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Topic: Colouring algorithm

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	Colouring algorithm (Site not responding. Last check: 2007-10-07)
		In computer science, the colouring algorithm is a simple look-up to virtual storage containing another stream to be parse d, to generate a function which acts on the original input.
		This is sometimes called "bootstrapping." A variant of the coloring algorithm is known as the graph coloring algorithm.
		The LLL Algorithm Papers on the LLL algorithm and its applications collected by François Koeune.
	www.serebella.com /encyclopedia/article-Colouring_algorithm.html (465 words)

	Sudoku Programmers :: View topic - Confused over Colouring
		Look at the contradiction in cells (3,1) and (8,2): the colour map for 6 says that one of them must be 6 but the colour map for 1 says that if one of them is 1 then they must both be 1.
		If one of the cells is already coloured, but the other is not, then the uncoloured cell is assigned the conjugate colour to the coloured one.
		If there are exactly two coloured cells in a row/column/box which are not already conjugates and their conjugates are the onlt two coloured cells in some other row/column/box then it is valid to merge the two colour pairs into one, again recolouring any cells from the discarded colour pair.
	www.setbb.com /phpbb/viewtopic.php?t=67&postdays=0&postorder=asc&start=15&mforum=sudoku (3134 words)

	colouring - Hutchinson encyclopedia article about colouring (Site not responding. Last check: 2007-10-07)
		Food additive used to alter or improve the colour of processed foods.
		Colourings include artificial colours, such as tartrazine and amaranth, which are made from petrochemicals, and the ‘natural’ colours such as chlorophyll, caramel, and carotene.
		Some of the natural colours are actually synthetic copies of the naturally occurring substances, and some of these, notably the synthetically produced caramels, may be injurious to health.
	encyclopedia.farlex.com /colouring (218 words)

	29a (Site not responding. Last check: 2007-10-07)
		Such an algorithm might be simpler than an algorithm for recognizing whether or not a graph is perfect, in view of precedents, and since what it would do is incomparable with perfect-graph recognition.
		This is an improvement on the non-robust algorithms of Hoang and Hertz which, assuming a graph is Meyniel, find a clique and colouring the same size.
		Edmonds' algorithm is much simpler than the Burlet-Fonlupt decomposition algorithm for recognizing Meyniel graphs, which was motivated by an interest in optimizing in Meyniel graphs, and which is used by Hoang and Hertz.
	www.aimath.org /WWN/perfectgraph/articles/html/29a (403 words)

	Graph Colouring Algorithm
		If it is not coloured, we colour it green and call ourselves recursively on that green vertex, swapping our primary colour (red) and secondary colour (green).
		If the vertex is already coloured, however, we add that edge to a queue of conflicted edges and try another edge which connects to our red vertex.
		Consider the result of the spanning tree algorithm on a square: you end up with a "U" shape, and the 1 conflicted edge is between a red vertex and a green vertex.
	www.cs.pdx.edu /~postj/graph/graph.html (629 words)

	No Title (Site not responding. Last check: 2007-10-07)
		Colouring a graph means assigning a colour to each vertex of the graph.
		For each one, confirm that the colouring the algorithm finds is proper, by checking that the colours on the two ends of each edge are different.
		Our ``colours'' will be the integers 0, 1, plus the ``special colour'' -1, which indicates that a vertex has not been assigned a colour.
	www.scar.utoronto.ca /~mendel/a58/98s/assignments/a2.html (2602 words)

	Citations: Iterated Greedy Graph Coloring and the Difficulty Landscape - Culberson (ResearchIndex)
		Given any k colouring of a graph, if the vertices are reordered so that vertices in each colour class are contiguous then it is trivial to prove that using the greedy procedure on this new ordering will result in another colouring with no....
		Turner reported the no choice algorithm and Brelaz s algorithm can almost find the optimal graph coloring for almost all random graphs [91] However, the no choice algorithm requires prior knowledge of the k colorability of the graph, and the chromatic number k is one of the inputs to the....
		This algorithm falls into the class of neighborhood search algorithms[20] which include TABU[9] and simulated annealing[11] However, the neighborhoods are different here, in that instead of forming a new partition by the....
	citeseer.ist.psu.edu /context/240980/229361 (2712 words)

	Mike Molloy's papers
		Colouring graphs when the number of colours is almost the maximum degree.
		Colouring Graphs whose Chromatic Number is Almost Their Maximum Degree.
		The Glauber dynamics on the colourings of a graph with large girth and maximum degree.
	www.cs.toronto.edu /%7Emolloy/webpapers/papers.html (493 words)

