Optimal Algorithms for Graphs and Images on a Shared Memory Mesh

In this paper we combine aspects of the PRAM and mesh models to achieve the benefits of each. Many fast algorithms are known for the PRAM, but it is unrealistic to build. Another well-known model is the 2-dimensional mesh, of which many have been built, but the algorithms are constrained by the mesh's diameter and bisection bandwidth, forcing algorithms for nontrivial problems to take time which is at least the square root of the number of processors. Here we use a minuscule amount of shared memory to make a practical design which is significantly faster than the standard mesh. We call this the shared memory mesh. We give several algorithms which show that a small amount of shared memory can significantly reduce the time for problems such as finding the median, labeling the connected components of an image, and labeling the connected components of a graph. Further, we show that many of these algorithms are optimal.