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An Empirical Assessment of Algorithms for Constructing a Minimum Spanning Tree
, 1994
"... We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algorit ..."
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Cited by 40 (4 self)
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We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through lowlevel implementation choices, to purely algorithmic issues. We discuss how to design a careful experimental comparison between various alternatives. Finally, we present the results from a study in which we used: multiple languages, compilers, and machines; all the major variants of the comparisonbased algorithms; and eight varieties of graphs in five families, with sizes of up to 0.5 million vertices (in sparse graphs) or 1.3 million edges (in dense graphs).
Towards A Discipline Of Experimental Algorithmics
"... The last 20 years have seen enormous progress in the design of algorithms, but very little of it has been put into practice, even within academia; indeed, the gap between theory and practice has continuously widened over these years. Moreover, many of the recently developed algorithms are very hard ..."
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Cited by 36 (8 self)
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The last 20 years have seen enormous progress in the design of algorithms, but very little of it has been put into practice, even within academia; indeed, the gap between theory and practice has continuously widened over these years. Moreover, many of the recently developed algorithms are very hard to characterize theoretically and, as initially described, suffer from large runningtime coefficients. Thus the algorithms and data structures community needs to return to implementation as the standard of value; we call such an approach Experimental Algorithmics. Experimental Algorithmics studies algorithms and data structures by joining experimental studies with the more traditional theoretical analyses. Experimentation with algorithms and data structures is proving indispensable in the assessment of heuristics for hard problems, in the design of test cases, in the characterization of asymptotic behavior of complex algorithms, in the comparison of competing designs for tractabl...
Algorithms and Experiments: The New (and Old) Methodology
 J. Univ. Comput. Sci
, 2001
"... The last twenty years have seen enormous progress in the design of algorithms, but little of it has been put into practice. Because many recently developed algorithms are hard to characterize theoretically and have large runningtime coefficients, the gap between theory and practice has widened over ..."
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Cited by 9 (4 self)
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The last twenty years have seen enormous progress in the design of algorithms, but little of it has been put into practice. Because many recently developed algorithms are hard to characterize theoretically and have large runningtime coefficients, the gap between theory and practice has widened over these years. Experimentation is indispensable in the assessment of heuristics for hard problems, in the characterization of asymptotic behavior of complex algorithms, and in the comparison of competing designs for tractable problems. Implementation, although perhaps not rigorous experimentation, was characteristic of early work in algorithms and data structures. Donald Knuth has throughout insisted on testing every algorithm and conducting analyses that can predict behavior on actual data; more recently, Jon Bentley has vividly illustrated the difficulty of implementation and the value of testing. Numerical analysts have long understood the need for standardized test suites to ensure robustness, precision and efficiency of numerical libraries. It is only recently, however, that the algorithms community has shown signs of returning to implementation and testing as an integral part of algorithm development. The emerging disciplines of experimental algorithmics and algorithm engineering have revived and are extending many of the approaches used by computing pioneers such as Floyd and Knuth and are placing on a formal basis many of Bentley's observations. We reflect on these issues, looking back at the last thirty years of algorithm development and forward to new challenges: designing cacheaware algorithms, algorithms for mixed models of computation, algorithms for external memory, and algorithms for scientific research.
A Machine Resolution of a FourColor Hoax
"... ring algorithm for maps. But it is reasonable to take the approach of first turning the map into a graph, and then use a coloring algorithm for graphs. 2. The Kempe FourColor Algorithm In 1879 Kempe gave an explicit method of 4coloring planar maps, which I summarize here from the point of view o ..."
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Cited by 6 (0 self)
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ring algorithm for maps. But it is reasonable to take the approach of first turning the map into a graph, and then use a coloring algorithm for graphs. 2. The Kempe FourColor Algorithm In 1879 Kempe gave an explicit method of 4coloring planar maps, which I summarize here from the point of view of planar graphs. Assume a planar graph G is given, with vertices labeled 1 through n; R, G, B, and Y denote the four colors red, green, blue, and yellow. Choose the first vertex in G call it v  having degree 5 or less (it follows quickly from Euler's formulas that every planar graph has such a vertex). Remove it. Color the remaining graph by induction on the vertex set. Then color v as follows: 1. If the neighbors of v have only 3 colors appearing among them, then there is a color left free for v. 2. If there are 4 neighbors of v and all 4 colors appear among them, use a Kempe chain to eliminate one of the colors from the set of neighbors. 3. If there are 5 neighbors of v and all 4 co
DISTRIBUTEDMEMORY PARALLEL ALGORITHMS FOR DISTANCE2 COLORING AND THEIR APPLICATION TO DERIVATIVE COMPUTATION
, 2010
"... The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. ..."
