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Topic: Sparse graph

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Diagonal matrix

Adjacency matrix

Adjacency list

Graph theory

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	Fast and effective algorithms for graph partitioning and sparse-matrix ordering
		Partitioning the graph of a sparse matrix to minimize the edge cut and distributing different partitions to different processors minimizes the communication overhead in parallel sparse-matrix vector multiplication [12].
		HEM, by absorbing the heavier edges, generates coarse graphs whose nodes are loosely connected (by the lighter remaining edges), thus ensuring that a partition of the coarse graph corresponds to a good partition of the original graph.
		The elimination graph is the graph of the partially factored sparse matrix in which all columns except those corresponding to the separator nodes have been eliminated.
	www.research.ibm.com /journal/rd/411/gupta.html (5952 words)

	Sparse graph code - Wikipedia, the free encyclopedia
		A Sparse graph code is a code which is represented by a sparse graph.
		Any linear code can be represented as a graph, where there are two sets of nodes - a set representing the transmitted bits and another set representing the constraints that the transmitted bits have to satisfy.
		The state of the art classical error-correcting codes are based on sparse graphs, achieving close to the Shannon limit.
	en.wikipedia.org /wiki/Sparse_graph_code (137 words)

	graph
		Formally, a graph is a set of vertices and a binary relation between vertices, adjacency.
		Moreover, a mathematical graph is not a comparison chart, nor a diagram with an x- and y-axis, nor a squiggly line on a stock report.
		GraphEd -- Graph Editor and Layout Program (C), graph manipulation (C++, C, Mathematica, and Pascal), build, traverse, top sort, etc. weighted, directed graphs (Java), JGraphT (Java) build, traverse, and display directed and undirected graphs, GEF - Graph Editing Framework (Java) a library to edit and display graphs.
	www.nist.gov /dads/HTML/graph.html (571 words)

	Project Report
		Sparse graph 2 (128X128): we produce these graphs with random function; the possibility that random two vertices are adjacent is 1/16.
		Sparse graph 2 (64X64): we produce these graphs with random function; the possibility that random two vertices are adjacent is 1/16.
		Sparse graph 2 (32X32): we produce these graphs with random function; the possibility that random two vertices are adjacent is 1/16.
	www.cs.wayne.edu /~lxiscas/csc575/ECE7660/ProjectReport.html (388 words)

	Sparse Matrices -- Recommendations and Resources (Site not responding. Last check: 2007-10-03)
		In the mathematical subfield of numerical analysis a sparse matrix is a matrix populated primarily with zeros.
		Sparse mode multicast is one mode which multicast can use to construct a tree for sending packets to the multicast subscribers.
		Sparse mode is ideal for groups where a very low percentage of the nodes (and their routers) will subscribe to the multicast.
	www.becomingapediatrician.com /health/139/sparse-matrices.html (934 words)

	Graphs (Site not responding. Last check: 2007-10-03)
		A graph G= consists of a set of vertices (also known as nodes) V and a set of edges (also known as arcs) E. An edge connects two vertices u and v; v is said to be adjacent to u.
		Graphs are useful for representing networks and maps of roads, railways, airline routes, pipe systems, telephone lines, electrical connections, prerequisites amongst courses, dependencies amongst tasks in a manufacturing system and a host of other data.
		Note that a rooted tree is a special kind of directed graph and that an unrooted tree is a special kind of undirected graph.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Graph (1162 words)

	The Graph Drawings
		Nine drawings of each graph were created, with the number of bends, crossings and the amount of perceived symmetry varied appropriately (Figures 1 and 2).
		To ensure that the subjects would not recognise that the same graphs were being used for all the drawings, each drawing had its nodes relabeled randomly: the same lettering (A - T) was used in each, but the relationships between the nodes was different.
		For each aesthetic, three drawings of each graph were produced, one with a small aesthetic measurement (few), one with an interim aesthetic measurement (some), and one with a large aesthetic measurement (many).
	www.jea.acm.org /ARTICLES/Vol2Nbr4/node5.html (319 words)

	Graph Definitions
		Connected Graph: A graph in which there is a path from every node to every other node in the graph.
		Dense Graph: Roughly speaking, a graph in which the number of edges is greater than or equal to V lg V, where V is the number of vertices.
		Sparse Graph: Roughly speaking, a graph in which the number of edges is < V lg V, where V is the number of vertices.
	www.cs.utk.edu /~parker/Courses/CS302-fall03/Notes/graph-defs.html (1629 words)

	The Artificial Unger Graph Coloring Applet
		In keeping with this theme, the graph has 49 vertices and are connected so the graph is a map of the continental United States of America (48 States and the District of Columbia).
		A complete graph is defined as one in which each vertex is connected to all the other vertices in the graph.
		To solve the general graph coloring problem with either algorithm, the applet starts with a color palette of n, where n is the number of vertices in the graph.
	www.duke.edu /~jmu2/color/gc.html (1773 words)

