424 research outputs found
Shortest Paths in Geometric Intersection Graphs
This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed.
In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique.
In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location.
Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions.
Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time.
In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time
Two-Dimensional Matter: Order, Curvature and Defects
Many systems in nature and the synthetic world involve ordered arrangements
of units on two-dimensional surfaces. We review here the fundamental role payed
by both the topology of the underlying surface and its detailed curvature.
Topology dictates certain broad features of the defect structure of the ground
state but curvature-driven energetics controls the detailed structured of
ordered phases. Among the surprises are the appearance in the ground state of
structures that would normally be thermal excitations and thus prohibited at
zero temperature. Examples include excess dislocations in the form of grain
boundary scars for spherical crystals above a minimal system size, dislocation
unbinding for toroidal hexatics, interstitial fractionalization in spherical
crystals and the appearance of well-separated disclinations for toroidal
crystals. Much of the analysis leads to universal predictions that do not
depend on the details of the microscopic interactions that lead to order in the
first place. These predictions are subject to test by the many experimental
soft and hard matter systems that lead to curved ordered structures such as
colloidal particles self-assembling on droplets of one liquid in a second
liquid. The defects themselves may be functionalized to create ligands with
directional bonding. Thus nano to meso scale superatoms may be designed with
specific valency for use in building supermolecules and novel bulk materials.
Parameters such as particle number, geometrical aspect ratios and anisotropy of
elastic moduli permit the tuning of the precise architecture of the superatoms
and associated supermolecules. Thus the field has tremendous potential from
both a fundamental and materials science/supramolecular chemistry viewpoint.Comment: Review article, 102 pages, 59 figures, submitted to Advances in
Physic
Methods and Measures for Analyzing Complex Street Networks and Urban Form
Complex systems have been widely studied by social and natural scientists in
terms of their dynamics and their structure. Scholars of cities and urban
planning have incorporated complexity theories from qualitative and
quantitative perspectives. From a structural standpoint, the urban form may be
characterized by the morphological complexity of its circulation networks -
particularly their density, resilience, centrality, and connectedness. This
dissertation unpacks theories of nonlinearity and complex systems, then
develops a framework for assessing the complexity of urban form and street
networks. It introduces a new tool, OSMnx, to collect street network and other
urban form data for anywhere in the world, then analyze and visualize them.
Finally, it presents a large empirical study of 27,000 street networks,
examining their metric and topological complexity relevant to urban design,
transportation research, and the human experience of the built environment.Comment: PhD thesis (2017), City and Regional Planning, UC Berkele
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