	[No title]
		We know linear time colouring algorithm for cographs, given their cotree (which can be constructed efficiently from graph).
		Colour this cograph optimally, using cograph colouring algorithm.
		Want algorithms that are "robust": whose performance degrades slowly for inputs close to restricted family (instead of completely breaking down for inputs outside the family).
	www.cs.toronto.edu /~fpitt/CSC366/20031/heuristics.txt (650 words)

	vbAccelerator - Optimised Colour Reduction Using Octrees
		Describes the working of the Octree colour quantisation algorithm be used to create optimised colour palettes of arbitrary depth (although typically used for 256 colours) from any image and compares the results with other colour reduction methods.
		Each object in the tree is set up so as well as storing the links to the next objects in the tree, it can also store a summation of red, green and blue values as well as the number of pixels the sum has been done over.
		Implementing It in VB Basically implementing this algorithm is something of a bitch, as is often the case with any types of Linked-List or Tree algorithm structure in VB.
	www.vbaccelerator.com /home/VB/Code/vbMedia/Image_Processing/Optimised_Colour_Reduction_using_Octrees/article.asp (645 words)

	Τμήμα Μηχ. Η/Υ & Πληροφορικής - Σεμινάρια
		The \textit{Frequencies Assignment Problem} on a wireless radio network is well formalized by variations of the \textit{vertex colouring problem} on the interference graph of this network, let us denote it by $G$.
		An interesting such variation is the colouring of the \textit{square} of $G$ (denoted by $G^2$).
		We provide a new colouring algorithm, called $MDsatur$, which colours any $SQP$ graph $G^2$ using at most $1.5\D(G) + c$ colours, where $c$ is a constant.
	www.ceid.upatras.gr /seminars/19_11_02.htm (186 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Complexity of an algorithm is a measure of the amount of time and/or space required by an algorithm for an input of a given size (n).
		In analysing an algorithm, rather than a piece of code, we will try and predict the number of times "the principle activity" of that algorithm is performed.
		If it is a graph colouring algorithm we might count the number of times we check that a coloured node is compatible with its neighbours.
	www.dcs.gla.ac.uk /~pat/52233/complexity.html (1198 words)

	Colloquium - Univ. of MT (Site not responding. Last check: 2007-10-07)
		Multicolouring is the assignment of sets of colours to the vertices of a graph, so that sets on adjacent vertices are disjoint.
		The application of graph colouring to frequency assignment in cellular networks has given a new impulse to the study of multicolouring for its own sake.
		While graph colouring is known to be hard in general, the situation can be different when we restrict ourselves to special classes of graphs.
	lennes.math.umt.edu /bigsky/colloq99.html (202 words)

	A Colourful Theory on Graphs
		In this model the vertices represent time intervals (maintenances and flights), the colours correspond to the airplanes, the precoloured part is the set of maintenances, and adjacency means that two represented time intervals overlap.
		Among others, this includes problems on graphs where feasible colourings exists whenever the size of every list is at least as large as the ‘chromatic number’; and on the largest possible number of vertices in partial colourings if the lists are not large enough to make the entire graph colourable.
		In mixed hypergraphs, two types of sets are distinguished, and vertex colourings are to be found where each set of the first type has two vertices of the same colour, while each set of the second type has two vertices of distinct colours.
	www.ercim.org /publication/Ercim_News/enw50/tuza.html (886 words)

	Colouring graphs
		Lars H: The Four Colour Theorem (or 4CT as specialists like to call it) is indeed famous for (amongst many other things) that it was the first mathematical theorem whose proof extensively relied on computer calculations.
		There is a polynomial algorithm for deciding whether a graph is 2-colourable (bipartite), but already the problem of deciding whether a general graph is 3-colourable is NP-complete.
		The algorithm used above is a branch-and-cut algorithm that makes a depth first search of the set of all possible ways of assigning to each vertex one colour from the given set, skipping branches of the search tree in which the same colour would be assigned to two neighbouring vertices.
	wiki.tcl.tk /10843 (919 words)

	A polynomial time list colouring algorithm for series-parallel graphs
		Given a graph and an assignment of lists to its nodes, a list colouring is an assignment of colours to the nodes of the graph, so that adjacent nodes receive different colours and each colour receives a colour from its list.
		The algorithm has complexity O(md^2), where m is the number of edges, and d is the maximum degree of the graph.
		A necessary and sufficient condition for the existence of a list colouring is given, which is based on a recursively defined function.
	www.cs.dal.ca /research/techreports/2002/CS-2002-02.shtml (177 words)

	e-Merge-ANT Publications
		The performance of the algorithm is assessed using soft k-coloring of random graphs having known chromatic number and edge probability ranging from moderate to high.
		This paper reports on an algorithm for anytime, stochastic, distributed constraint optimization that uses iterated, peer-to-peer interaction to try to achieve rapid, approximate solutions to large constraint problems in which the constraint variables are naturally distributed.
		The algorithm is designed to quickly reduce the number of colour conflicts in large, sparse graphs in a scalable, robust, low-cost manner.
	ants.kestrel.edu /papers (713 words)

	[No title]
		G can be coloured using 2 colours } is in P! Approximation algorithm: input G=(V,E) sort V in order of non-increasing degree for each v in V (in sorted order): colour v with the first available colour that does not conflict with any of v's neighbours Example.
		This helps explain the success of the greedy colouring algorithm, since many graphs encountered in practice are "close" to co-graphs.
		This algorithm has a worst-case running time that is exponential, but for most inputs encountered in practice, it does much better (including much better than the complicated polytime algorithm).
	www.cs.toronto.edu /~fpitt/CSC363/20049/lectures/LN13.txt (696 words)

