Integer Programming Formulations for Minimum Spanning Forests and Connected Components in Sparse Graphs

The cancer remained as one of the major cause of mortality worldwide. Instead of traditional excitional biopsy, it will be ideal to put forward the cancer detection using invasive techniques, it can make the treatment more effective and simple. Here a self sufficient classification method has been introduced that combines both spectral and spatial information on hyper-spectral images of an animal tissue in distinguishing cancerous via healthy. An automated algorithm based on an optimal band selection and Minimum Spanning Forest (MSF) algorithm has been proposed for classification of hyperspectral images. The segmentation has been achieved by collecting the features from the selected markers in the optimal band. In optimal band selection, the unnecessary band may get eliminated and selects a range where there is a probability of having abnormality and the markers are being selected using the intensity feature available in the selected band. From these numerous markers available, the problematic region if available, may be differentiated using the MSF algorithm. The MSF is utilizing the pixel distribution of every marker. The SVM classifier is finally used to distinguish between the normal or abnormal based on the MSF features. The corresponding MSF based advanced cancer detection scheme is accurate and efficient, making it as a good noninvasive tool. It is not only applicable in early cancer detection, but also as an interpretive tool in tumor margin resection.

Minimum Spanning Forest (MSF) is a graph-based technique used for segmenting and classification of images. In this article, a new method based on MSF is introduced that can be used to supervised classification of hyperspectral images. For a given hyperspectral image, a pixel-based classification, such as Support Vector Machine (SVM) or Maximum Likelihood (ML) is performed. On the other hand, dimensionality reduction is carried out by Principal Components Analysis (PCA) and the first eight components are considered as the reference data. The most reliable pixels, which are obtained from the result of pixel-based classifiers, are used as markers in the construction of MSF. In the next stage, three MSF’s are created after considering three distinct criteria of similarity (dissimilarity). Ultimately, using the majority voting rule, the obtained classification maps are combined and the final classification map is formed. The simulation results presented on an AVRIS image of the vegetation area indicate that the proposed technique enhanced classification accuracy and provides an accurate classification map.

Massively parallel computation (MPC) algorithms for graph problems have witnessed a resurgence of interest in recent years. Despite major progress for numerous graph problems however, the complexity of the sparse graph connectivity problem in this model has remained elusive: While classical logarithmic-round PRAM algorithms for finding connected components in any n-vertex graph have been known for more than three decades (and imply the same bounds for MPC model), no o(log n)-round MPC algorithms are known for this task with truly sublinear in n memory per machine (which is the only interesting regime for sparse graphs with O(n) edges). It is conjectured that an o(log n)-round algorithm for connectivity on general sparse graphs with n1-Ω (1) per-machine memory may not exist, a conjecture that also forms the basis for multiple conditional hardness results on the round complexity of other problems in the MPC model.We take an opportunistic approach towards the sparse graph connectivity problem by designing an algorithm with improved performance in terms of the connectivity structure of the input graph. Formally, we design an MPC algorithm that finds all connected components with spectral gap at least λ in a graph in O(log log n + log(1/λ)) MPC rounds and nδ memory per machine for any constant δ ∈ (0,1). While this algorithm still requires Θ(log n) rounds in the worst-case, it achieves an exponential round reduction on well-connected'' components with λ ≥ 1/polylog(n) using only nδ memory per machine and l(n) total memory, and still operates in o(log n)l rounds even when λ = 1/no(1).En-route to our main result, we design a new distributed data structure for performing independent random walks from all vertices simultaneously, as well as a new leader-election algorithm with exponentially faster round complexity on random graphs.

Provided that one is willing to use randomness and to tolerate an approximate answer, many computational problems admit ultrafast algorithms that run in less than linear time in the length of the input. In many interesting cases, even algorithms that run in constant time are known, whose efficiency depends only on the accuracy of the approximation and not on the length of the inputs. Algorithms for graph problems on dense graphs are especially efficient and simple. I will describe an algorithm that estimates the size of the maximum cut in a dense graph, and its specialization to the task of distinguishing bipartite dense graphs from dense graphs that are far from bipartite. Results explaining the simplicity of such algorithms will also be discussed. Some sublinear-time algorithms are also known for graph problems in sparse graphs, but they are typically more elaborate. I will describe a simple but very clever algorithm that approximates the number of connected components of a given graph, and its generalization to the problem of approximating the weight of the minimum spanning tree of a given weighted graph. The algorithm runs in time dependent only on the maximum degree, the required quality of approximation, and the range of weights, but the running time is independent of the number of vertices.

