As finite element discretizations ever grow in size to address real-world problems, there is an increasing need for fast algorithms. Nowadays there are many GPU/CPU parallel approaches to solve such problems. Multigrid methods can be used to solve large-scale problems, or even better they can be used to precondition the conjugate gradient method, yielding better results in general. Capabilities of multigrid algorithms rely on the effectiveness of the inter-grid transfer operators. In this thesis we focus on the aggregation approach, discussing how different aggregation strategies affect the convergence rate. Based on these discussions, we propose an alternative parallel aggregation algorithm to improve convergence. We also provide numerous experimental results that compare different aggregation approaches, multigrid methods, and conjugate gradient iteration counts, showing that our proposed algorithm performs better in serial and parallel. / Modeling real-world problems incurs a high computational cost because these mathematical models involve large-scale data manipulation. Thus we need fast and efficient algorithms. Nowadays there are many high-performance approaches for these problems. One such method is called the Multigrid algorithm. This approach models a physical domain using a hierarchy of grids, and so the effectiveness of these approaches relies on how well data can be transferred from grid to grid. In this thesis, we focus on the aggregation approach, which clusters a grid’s vertices according to its connections. We also provide an alternative parallel aggregation algorithm to give a faster solution. We show numerous experimental results that compare different aggregation approaches and multigrid methods, showing that our proposed algorithm performs better in serial and parallel than other popular implementations.

A fundamental issue in many distributed computing problems is the need for nodes to distinguish themselves from their neighbors in a process referred to as symmetry breaking. Many well-known problems such as Maximal Independent Set (MIS), t-Ruling Set, Maximal Matching, and (\Delta+1)-Coloring, belong to the class of problems that require symmetry breaking. These problems have been studied extensively in the LOCAL model, which assumes arbitrarily large message sizes, but not as much in the CONGEST and k-machine models, which assume messages of size O(log n) bits. This dissertation focuses on finding upper and lower bounds for symmetry breaking problems, such as MIS and t-Ruling Set, in these congested models. Chapter 2 shows that an MIS can be computed in O(sqrt{log n loglog n}) rounds for graphs with constant arboricity in the CONGEST model. Chapter 3 shows that the t-ruling set problem, for t \geq 3, can be computed in o(log n) rounds in the CONGEST model. Moreover, it is shown that a 2-ruling set can be computed in o(log n) rounds for a large range of values of the maximum degree in the graph. In the k-machine model, k machines must work together to solve a problem on an arbitrary n-node graph, where n is typically much larger than k. Chapter 4 shows that any algorithm in the BEEP model (which assumes 'primitive' single bit messages) with message complexity M and round complexity T can be simulated in O(t(M/k^2 + T) poly(log n)) rounds in the k-machine model. Using this result, it is shown that MIS, Minimum Dominating Set (MDS), and Minimum Connected Dominating Set (MCDS) can all be solved in O(poly(log n) m/k^2) rounds in the k-machine model, where 'm' is the number of edges in the input graph. It is shown that a 2-ruling set can be computed even faster, in O((n/k^2+ k) poly(log n)) rounds, in the k-machine model. On the other hand, using information theoretic techniques and a reduction to a communication complexity problem, an \Omega(n/(k^2 poly(log n))) rounds lower bound for MIS in the k-machine model is also shown. As far as we know, this is the first example of a lower bound in the k-machine model for a symmetry breaking problem. Chapter 5 focuses on the Max Clique problem in the CONGEST model. Max Clique is trivially solvable in one round in the LOCAL model since each node can share its entire neighborhood with all neighbors in a single round. However, in the CONGEST model, nodes have to choose what to communicate and along what communication links. Thus, in a sense, they have to break symmetry and this is forced upon them by the bandwidth constraints. Chapter 5 shows that an O(n^{3/5})-approximation to Max Clique in the CONGEST model can be computed in O(1) rounds. This dissertation ends with open questions in Chapter 6.

Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G. Among all such minimum vertex-colorings of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-χ-coloring of G. The number of color classes that are dominating sets in a dominating-χ-coloring of G is defined to be the dominating-χ-color number of G. In this paper, we continue to investigate the dominating-χ-color number of a graph first defined and studied in [1].