New Parallel Algorithms for Planarity Testing

Abstract

Planar graphs (defined as graphs in which no edges cross) have special properties and are often used in applications such as circuit design or transportation networks. While many linear work implementations of planarity testing algorithms exist, to our best knowledge, there is no practical implementation of a parallel planarity testing algorithm. In this thesis, we will describe and analyze two new parallel algorithms for planarity testing, both derived from the Boyer-Myrvold algorithm. First, we will present a divide-and-conquer approach, where the graph's edges are evenly distributed among worker threads. Each thread independently executes the sequential Boyer-Myrvold algorithm on its designated subgraph. Then, pairs of subgraphs are merged by embedding the edges between subgraphs with modified Boyer-Myrvold methods. The primary challenge of the divide-and-conquer approach is the merge step as determining the relative positions of subgraphs is a complicated and difficult process. Next, we describe the design and implementation of a new and simpler parallel algorithm. This algorithm modifies the Boyer-Myrvold algorithm by processing vertices in layers from the bottom-up (rather than sequentially by reverse DFI order). The computation in each layer is parallelized. On planar graphs, this algorithm achieves 2.4--2.7 times speedup over the sequential algorithm when run on 16 cores. On non-planar graphs, the performance gain is even more significant, with speedups ranging from 9 to 22 times on 16 cores.