In this paper, the concept of minimum spanning forest (MSF) is utilized to formulate the restoration problem where each spanning tree in a forest is a self-sustained islanded grid (SSIG). Specifically, a weight is assigned to each edge in a distribution system based on several factors such as their exposure to vegetation, span length, location and structures supporting them. Then, a MSF is obtained for the given network by switching off the edges with higher weights to form several SSIGs each energized by a master unit (micro-turbine) in coordination with several intermittent DGs (wind, solar) and energy storage system (ESS) while achieving the resilience objectives. Numerical results based on a modified IEEE 123-node test case validates the effectiveness of the proposed approach.

A new method for supervised hyperspectral data classification is proposed. In particular, the notion of Stochastic Minimum Spanning Forests (MSFs) is introduced. For a given hyper-spectral image, a pixelwise classification is first performed. From this classification map, M marker maps are generated by randomly selecting pixels and labeling them as markers for the construction of MSFs. The next step consists in building an MSF from each of the M marker maps. Finally, all the M realizations are aggregated with a maximum vote decision rule, resulting in a final classification map. The experimental results presented on an AVIRIS image of the vegetation area show that the proposed approach yields accurate classification maps, and thus is attractive for hyperspectral data analysis.

Roads fragment habitat and pose a variety of threats to wildlife by restricting their physical movements or causing direct mortality. Wildlife crossing structures can mitigate some of these effects by simultaneously connecting fragmented habitat and reducing collisions between animals and traffic. This research develops two classes of spatial models for optimally locating crossing structures in situations where the goal is to connect discrete habitat patches that are fragmented by roads. The first group of models uses variations of minimum spanning forest (MSF) problems to select crossing structure locations in scenarios where the goal is to connect all habitat patches in the landscape. This article explores three variations: distance-based, intersection-weighted, and component-based MSFs. The second set of models includes newly developed integer programs with objectives that maximize interpatch connectivity given a fixed number of crossing structures. One of these integer programs maximizes overall connectivity in a landscape, and the other maximizes the connectivity of patches to a core reserve. We illustrated the models using hypothetical landscapes and applied them to a case study where the goal is to connect vernal pool habitat for amphibians. Results of the applications guide a discussion of the relative merits of each approach for meeting various conservation goals related to improving habitat connectivity in fragmented landscapes. More broadly, this article develops an analytical framework for dealing with habitat connectivity issues due to road network fragmentation as well as a new neighborhood graph and several spatial models.

Remmel and Williamson recently defined a class of directed graphs, called filtered digraphs, and described a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions [12]. Filtered digraphs include many specialized graphs such as complete k-partite graphs. The Remmel-Williamson bijections provide explicit formulas for various multivariate generating functions for the oriented spanning forests which arise in this context. In this paper, we prove another important property of these bijections, namely, that it allows one to construct efficient algorithms for ranking and unranking spanning trees or spanning forests of filtered digraphs G. For example, we show that if G=(V,E) is a filtered digraph and SP(G) is the collection of spanning trees of G, then our algorithm requires O(|V|) operations of sum, difference, product, quotient, and comparison of numbers less than or equal |SP(G)| to rank or unrank spanning trees of G.

