Efficient Out-of-Core Algorithms for Linear Relaxation Using Blocking Covers

摘要

When a numerical computation fails to fit in the primary memory of a serial or parallel computer, a so-called “out-of-core” algorithm, which moves data between primary and secondary memories, must be used. In this paper, we study out-of-core algorithms for sparse linear relaxation problems in which each iteration of the algorithm updates the state of every vertex in a graph with a linear combination of the states of its neighbors. We give a general method that can save substantially on the I/O traffic for many problems. For example, our technique allows a computer withMwords of primary memory to performT=Ω(M1/5) cycles of a multigrid algorithm for a two-dimensional elliptic solver over an n-point domain using onlyΘ(nT/M1/5) I/O transfers, as compared with the naive algorithm which requiresΩ(nT) I/O's. Our method depends on the existence of a “blocking” cover of the graph that underlies the linear relaxation. A blocking cover has the property that the subgraphs forming the cover have large diameters once a small number of vertices have been removed. The key idea in our method is to introduce a variable for each removed vertex for each time step of the algorithm. We maintain linear dependences among the removed vertices, thereby allowing each subgraph to be iteratively relaxed without external communication. We give a general theorem relating blocking covers to I/O-efficient relaxation schemes. We also give an automatic method for finding blocking covers for certain classes of graphs, including planar graphs andd-dimensional simplicial graphs with constant aspect ratio (i.e., graphs that arise from dividingd-space into “well-shaped” polyhedra). As a result, we can performTiterations of linear relaxation on anyn-vertex planar graph using onlyΘ(n+nT lgn/M1/4) I/O's or on anyn-noded-dimensional simplicial graph with constant aspect ratio using onlyΘ(n+nT lg n/MΩ(1/d)) I/O's.