	Mandelbrots and Julia Sets
		When using MandelbrotDriver.py the pattern type and colouring algorithm are specified using the '−t' parameter.
		The stars colouring algorithm iterates the same equation as the standard scheme, but it also records the values of Z during the iteration.
		In the bands colouring algorithm, instead of calculating the smallest value of Z encountered during the iteration we monitor the real and imaginary parts independently, finding the smallest values of r
	www.eddaardvark.co.uk /python_patterns/colour_schemes.html (470 words)

	Τμήμα Μηχ. Η/Υ & Πληροφορικής - Σεμινάρια
		We compare the quality of the colourings achieved by these algorithms, with the colourings obtained by our algorithms and with the results obtained from two well-known {\em greedy colouring heuristics}.
		The most interesting conclusions of our experimental study are: 1) all colouring algorithms considered here have {\em almost optimal} performance on the squares of ``{\em non-extremal}" planar graphs.
		2) all known colouring algorithms especially designed for colouring SQPG, give {\em significantly better} results, even on hard to colour graphs, when the vertices of the input graph are {\em randomly named}.
	www.ceid.upatras.gr /seminars/02_12_03.htm (301 words)

	McGill University School of Computer Science: CS250B Section 2
		This algorithm is clearly polynomial-time because it considers each edge and each vertex in the graph a constant number of times.
		Algorithm H(P,I) Input: Algorithm P and input I. Output: True if P with input I eventually halts and false if it loops forever.
		Therefore, there is no such algorithm H that determines whether or not a program terminates given some set of input.
	cgm.cs.mcgill.ca /~msuder/courses/250/lectures/complexity (1276 words)

	Some recent papers that are available on-line:
		Random Structures and Algorithms 26, John Wiley and Sons, 319-358.
		Random Structures and Algorithms 25, John Wiley and Sons, 432-449.
		Greedy algorithms for the shortest common superstring that are asymptotically optimal
	www.math.cmu.edu /%7Eaf1p/papers.html (2371 words)

	uk-dave.com - Projects - Java - Colour Tracker (Site not responding. Last check: 2007-10-07)
		Blob colouring is a technique used to find regions of similar colour in an image.
		In implementing the blob colouring algorithm, a two-dimensional integer array is used whose size is equal to that of the image.
		Using the second method of mapping equivalent regions, the blob colouring algorithm only took on average 4 seconds rather than 15 minutes to process the test image (including time to flatten the tree and re-calculate the region array).
	www.uk-dave.com /projects/java/colourtracker.php (2196 words)

	Graph Colouring via the Probabilistic Method
		Colouring a graph with the minimum number of colours is a classical problem in graph theory and has many applications.
		The purpose of the talk is to present a naive algorithm for colouring a certain type of graphs and explain how to analyze it with elementary probabilistic tools that we will describe first.
		We colour a graph G such that every pair of adjacent vertices receive different colours.
	pauillac.inria.fr /algo/seminars/sem96-97/reed.html (809 words)

	CS 762 (Graph-Theoretic Algorithms), Fall 2005: Lecture Summaries
		The number of colours used is the maximal indegree + 1.
		If we have k layers, then we can colour with k colours and have a clique of size k, so both is optimal.
		This colouring can be found with the greedy-heuristic if we use as vertex ordering one where every vertex has indegree 5.
	www.student.cs.uwaterloo.ca /~cs762/Summaries/index.php (2348 words)

	Complexity Theory - NP and NP-completeness
		Output: 1 if each node of G can be assigned one of three colours in such a way that no two nodes joined by an edge are given the same colour; 0 otherwise.
		That W does not assign the same colour to two nodes in I which are joined by an edge of I.
		To solve a decision problem `all' that is needed is an algorithm to determine if there is a genuine witness, W is in Wn for any given input instance I of size n.
	www.csc.liv.ac.uk /~ped/teachadmin/algor/npcomp.html (1801 words)

	[No title]
		This can be used to look up the colour in a colour map which, rather than consisting of discrete list of colours, consists of a short program which accepts a single real number and outputs three real numbers (red, green and blue).
		A new algorithm could be to calculate the pixel which is furthest away from all the other pixels already calculated.
		Better still, if you store the output from (4), you can change the colouring algorithm without recalculation, although this could be impractical because of the amount of data that needs to be stored, especially if you start storing lots of values of "z" rather than just the last one.
	www.xmission.com /pub/lists/fractint/archive/v01.n066 (4576 words)

	ALCOMFT-TR-03-169 (Site not responding. Last check: 2007-10-07)
		We compare the quality of the colourings achieved by these algorithms, with the colourings obtained by our algorithms and with the results obtained from two well-known greedy colouring heuristics.
		2) all known colouring algorithms especially designed for colouring SQPG, give significantly better results, even on hard to colour graphs, when the vertices of the input graph are randomly named.
		MDsatur colours the tested graphs with 1.1 OPT colours in most of the cases, even on hard instances, where OPT denotes the number of colours in an optimal colouring.
	www.brics.dk /ALCOM-FT/TR/ALCOMFT-TR-03-169.html (277 words)

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