	Graph introduction
		For a graph with V vertices, a V x V matrix is used where each element V[i, j] is a Boolean that says whether there is an edge from vertex i to vertex j.
		It is better to use the matrix for dense graphs where the number of edges is close to the number of elements in the matrix and to use an adjacency list when the graph is sparse.
		The goal of a graph partitioner, also called mesh partitioner, is to separate the vertices of a graph into almost equal-sized components such that the number of edges between components is minimized.
	pagebox.net /graph.html (842 words)

	Dense graph - Wikipedia, the free encyclopedia
		In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges.
		The distinction between sparse and dense graphs is rather vague.
		One possibility is to choose a number k with 1 < k < 2 and to define sparse graph to be a graph with
	en.wikipedia.org /wiki/Sparse_graph (205 words)

	Graph Quality
		These graphs were submitted by students as part of assignments and project reports at several universities.
		The graph and its caption should be as self-explanatory as possible without becoming crowded or clotted.
		If your graph is dense with data it will usually require more explanation than a sparse graph, either in writing (in a report or articles) or verbally (in a presentation).
	silver.neep.wisc.edu /~lakes/AbGraph.html (1409 words)

	CS 1501
		The purpose of this assignment is to familiarize you with graphs and graph algorithms, and to demonstrate the differences between the adjacency matrix and adjacency list implementations.
		In the sparse graph, a reasonable approach is to generate V or VlgV random vertex pairs and "add" each edge to the adjacency list and adjacency matrix (making sure the pair has not already been generated – so you may have to try more than the required number of edges).
		Some of the algorithms may run very quickly (especially for sparse graphs) and may be difficult to get good timing results for.
	www.cs.pitt.edu /~kirk/cs1501/assignments/graphs/06-1/list-mat.html (1019 words)

	Construction of Graphs and Digraphs
		Construct the graph G with vertex set V = {@ v_1, v_2,..., v_p @} (where v_i = i for each i if the first form of the constructor is used, or the ith element of the enumerated or indexed set S otherwise), and edge set E = { e_1, e_2,..., e_q }.
		The edges of each graph H of S will be included among the edges of G. An edge, set of edges, or edge-set of a digraph H on p vertices.
		A random graph on p vertices such that the probability that an arbitrary pair of vertices is adjacent is given by the real number r, where r lies in the interval [0, 1].
	www.umich.edu /~gpcc/scs/magma/text1179.htm (1921 words)

	Graph Magics
		Kruskal's alogrithm basically runs in O(NM), and in O(MlogN) with a good implementation of the algorithm (N is the number of nodes and M is the number of edges).
		For sparse graphs a complexity of O(NlogN) may be obtained.
		Providing that the graph is sparse, each node has few neighbors and thus the average complexity is O(NlogN).
	www.graph-magics.com /articles/min_spantree.php (728 words)

	Algorithms in C++ Part 5: Graph Algorithms, 3rd Edition - $38.24 (Site not responding. Last check: 2007-10-03)
		The focus this time is on graph algorithms, which are increasingly critical for a wide range of applications, such as network connectivity, circuit design, scheduling, transaction processing, and resource allocation.
		Graphs and graph algorithms are pervasive in modern computing applications.
		Basic properties of graphs and graph algorithms are developed from first principles, but full understanding often can lead to deep and difficult mathematics.
	www.awprofessional.com /title/0201361183 (2467 words)

	Sparse Graphs
		This has obvious advantage for graphs with a low edge density, in that it allows to construct much larger graphs than if they were to be represented by means of an adjacency matrix (the dense representation).
		Another advantage of the sparse representation is for graph algorithms which are linear in the number of edges.
		Without being exhaustive, we will list here the functions for which the graph's sparse representation remains unchanged; in case of doubt some tools are provided below to help determine what the representation of a graph is. Note that almost all graph representation conversions involve going from a sparse representation to a dense representation.
	www.umich.edu /~gpcc/scs/magma/text1180.htm (487 words)

	1.4.12 Planarity Detection and Embedding (Site not responding. Last check: 2007-10-03)
		Graphs arising in many applications, such as road networks or printed circuit boards, are naturally planar because they are defined by surface structures.
		Planar graphs have a variety of nice properties, which can be exploited to yield faster algorithms for many problems on planar graphs.
		Since every subgraph of a planar graph is planar, this means that there is always a sequence of low-degree vertices whose deletion from G eventually leaves the empty graph.
	www.cs.sunysb.edu /~algorith/files/planar-drawing.shtml (293 words)

	Graphs with a Sparse Representation (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-03)
		As mentioned in the Introduction of this chapter it is possible to construct graphs having a sparse representation.
		This has an obvious advantage for graphs with a low edge density, in that it allows to construct much larger graphs than if they were represented by means of an adjacency matrix (the dense representation).
		Another advantage of the sparse representation is for graph algorithms which are linear in the number of edges (the planarity tester and the triconnectivity tester), and more generally, for those algorithms based on the adjacency list representation (the flow-based algorithms and the shortest-paths algorithms).
	www.math.lsu.edu.cob-web.org:8888 /magma/text1374.htm (692 words)

