Proximity problems and point set embeddability problems are fundamental and well-studied in computational geometry and graph drawing. Examples of such problems that are of particular interest to us in this dissertation include: finding the closest pair among a set P of points, finding the k-nearest neighbors to each point p in P, answering reverse k-nearest neighbor queries, computing the Yao graph, the Semi-Yao graph and the Euclidean minimum spanning tree of P, and mapping the vertices of a planar graph to a set P of points without inducing edge crossings. In this dissertation, we consider so-called kinetic version of these problems, that is, the points are allowed to move continuously along known trajectories, which are subject to change. We design a set of data structures and a mechanism to efficiently update the data structures. These updates occur at critical, discrete times. Also, a query may arrive at any time. We want to answer queries quickly without solving problems from scratch, so we maintain solutions continuously. We present new techniques for giving kinetic solutions with better performance for some these problems, and we provide the first kinetic results for others. In particular, we provide: • A simple kinetic data structure (KDS) to maintain all the nearest neighbors and the closest pair. Our deterministic kinetic approach for maintenance of all the nearest neighbors improves the previous randomized kinetic algorithm. • An exact KDS for maintenance of the Euclidean minimum spanning tree, which improves the previous KDS. • The first KDS's for maintenance of the Yao graph and the Semi-Yao graph. • The first KDS to consider maintaining plane graphs on moving points. • The first KDS for maintenance of all the k-nearest neighbors, for any k ≥ 1. • The first KDS to answer the reverse k-nearest neighbor queries, for any k ≥ 1 in any fixed dimension, on a set of moving points. / Graduate

In this dissertation, we investigate a number of geometric problems in different settings. We present lower bounds and approximation algorithms for geometric problems in sequential and distributed settings. For the sequential setting, we prove the first hardness of approximation results for the following problems: \begin{itemize} \item Red-Blue Geometric Set Cover is APX-hard when the objects are axis-aligned rectangles. \item Red-Blue Geometric Set Cover cannot be approximated to within $2^{\log^{1-1/{(\log\log m)^c}}m}$ in polynomial time for any constant $c < 1/2$, unless $P=NP$, when the given objects are $m$ triangles or convex objects. This shows that Red-Blue Geometric Set Cover is a harder problem than Geometric Set Cover for some class of objects. \item Boxes Class Cover is APX-hard. \end{itemize} We also define MaxRM-3SAT, a restricted version of Max3SAT, and we prove that this problem is APX-hard. This problem might be interesting in its own right.\\ In the distributed setting, we define a new model, the fixed-link model, where each processor has a position on the plane and processors can communicate to each other if and only if there is an edge between them. We motivate the model and study a number of geometric problems in this model. We prove lower bounds on the communication complexity of the problems in the fixed-link model and present approximation algorithms for them. We prove lower bounds on the number of expected bits required for any randomized algorithm in the fixed-link model with $n$ nodes to solve the following problems, when the communication is in the asynchronous KT1 model: \begin{itemize} \item $\Omega(n^2/\log n)$ expected bits of communication are required for solving Diameter, Convex Hull, or Closest Pair, even if the graph has only a linear number of edges. \item $\Omega( min\{n^2,1/\epsilon\})$ expected bits of communications are required for approximating Diameter within a $1-\epsilon$ factor of optimal, even if the graph is planar. \item $\Omega(n^2)$ bits of communications is required for approximating Closest Pair in a graph on an $[n^c] \times [n^c]$ grid, for any constant $c>1+1/(2\lg n)$, within $\frac{n^{c-1/2}}{4}-\epsilon$ factor of optimal, even if the graph is planar. \end{itemize} We also present approximation algorithms in geometric communication networks with $n$ nodes, when the communication is in the asynchronous CONGEST KT1 model: \begin{itemize} \item An $\epsilon$-kernel, and consequently $(1-\epsilon)$-\diamapprox~ and \ep -Approximate Hull with $O(\frac{n}{\sqrt{\epsilon}})$ messages plus the costs of constructing a spanning tree. \item An $\frac{n^c}{\sqrt{\frac{k}{2}}}$-Approximate Closest Pair on an $[n^c] \times [n^c]$ grid , for a constant $c>1/2$, plus the cost of computing a spanning tree, for any $k\leq {n-1}$. \end{itemize} We also define a new version of the two-party communication problem, Path Computation, where two parties communicate through a path. We prove a lower bound on the communication complexity of this problem. / Graduate