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Structure of networks
Published in Karthik Raman, An Introduction to Computational Systems Biology, 2021
This represents sets of nodes, where every pair of nodes have a path (not an edge) between them. A graph may have more than one connected component. In the case of directed graphs, there is the equivalent notion of a strongly connected component, where every pair of nodes are connected to each other in both directions. That is, for every pair of vertices {A,B} in a strongly connected component, there is a path from A to B, as well as B to A. The graph is weakly connected if the underlying undirected graph is connected. Connected components are maximal, i.e. they denote the maximal connected sub-graph of a graph or the maximal strongly connected sub-graph of a graph.
Efficient and Accurate Algorithms in Machine Vision Inspection
Published in Don Potter, Manton Matthews, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2020
One approach to find bounding boxes of segments is using connected component technique. Finding connected components in an image is a classic problem which has been intensively studied in the community of image processing and computer vision. We evaluated a number of connected component algorithms before we decided to develop our own algorithm to meet this requirement of this application. Rosenfeld and Pfaltz first introduced an algorithm that makes only 2 passes through the image and a large global table for recording equivalences. In 1981, Haralick proposed an iterative algorithm[]. This algorithm uses no auxiliary storage to produce the labeled image from the binary image, therefore it would be useful in environments whose storage is severely limited[HaS83]. However the number of scan times through the image is determined by the complexity of the connected components in this image. Lumia, Shapiro and Zuniga[LSZ83] developed a space- efficient two-pass algorithm that uses a local equivalence table.
Dimensionality Reduction
Published in Taylor Arnold, Michael Kane, Bryan W. Lewis, A Computational Approach to Statistical Learning, 2019
Taylor Arnold, Michael Kane, Bryan W. Lewis
For simplicity, assume that we have a similarity matrix S defined by an unweighted adjacency matrix. A connected component of a graph is any set of nodes that contains all of its neighbors; in other words, it is the set of nodes that can all be reached by following along edges of the graph. If we take an n-dimensional vector v and set it to a constant on a connected component and zero otherwise, notice that the quadratic form in Equation 9.33 will be equal to zero. There is no pair i and j for which both Si,j and (vi - vj) are both non-zero. By construction, the quadratic form defined by L can never be less than zero, and therefore v is a minimal eigenvector. The corresponding eigenvalue is 0. So, a graph with m connected components has a graph Laplacian whose m smallest eigenvalues are equal to zero.
Improved formulations of the joint order batching and picker routing problem
Published in International Journal of Production Research, 2022
Here, we present another type of strengthened connectivity constraints, which we call basic cuts. can be obtained by the following operations. For any order o, let denote a set of (which represents a subaisle) that contains at least one picking location . Let be a set of vertices associated with . Then, any with both ends in will be added to . By executing a depth-first search algorithm, we can discover all connected components of the graph . Figure 5 shows an example of the two connected components induced by an order. Let S be a vertex set of a connected component that does not contain the origin. All picking locations within the subaisle that has both ends in S will then be added to S. Then S will be added to . We call the strengthened connectivity constraints induced by basic cuts.
On the Use of Graph Theory to Interpret the Output Results from a Monte-Carlo Depletion Code
Published in Nuclear Science and Engineering, 2021
One of the first basic questions one could ask when dealing with a graph of any sort has to do with the number of separate pieces it contains. A graph can indeed be constituted by different pieces or connected components that do not interact with one another. For undirected graphs (i.e., no notion of direction is defined on the edges), this has a straightforward meaning, and the number of components simply reflects the fact that the object is made up of separate independent pieces that can be analyzed separately. The question is more subtle when working with directed graphs, for which connectedness can bear two separate meanings, embodied in the notions of strong and weak components of a graph. Weakly connected components of a directed graph correspond to the intuitive notion of separate pieces that the graph is made of. But, when the graph is directed, a situation might arise where regions (i.e., subsets of vertices) can be accessed by following a path along the edges of the graph but cannot be left because there is no directed way out of the subset of vertices. In this latter situation, the subset of vertices is said to form a strongly connected component of the graph.
Object based classification using multisensor data fusion and support vector algorithm
Published in International Journal of Image and Data Fusion, 2018
In this analysis, the image is divided into foreground (object) and background pixels. The connected regions of foreground pixels (called blobs) have to be extracted. This process of labelling regions of connected foreground pixels is called ‘connected component labelling’ (Bailey and Johnston. 2008). Connected components labelling scans an image and groups its pixels into components based on pixel connectivity, i.e. all pixels in a connected component share similar pixel intensity values and are in some way connected with each other. The pixels that are connected to each other and are not connected with other pixels are identified, categorised and marked as the connected components. Pixel’s numbers and tags of a component are saved in array matrix of that image and in corresponding elements of each pixel. According to the approach of connected component analysis, which may be 4-connectivity, 8- connectivity, we can categorise the pixels of an image to connected components into class labels.