Topic

# Vertex (geometry)

About: Vertex (geometry) is a(n) research topic. Over the lifetime, 18765 publication(s) have been published within this topic receiving 294216 citation(s). The topic is also known as: 0-polytope & 0-simplex.

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TL;DR: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point.

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Abstract: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.

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25,414 citations

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Grzegorz Malewicz, Matthew H. Austern

^{1}, Aart J. C. Bik^{1}, James C. Dehnert^{1}+3 more•Institutions (1)06 Jun 2010-

TL;DR: A model for processing large graphs that has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier.

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Abstract: Many practical computing problems concern large graphs. Standard examples include the Web graph and various social networks. The scale of these graphs - in some cases billions of vertices, trillions of edges - poses challenges to their efficient processing. In this paper we present a computational model suitable for this task. Programs are expressed as a sequence of iterations, in each of which a vertex can receive messages sent in the previous iteration, send messages to other vertices, and modify its own state and that of its outgoing edges or mutate graph topology. This vertex-centric approach is flexible enough to express a broad set of algorithms. The model has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier. Distribution-related details are hidden behind an abstract API. The result is a framework for processing large graphs that is expressive and easy to program.

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3,556 citations

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26 Oct 2005-

Abstract: Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important advantages: it captures well the community structure in a network, it can be computed efficiently, it works at various scales, and it can be used in an agglomerative algorithm to compute efficiently the community structure of a network. We propose such an algorithm which runs in time O(mn2) and space O(n2) in the worst case, and in time O(n2log n) and space O(n2) in most real-world cases (n and m are respectively the number of vertices and edges in the input graph).

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2,070 citations

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TL;DR: This paper presents and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, they find ak-way partitioning of the smaller graph, and then they uncoarsen and refine it to construct ak- way partitioning for the original graph.

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Abstract: In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find ak-way partitioning of the smaller graph, and then we uncoarsen and refine it to construct ak-way partitioning for the original graph. These algorithms compute ak-way partitioning of a graphG= (V,E) inO(|E|) time, which is faster by a factor ofO(logk) than previously proposed multilevel recursive bisection algorithms. A key contribution of our work is in finding a high-quality and computationally inexpensive refinement algorithm that can improve upon an initialk-way partitioning. We also study the effectiveness of the overall scheme for a variety of coarsening schemes. We present experimental results on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that this new scheme produces partitions that are of comparable or better quality than those produced by the multilevel bisection algorithm and requires substantially smaller time. Graphs containing up to 450,000 vertices and 3,300,000 edges can be partitioned in 256 domains in less than 40 s on a workstation such as SGI's Challenge. Compared with the widely used multilevel spectral bisection algorithm, our new algorithm is usually two orders of magnitude faster and produces partitions with substantially smaller edge-cut.

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1,619 citations

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Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

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1,283 citations