree becomes balanced while maintaining the hierarchical structure. Consequently, the balanced search tree achieves the optimal time complexity of O(log n) for the searching purpose.

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