Locally minimal uniformly oriented shortest networks

AbstractThe Steiner problem in a λ-plane is the problem of constructing a minimum length network interconnecting a given set of nodes (called terminals), with the constraint that all line segments in the network have slopes chosen from λ uniform orientations in the plane. This network is referred to as a minimum λ-tree. The problem is a generalization of the classical Euclidean and rectilinear Steiner tree problems, with important applications to VLSI wiring design.A λ-tree is said to be locally minimal if its length cannot be reduced by small perturbations of its Steiner points. In this paper we prove that a λ-tree is locally minimal if and only if the length of each path in the tree cannot be reduced under a special parallel perturbation on paths known as a shift. This proves a conjecture on necessary and sufficient conditions for locally minimal λ-trees raised in [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222]. For any path P in a λ-tree T, we then find a simple condition, based on the sum of all angles on one side of P, to determine whether a shift on P reduces, preserves, or increases the length of T. This result improves on our previous forbidden paths results in [M. Brazil, D.A. Thomas, J.F. Weng, Forbidden subpaths for Steiner minimum networks in uniform orientation metrics, Networks 39 (2002) 186–222]

Algorithms for cartographic visualization

Maps are effective tools for communicating information to the general public and help people to make decisions in, for example, navigation, spatial planning and politics. The mapmaker chooses the details to put on a map and the symbols to represent them. Not all details need to be geographic: thematic maps, which depict a single theme or attribute, such as population, income, crime rate, or migration, can very effectively communicate the spatial distribution of the visualized attribute. The vast amount of data currently available makes it infeasible to design all maps manually, and calls for automated cartography. In this thesis we presented efficient algorithms for the automated construction of various types of thematic maps. In Chapter 2 we studied the problem of drawing schematic maps. Schematic maps are a well-known cartographic tool; they visualize a set of nodes and edges (for example, highway or metro networks) in simplified form to communicate connectivity information as effectively as possible. Many schematic maps deviate substantially from the underlying geography since edges and vertices of the original network are moved in the simplification process. This can be a problem if we want to integrate the schematized network with a geographic map. In this scenario the schematized network has to be drawn with few orientations and links, while critical features (cities, lakes, etc.) of the base map are not obscured and retain their correct topological position with respect to the network. We developed an efficient algorithm to compute a collection of non-crossing paths with fixed orientations using as few links as possible. This algorithm approximates the optimal solution to within a factor that depends only on the number of allowed orientations. We can also draw the roads with different thicknesses, allowing us to visualize additional data related to the roads such as trafic volume. In Chapter 3 we studied methods to visualize quantitative data related to geographic regions. We first considered rectangular cartograms. Rectangular cartograms represent regions by rectangles; the positioning and adjacencies of these rectangles are chosen to suggest their geographic locations to the viewer, while their areas are chosen to represent the numeric values being communicated by the cartogram. One drawback of rectangular cartograms is that not every rectangular layout can be used to visualize all possible area assignments. Rectangular layouts that do have this property are called area-universal. We show that area-universal layouts are always one-sided, and we present algorithms to find one-sided layouts given a set of adjacencies. Rectangular cartograms often provide a nice visualization of quantitative data, but cartograms deform the underlying regions according to the data, which can make the map virtually unrecognizable if the data value differs greatly from the original area of a region or if data is not available at all for a particular region. A more direct method to visualize the data is to place circular symbols on the corresponding region, where the areas of the symbols correspond to the data. However, these maps, so-called symbol maps, can appear very cluttered with many overlapping symbols if large data values are associated with small regions. In Chapter 4 we proposed a novel type of quantitative thematic map, called necklace map, which overcomes these limitations. Instead of placing the symbols directly on a region, we place the symbols on a closed curve, the necklace, which surrounds the map. The location of a symbol on the necklace should be chosen in such a way that the relation between symbol and region is as clear as possible. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. We developed algorithms to compute necklace maps and demonstrated our method with experiments using various data sets and maps. In Chapter 5 and 6 we studied the automated creation of ow maps. Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of ow between a source and a target. Good ow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We developed a new algorithm for drawing ow trees, ow maps with a single source. Unlike existing methods, our method merges lines smoothly and avoids self-intersections. Our method is based on spiral trees, a new type of Steiner trees that we introduced. Spiral trees have an angle restriction which makes them appear smooth and hence suitable for drawing ow maps. We study the properties of spiral trees and give an approximation algorithm to compute them. We also show how to compute ow trees from spiral trees and we demonstrate our approach with extensive experiments

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Algorithmic embeddings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 233-242).We present several computationally efficient algorithms, and complexity results on low distortion mappings between metric spaces. An embedding between two metric spaces is a mapping between the two metric spaces and the distortion of the embedding is the factor by which the distances change. We have pioneered theoretical work on relative (or approximation) version of this problem. In this setting, the question is the following: for the class of metrics C, and a host metric M', what is the smallest approximation factor a > 1 of an efficient algorithm minimizing the distortion of an embedding of a given input metric M E C into M'? This formulation enables the algorithm to adapt to a given input metric. In particular, if the host metric is "expressive enough" to accurately model the input distances, the minimum achievable distortion is low, and the algorithm will produce an embedding with low distortion as well. This problem has been a subject of extensive applied research during the last few decades. However, almost all known algorithms for this problem are heuristic. As such, they can get stuck in local minima, and do not provide any global guarantees on solution quality. We investigate several variants of the above problem, varying different host and target metrics, and definitions of distortion.(cont.) We present results for different types of distortion: multiplicative versus additive, worst-case versus average-case and several types of target metrics, such as the line, the plane, d-dimensional Euclidean space, ultrametrics, and trees. We also present algorithms for ordinal embeddings and embedding with extra information.by Mihai Bădoiu.Ph.D