Max flow vitality in general and stst-planar graphs

The \emph{vitality} of an arc/node of a graph with respect to the maximum flow between two fixed nodes $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only $2(n-1)$ max flow instances and, In $st$-planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in $O(n)$ worst-case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in time proportional to the size of the set.Comment: 12 pages, 3 figure

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure

Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs

This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs. In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest. In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures

Computing Lengths of Shortest Non-Crossing Paths in Planar Graphs

Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.Comment: 17 pages, 11 figure

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

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

Proceedings of the GIS Research UK 18th Annual Conference GISRUK 2010

This volume holds the papers from the 18th annual GIS Research UK (GISRUK). This year the conference, hosted at University College London (UCL), from Wednesday 14 to Friday 16 April 2010. The conference covered the areas of core geographic information science research as well as applications domains such as crime and health and technological developments in LBS and the geoweb. UCL’s research mission as a global university is based around a series of Grand Challenges that affect us all, and these were accommodated in GISRUK 2010. The overarching theme this year was “Global Challenges”, with specific focus on the following themes: * Crime and Place * Environmental Change * Intelligent Transport * Public Health and Epidemiology * Simulation and Modelling * London as a global city * The geoweb and neo-geography * Open GIS and Volunteered Geographic Information * Human-Computer Interaction and GIS Traditionally, GISRUK has provided a platform for early career researchers as well as those with a significant track record of achievement in the area. As such, the conference provides a welcome blend of innovative thinking and mature reflection. GISRUK is the premier academic GIS conference in the UK and we are keen to maintain its outstanding record of achievement in developing GIS in the UK and beyond

Annual Report 2012 / Institute for Pulsed Power and Microwave Technology = Jahresbericht 2012 / Institut für Hochleistungsimpuls- und Mikrowellentechnik. (KIT Scientific Reports ; 7643)

The Institute for Pulsed Power and Microwave Technology (Institut für Hochleistungsimpuls- und Mikrowellentechnik - IHM) is doing research in the areas of pulsed power and high power microwave technologies. Both, research and development of high power sources as well as related applications are in the focus. Applications for pulsed power technologies are ranging from material processing to bioelectrics. Microwave technologies are focusing on RF sources for electron cyclotron resonance heating and on applications for material processing at microwave frequencies