A unit disk graph is the intersection graph of a set of unit diameter disks in the plane. In this paper we consider liar’s domination problem on unit disk graphs, a variant of dominating set problem. We call this problem as Euclidean liar’s domination problem. In the Euclidean liar’s domination problem, a set P = {p1, p2, . . . , pn} of n points (disk centers) are given in the Euclidean plane. ...

It is challenging to find a small set of data points, so-called exemplars or landmarks, that are nicely representative of itself and other data points. Affinity propagation (AP) is an effective algorithm that identifies exemplars among data points by recursively sending realvalued messages between pairs of data points. AP calculates the message using the similarity among data points. Hence, the...

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor Cdel ≈ 2.42; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor ρ = 1 + ...

The maximum detour and spanning ratio of an embedded graph G are values that measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n logn) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works [5,10]. We also ge...

Humans and other animals build up spatial knowledge of the environment on the basis of visual information and path integration. We compare three hypotheses about the geometry of this knowledge of navigation space: (a) 'cognitive map' with metric Euclidean structure and a consistent coordinate system, (b) 'topological graph' or network of paths between places, and (c) 'labelled graph' incorporat...

Federated Learning (FL) can combine multiple clients for training and keep client data local, which is a good way to protect privacy. There are many excellent FL algorithms. However, most of these only process with regular structures, such as images videos. They cannot non-Euclidean spatial data, that is, irregular data. To address this problem, we propose Learning-Based Graph Convolutional Net...

Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data these are typically represented Euclidean domains. Nevertheless, there is an increasing number of applications where collected non-Euclidean domains and as graph-structured with high-dimensional features interdependency among nodes. complexity ha...

Structural and Algorithmic Properties of Static and Mobile Random Geometric Graphs by Alexandre de Oliveira Stauffer Doctor of Philosophy in Computer Science and the Designated Emphasis in Communication, Computation and Statistics University of California, Berkeley Professor Alistair Sinclair, Chair We study fundamental problems for static and mobile networks. First, we consider the random geom...

The clustered shortest-path tree problem (CluSPTP) is an extension of the classical single-source problem, in which, given a graph with set nodes partitioned into predefined, mutually exclusive and exhaustive clusters, we are looking for spanning from source to all other graph, property that each cluster should induce connected subtree. CluSPTP belongs class generalized combinatorial optimizati...

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-...