Simetrik olmayan matrisler için spektral bölümlemeyi yeniden değerlendirme

Parallel solutions to scientific problems having graph representation require efficient tasks and partitioning data. In this thesis, various parallel graph partitioning algorithms are studied. While these algorithms are applicable to both directed and undirected graphs, we focus on the directed case whose matrix representations are sparse and unsymmetric arising in linear system of equations representing various application domains such as computational fluid dynamics and thermal problems. Strategies inspected in this study are ParMETIS with the Multilevel Kernighan-Lin algorithm and the spectral partitioning algorithm with k-means clustering (SPEC) as well as the recursive spectral partitioning algorithm in CHACO. We have implemented SPEC in C programming language using PETSc and SLEPc libraries, whereas CHACO and ParMETIS are called from PETSc. Weighted partitioning is done under the consideration of the edge weights of the graph. SPEC is compared with the libraries only when the unweighted partitioning is made due to the limitations of the libraries for weighted partitioning. Hence, for weighted partitioning, only various eigensolver tolerances in SLEPc are studied in terms of the edge-cut and partitioning time. Another study is performed for the spectral partitioning algorithm based on eigensolver tolerance used with the k-means algorithm in MATLAB. The comparison is based on the quality of the partitioning (edge-cut and partition imbalance) and the number of iterations. The quality of partitioning is determined by the edge-cut and the load imbalance, which could be based on the edge and vertex imbalance ratios of partitions depending on the application. Since the adjacency matrix of a graph is structurally symmetric, the eigenvalue problem can only be solved approximately when the matrix is unsymmetric. Thus, only approximate results are provided in this study. It is deduced that using SPEC performs better than the existing software libraries when the number of cut edges is compared in unweighted partitioning of unsymmetric matrices.