New Applications of Nearest-Neighbor Chains: Euclidean TSP and Motorcycle Graphs

We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0

New Applications of the Nearest-Neighbor Chain Algorithm

The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together

Graph Analysis and Applications in Clustering and Content-based Image Retrieval

About 300 years ago, when studying Seven Bridges of Königsberg problem - a famous problem concerning paths on graphs - the great mathematician Leonhard Euler said, “This question is very banal, but seems to me worthy of attention”. Since then, graph theory and graph analysis have not only become one of the most important branches of mathematics, but have also found an enormous range of important applications in many other areas. A graph is a mathematical model that abstracts entities and the relationships between them as nodes and edges. Many types of interactions between the entities can be modeled by graphs, for example, social interactions between people, the communications between the entities in computer networks and relations between biological species. Although not appearing to be a graph, many other types of data can be converted into graphs by cer- tain operations, for example, the k-nearest neighborhood graph built from pixels in an image. Cluster structure is a common phenomenon in many real-world graphs, for example, social networks. Finding the clusters in a large graph is important to understand the underlying relationships between the nodes. Graph clustering is a technique that partitions nodes into clus- ters such that connections among nodes in a cluster are dense and connections between nodes in different clusters are sparse. Various approaches have been proposed to solve graph clustering problems. A common approach is to optimize a predefined clustering metric using different optimization methods. However, most of these optimization problems are NP-hard due to the discrete set-up of the hard-clustering. These optimization problems can be relaxed, and a sub-optimal solu- tion can be found. A different approach is to apply data clustering algorithms in solving graph clustering problems. With this approach, one must first find appropriate features for each node that represent the local structure of the graph. Limited Random Walk algorithm uses the random walk procedure to explore the graph and extracts ef- ficient features for the nodes. It incorporates the embarrassing parallel paradigm, thus, it can process large graph data efficiently using mod- ern high-performance computing facilities. This thesis gives the details of this algorithm and analyzes the stability issues of the algorithm. Based on the study of the cluster structures in a graph, we define the authenticity score of an edge as the difference between the actual and the expected number of edges that connect the two groups of the neighboring nodes of the two end nodes. Authenticity score can be used in many important applications, such as graph clustering, outlier detection, and graph data preprocessing. In particular, a data clus- tering algorithm that uses the authenticity scores on mutual k-nearest neighborhood graph achieves more reliable and superior performance comparing to other popular algorithms. This thesis also theoretically proves that this algorithm can asymptotically find the complete re- covery of the ground truth of the graphs that were generated by a stochastic r-block model. Content-based image retrieval (CBIR) is an important application in computer vision, media information retrieval, and data mining. Given a query image, a CBIR system ranks the images in a large image database by their “similarities” to the query image. However, because of the ambiguities of the definition of the “similarity”, it is very diffi- cult for a CBIR system to select the optimal feature set and ranking algorithm to satisfy the purpose of the query. Graph technologies have been used to improve the performance of CBIR systems in var- ious ways. In this thesis, a novel method is proposed to construct a visual-semantic graph—a graph where nodes represent semantic concepts and edges represent visual associations between concepts. The constructed visual-semantic graph not only helps the user to locate the target images quickly but also helps answer the questions related to the query image. Experiments show that the efforts of locating the target image are reduced by 25% with the help of visual-semantic graphs. Graph analysis will continue to play an important role in future data analysis. In particular, the visual-semantic graph that captures important and interesting visual associations between the concepts is worthyof further attention

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Extraction of routing relevant geodata using telemetry sensor data of agricultural vehicles

The mechanization of processes in agriculture is growing within the last hundred years. Within the last decades, the information technology in this sector constantly grew and data is one of the key factors for process optimization. For planning and execution of logistic processes and harvest campaigns, road maps and field geometries are essential to guide the large vehicles to their work places and to optimize harvest chains. Through nowadays available telemetry data of these vehicles, a new data source generates new opportunities. Mining geographic data from these movement data, that can improve agricultural work processes is one of the main objectives of this thesis. As a first step, data cleaning processes, and further preprocessing steps are shown. With classification algorithms, the continuous movement data will be separated into different work processes. Based on this data, algorithms to generate geographic geographic features, such as field boundaries have been analyzed and improved. As quality metric to compare the results, the Jaccard-Distance has been established. With the classified road representing measurements, the rural road networks were created and the results of different algorithmic approaches have been compared. The usability of volunteered geographic information to route the heterogeneous set of agricultural vehicles is shown in a third step. Due to the fact, that routes for e.g. harvesters are not ending at the field boundary, solutions for infield route graph generation have been given. The presented components provide the content and the services within a framework structure. The concluding prototype, a web based routing system demonstrates the interaction of all components and provides a consecutive routing from farm to field and within the field

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Operational Research IO2017, Valença, Portugal, June 28-30

This proceedings book presents selected contributions from the XVIII Congress of APDIO (the Portuguese Association of Operational Research) held in Valença on June 28–30, 2017. Prepared by leading Portuguese and international researchers in the field of operations research, it covers a wide range of complex real-world applications of operations research methods using recent theoretical techniques, in order to narrow the gap between academic research and practical applications. Of particular interest are the applications of, nonlinear and mixed-integer programming, data envelopment analysis, clustering techniques, hybrid heuristics, supply chain management, and lot sizing and job scheduling problems. In most chapters, the problems, methods and methodologies described are complemented by supporting figures, tables and algorithms. The XVIII Congress of APDIO marked the 18th installment of the regular biannual meetings of APDIO – the Portuguese Association of Operational Research. The meetings bring together researchers, scholars and practitioners, as well as MSc and PhD students, working in the field of operations research to present and discuss their latest works. The main theme of the latest meeting was Operational Research Pro Bono. Given the breadth of topics covered, the book offers a valuable resource for all researchers, students and practitioners interested in the latest trends in this field.info:eu-repo/semantics/publishedVersio