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On Approximately Counting Colourings of Small Degree Graphs
 SIAM Journal on Computing
, 1998
"... We consider approximate counting of colourings of an nvertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colourings would mix rapidly provided the number of colours, k, exceeded the maximum degree \Delta of the grap ..."
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Cited by 14 (6 self)
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We consider approximate counting of colourings of an nvertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colourings would mix rapidly provided the number of colours, k, exceeded the maximum degree \Delta of the graph by a factor of at least 2. Lack of improvements on this bound led to the conjecture that k 2\Delta was a natural barrier. We disprove this conjecture in the simplest case of 5colouring graphs of maximum degree 3. Our proof involves a computerassisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colourings using the method of path coupling. We outline an extension to 7colourings of trianglefree 4regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikly to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper kcolo...
Beating the 2∆ Bound for Approximately Counting Colourings: A ComputerAssisted Proof of Rapid Mixing
 IN 9TH ANNUAL SYMPOSIUM ON DISCRETE ALGORITHMS, ACMSIAM
, 1998
"... We consider random walks on graph colourings of an nvertex graph. It has been shown by Jerrum and by Salas and Sokal that a simple random walk would mix rapidly provided the number of colours, k, exceeded the maximum degree ∆ of the graph by a factor of at least 2. Lack of improvements on this boun ..."
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Cited by 13 (3 self)
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We consider random walks on graph colourings of an nvertex graph. It has been shown by Jerrum and by Salas and Sokal that a simple random walk would mix rapidly provided the number of colours, k, exceeded the maximum degree ∆ of the graph by a factor of at least 2. Lack of improvements on this bound led to a conjecture that k 2∆ was a natural barrier. We disprove this conjecture in the simplest case of 5colouring graphs of maximum degree 3. Our proof involves a novel computerassisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colourings using the method of path coupling. We outline an extension to 7colourings of trianglefree 4regular graphs.
On Approximately Counting Colorings Of Small Degree Graphs
 SIAM Journal on Computing
, 1999
"... . We consider approximate counting of colorings of an nvertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree # of the graph by a ..."
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Cited by 5 (1 self)
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. We consider approximate counting of colorings of an nvertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree # of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5coloring graphs of maximum degree 3. Our proof involves a computerassisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colorings using the method of path coupling. We outline an extension to 7colorings of trianglefree 4regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikely to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper kcolorings of graphs with maximum degree # is #Pcomplete whe...
Accurate and Precise Computation using Analog VLSI, with Applications to Computer Graphics and Neural Networks
, 1993
"... This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI ..."
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Cited by 3 (1 self)
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This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI. The nature of the design methodology focuses on defining goals for circuit behavior to be met as part of the design process. To increase the accuracy of analog computation, we develop techniques for creating compensated circuit building blocks, where compensation implies the cancellation of device variations, offsets, and nonlinearities. These compensated building blocks can be used as components in larger and more complex circuits, which can then also be compensated. To this end, we develop techniques for automatically determining appropriate parameters for circuits, using constrained optimization. We also fabricate circuits that implement multidimensional gradient estimation for a grad...
CrossEntropy Randomized Motion Planning
"... Abstract—This paper is concerned with motion planning for nonlinear robotic systems operating in constrained environments. Motivated by recent developments in samplingbased motion planning and Monte Carlo optimization we propose a general randomized path planning method based on sampling in the spa ..."
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Cited by 2 (0 self)
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Abstract—This paper is concerned with motion planning for nonlinear robotic systems operating in constrained environments. Motivated by recent developments in samplingbased motion planning and Monte Carlo optimization we propose a general randomized path planning method based on sampling in the space of trajectories. The idea is to construct a probability distribution over the set of feasible paths and to perform the search for an optimal trajectory through importance sampling. At the core of the approach lies the crossentropy method for estimation of rareevent probabilities. The algorithm recursively approximates the optimal sampling distribution which guides the set of sampled trajectories towards regions of progressively lower cost until converging to a delta distribution at the optimum. Our main goal is to provide a framework for consistent adaptive sampling correlating the spatial structure of trajectories and their computed costs. The approach is illustrated with two simple examples–a point mass vehicle and the Dubins car, and is then applied to a simulated helicopter flying optimally in a 3D terrain. I.
Ludmila Alaverdian
"... this paper, we describe our development of a novel family of capillary DNA sequencing instruments carried out during the last five years at SUNY SB and Biophotonics Corporation and supported by the NIH (NHGRI, NCI). Our goal has been the implementation of capillary DNA sequencing instruments with ul ..."
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this paper, we describe our development of a novel family of capillary DNA sequencing instruments carried out during the last five years at SUNY SB and Biophotonics Corporation and supported by the NIH (NHGRI, NCI). Our goal has been the implementation of capillary DNA sequencing instruments with ultrahigh sensitivity, large dynamic range, and a convenient modular architecture. The enhanced performance of our instruments is based on two technical innovations: the multilaser excitation of fluorescence of labeled DNA fragments and the colorblind detection based on the singlephoton counting technique. We shall describe the design and the results of performance testing for several versions of ultrasensitive singlecapillary DNA sequencing instruments employing singlephoton detection. We shall also discuss our approach to extending the same operational principles to multilane instruments