The classification methods based on minimum spanning forest (MSF) have yielded impressive results for hyperspectral image. However, previous methods exist several drawbacks, i.e., marker selection methods are easily affected by boundary noise pixels, dissimilarity measure methods between pixels are inaccurate, and also image segmentation process is not robust, since they have not effectively utilized spatial information. To this end, in this paper, novel gradient-based marker selection technique, dissimilarity measures, and adaptive connection weighting method are proposed by making full use of spatial information in hyperspectral image. Concretely, for a given hyperspectral image, a pixel-wise classification is firstly performed, and meanwhile the gradient map is generated by a morphology-based algorithm. Secondly, the most reliable pixels are selected as the markers from the classification map, and then the boundary noise pixels are excluded from the marker map by using the gradient map. Thirdly, several new dissimilarity measures are proposed by incorporating gradient information or probability information of pixels. Furthermore, in the growth procedure of MSF, the connection weighting between pixels is adjusted adaptively to improve the robustness of the MSF algorithm. Finally, when building the final classification map by using the majority voting rule, the labels of the training samples are used to dominate the label prediction. Experimental results are performed on two hyperspectral image sets Indian Pines and University of Pavia with different resolutions and contexts. The proposed approach yields higher classification accuracies compared to previously proposed classification methods, and provides accurate segmentation maps.

The total weight of the minimum spanning is the smallest in the connected graph. It can be used to solve many practical problems in urban life. Prims algorithm and Kruskals algorithm are greedy algorithms for solving the minimum spanning tree problem. But they make greedy choices in different ways. The paper focuses on two greedy algorithms for solving the minimum spanning tree problem. The author will evaluate each algorithms complexity and determine their most suitable condition as well. The author compares the running process of the two algorithms and analyzes the relationship between their algorithm complexity and the number of edges, which can describe the sparsity of the graphs. The result shows that Kruskals complexity is related to the number of edges, so it is better for sparse graphs. Prims complexity is related to the number of vertices, so it is better at analyzing connected graphs with lots of vertices, that is, dense graphs. As a consequence, Prims algorithm is better for dense graphs.

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

We consider the implementation of meldable priority queues with integer keys in the RAM model. We present two new general techniques for transforming non-meldable priority queues into meldable ones. These transformations can be described symbolically as:non-meldable priority queue +union-find ← meldable priority queuenon-meldable priority queue +slow meldable priority queue ←faster meldable priority queueUsing the first transformation to combine a recent non-meldable RAM priority queue of Thorup with the classical union-find data structure we obtain a meldable RAM priority queue with an amortized cost of O(log log n·α(n)) per operation, where α(n) = α(n, n) is the inverse Ackermann function. Using instead a randomized priority queue of Han and Thorup we obtain an expected amortized cost of O(√(log log n) · α(n)) per operation. The second transformation yields slower meldable priority queues, but the obtained queues can support the insert, find-min and decrease-key operations in constant time. In particular, by combining a randomized atomic-heap of Thorup with, e.g., the classical Fibonacci heaps of Fredman and Tarjan, we obtain, for every fixed e > 0, a meldable priority queue with an expected amortized cost of O(1) for each insert, find-min and decrease-key operation, and an expected amortized cost of O((log n)1/2+e) for each delete or meld operation.Using the meldable priority queues of the first type, we obtain improved algorithms for finding minimum directed spanning trees in graphs with integer edge weights: a deterministic O(m · log log n · α(n)) time algorithm and a randomized O(m · √(log log n) · α(n)) expected time algorithm. These bounds improve, for very sparse graphs, on the O(m + n log n) running time of an algorithm by Gabow, Galil, Spencer and Tarjan that works for arbitrary edge weights.

We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the congested clique model. The input graph in the congested clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange O(log n) bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka's algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in O(1) rounds to its two restricted instances. The former instance has a graph with maximal degree O(log log n) as the input - here our sample-combining technique helps. In the latter instance, a partition of the input graph into O(n/ log log n) connected components is known. This gives an opportunity to apply previous algorithms to determine connected components in O(1) rounds. Our result addresses a problem proposed by Lotker et al. [SPAA 2003; SICOMP 2005] and improves over previous O(log* n) algorithm of Ghaffari et al. [PODC 2016], and O(log log log n) algorithm of Hegeman et al. [PODC 2015]. It also determines Θ(1) round complexity in the congested clique for MST, as well as other graph problems, including bipartiteness, cut verification, s-t connectivity, and cycle containment.

Verifying minimum spanning tree algorithms with Stone relation algebras

Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths (APSP) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi- and many-core architectures. It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures. In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortestpaths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.