	SCOTCH: Static Mapping, Graph, Mesh and Hypergraph Partitioning, and Sparse Matrix Ordering Package
		Its purpose is to apply graph theory, with a divide and conquer approach, to scientific computing problems such as graph and mesh partitioning, static mapping, and sparse matrix ordering, in application domains ranging from structural mechanics to operating systems or bio-chemistry.
		The source and target graphs may have any topology, and their vertices and edges may be weighted.
		Its running time is linear in the number of edges of the source graph, and logarithmic in the number of vertices of the target graph for mapping computations.
	www.labri.fr /perso/pelegrin/scotch (945 words)

	sparse graph (Site not responding. Last check: 2007-10-03)
		Definition: A graph in which the number of edges is much less than the possible number of edges.
		See also dense graph, complete graph, adjacency-list representation.
		Note: A directed graph can have at most n(n-1) edges, where n is the number of vertices.
	www.nist.gov /dads/HTML/sparsegraph.html (177 words)

	Sparse vs. Dense Graphs (Site not responding. Last check: 2007-10-03)
		Informally, a graph with relatively few edges is sparse, and a graph with many edges is dense.
		Definition (Sparse Graph) A sparse graph is a graph
		A graph that is not sparse is said to be dense:
	www.brpreiss.com /books/opus4/html/page534.html (123 words)

	AbstractSparseGraph (JUNG 1.7.5 API) (Site not responding. Last check: 2007-10-03)
		It is appropriate for sparse graphs (those in which each vertex has only a few neighbors).
		For dense graphs (those in which each vertex is connected to most other vertices), a different implementation (for example, one which represents a graph via a matrix) may be more appropriate.
		Removes the edge from the graph, and notifies that the edge and its incident vertices that it has been removed.
	jung.sourceforge.net /doc/api/edu/uci/ics/jung/graph/impl/AbstractSparseGraph.html (707 words)

	Encyclopedia of Sparse Graph Codes
		This is a database of sparse graph codes.
		This is Margulis's Cayley graph construction with p=11.
		The graph is called the Petersen graph, so maybe a good name for this code would be the (15,6) Petersen code.
	www.inference.phy.cam.ac.uk /mackay/codes/data.html (1355 words)

	Sparse Graphs
		As mentioned in the Introduction of this chapter it is now possible to construct graphs having a sparse representation.
		Without being exhaustive, we will list here the functions for which the graph's sparse representation needs no internal conversion; in case of doubt some tools are provided below to help determine what the representation of a graph is. Note that almost all graph representation conversions involve going from a sparse representation to a dense representation.
		Note that when conversion of the graph's representation into the alternative one occurs the original representation is not deleted.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1246.htm (522 words)

	Combinatorica (Site not responding. Last check: 2007-10-03)
		Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica is the definitive guide to Combinatorica, perhaps the most widely used software for teaching and research in discrete mathematics.
		Graph Database -- We have assembled a collection of in Combinatorica format for testing conjectures and algorithms.
		The latest release of the package, data bases of interesting graphs, and additional files which may be of interest are available by anonymous FTP from ftp.cs.sunysb.edu.
	www.cs.sunysb.edu /~skiena/combinatorica (962 words)

	Sparse graph code (Site not responding. Last check: 2007-10-03)
		a multi-threaded design (linked list searches, graph based algorithms, even sparse matrix operations...
		A Sparse graph code is a code where the graph that represents it is sparse.
		The state of the art classical error-correcting codes are based on sparse graphs, achieving close to the Nyquist-Shannon limits.
	publicliterature.org /en/wikipedia/s/sp/sparse_graph_code.html (115 words)

	METIS - Family of Multilevel Partitioning Algorithms \| Karypis Lab
		METIS is a family of programs for partitioning unstructured graphs and hypergraphs and computing fill-reducing orderings of sparse matrices.
		The underlying algorithms used by METIS are based on the state-of-the-art multilevel paradigm that has been shown to produce high quality results and scale to very large problems.
		METIS is a set of serial programs for partitioning graphs, partitioning finite element meshes, and producing fill reducing orderings for sparse matrices.
	www-users.cs.umn.edu /~karypis/metis/metis (229 words)

	GraphStep: A System Architecture for Sparse-Graph Algorithms (FCCM 2006) (Site not responding. Last check: 2007-10-03)
		The graph structures are large, and the applications need regular access to a large, data-dependent portion of the graph for each operation (e.g.
		On conventional microprocessors, the graph structures exceed on-chip cache capacities, making main-memory bandwidth and latency the key performance limiters.
		To avoid this ``memory wall,'' we introduce a concurrent system architecture for sparse graph algorithms that places graph nodes in small distributed memories paired with specialized graph processing nodes interconnected by a lightweight network.
	www.cs.caltech.edu /research/ic/abstracts/graphstep_fccm2006.html (262 words)

	Graphs and Representations
		Good for sparse graphs and is generally preferred.
		This is as small as we can imagine if G does not have a special (computable) structure.
		Many other graph variants are easily incorporated into the adjacency-lists representation.
	www.cs.fsu.edu /~cop4531/slideshow/chapter23/23-1.html (220 words)

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