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Cited by 5 (4 self)
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The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. We present efficient, distributedmemory, parallel heuristic algorithms for this NPhard problem as well as for two related problems used in the computation of Jacobians and Hessians. Parallel speedup is achieved through graph partitioning, speculative (iterative) coloring, and a BSPlike organization of parallel computation. Results from experiments conducted on a PC cluster employing up to 96 processors and using largesize realworld as well as synthetically generated test graphs show that the algorithms are scalable. In terms of quality of solution, the algorithms perform remarkably well—the number of colors used by the parallel algorithms was observed to be very close to the number used by the sequential counterparts, which in turn are quite often near optimal. Moreover, the experimental results show that the parallel distance2 coloring algorithm compares favorably with the alternative approach of solving the distance2 coloring problem on a graph G by first constructing the square graph G² and then applying a parallel distance1 coloring algorithm on G2. Implementations of the algorithms are made available via the Zoltan loadbalancing library.
Functional Unit Maps for DataDriven Visualization of HighDensity EEG Coherence
"... Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coh ..."
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Cited by 3 (0 self)
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Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coherence visualizations are generally either hypothesisdriven, or datadriven graph visualizations which are cluttered. In this paper, a new method is presented for datadriven visualization of highdensity EEG coherence, which strongly reduces clutter and is referred to as functional unit (FU) map. Starting from an initial graph, with vertices representing electrodes and edges representing significant coherences between electrode signals, we define an FU as a set of electrodes represented by a clique consisting of spatially connected vertices. In an FU map, the spatial relationship between electrodes is preserved, and all electrodes in one FU are assigned an identical gray value. Adjacent FUs are visualized with different gray values and FUs are connected by a line if the average coherence between FUs exceeds a threshold. Results obtained with our visualization are in accordance with known electrophysiological findings. FU maps can be used as a preprocessing step for conventional analysis.
Kempe equivalence of colorings
, 2005
"... Several basic theorems about the chromatic number of graphs can be extended to results in which, in addition to the existence of a kcoloring, it is also shown that all kcolorings of the graph in question are Kempe equivalent. Here, it is also proved that for a planar graph with chromatic number le ..."
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Cited by 2 (1 self)
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Several basic theorems about the chromatic number of graphs can be extended to results in which, in addition to the existence of a kcoloring, it is also shown that all kcolorings of the graph in question are Kempe equivalent. Here, it is also proved that for a planar graph with chromatic number less than k, allkcolorings are Kempe equivalent. 1
Enumeration of Maximal Planar Graphs with minimum degree Five
"... . In this paper, we give a process to enumerate all maximal planar graphs with minimum degree #ve, with a #xed number of vertices. These graphs are called MPG5 #6#. Wehave used the algorithm #3# to generate these graphs. For the MPG5enumeration wehave used the general method due to Brenda McKay ..."
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. In this paper, we give a process to enumerate all maximal planar graphs with minimum degree #ve, with a #xed number of vertices. These graphs are called MPG5 #6#. Wehave used the algorithm #3# to generate these graphs. For the MPG5enumeration wehave used the general method due to Brenda McKay in #5#; i.e. by computing the orbits set of automorphism group and canonical forms. 1 Introduction We restrict ourselves to undirected, connected and planar graphs G whichhave no loops or multiple edges. Given a simple graph G =#X;E# with node set X and edge set E. A graph G is said to be embeddable on a surface S if it can be drawn on S that its edges intersect only at their end vertices; and planar if it can be embedded on a plane. If we consider a planar graph with no loops or faces, bounded bytwo edges, it may be possible to add a new edge to the given representation of S, such that these properties are preserved. When no such adjunction can be made, we call G maximal planar. A plana...
Functional Unit Maps for DataDriven Visualization of HighDensity EEG Coherence
"... Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coh ..."
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Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coherence visualizations are generally either hypothesisdriven, or datadriven graph visualizations which are cluttered. In this paper, a new method is presented for datadriven visualization of highdensity EEG coherence, which strongly reduces clutter and is referred to as functional unit (FU) map. Starting from an initial graph, with vertices representing electrodes and edges representing significant coherences between electrode signals, we define an FU as a set of electrodes represented by a clique consisting of spatially connected vertices. In an FU map, the spatial relationship between electrodes is preserved, and all electrodes in one FU are assigned an identical gray value. Adjacent FUs are visualized with different gray values and FUs are connected by a line if the average coherence between FUs exceeds a threshold. Results obtained with our visualization are in accordance with known electrophysiological findings. FU maps can be used as a preprocessing step for conventional analysis. Categories and Subject Descriptors (according to ACM CCS):
Four Coloring for a subset of Maximal Planar Graphs with minimum degree five
"... In this paper, we present some results on maximal planar graphs with minimum degree five, denoted by MPG5 graphs [6]. We consider a subset of MPG5 graphs, called the Z graphs, for which all vertices of degree superior to five are not adjacent. We give a vertex four coloring for every Z graph. ..."
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In this paper, we present some results on maximal planar graphs with minimum degree five, denoted by MPG5 graphs [6]. We consider a subset of MPG5 graphs, called the Z graphs, for which all vertices of degree superior to five are not adjacent. We give a vertex four coloring for every Z graph.