Lecture 25 : Element - wise Sampling of Graphs and Linear Equation Solving , Cont

Last time, we talked about a very simple solver for Laplacian-based systems of linear equations, i.e., systems of linear equations of the form Ax = b, where the constraint matrix A is the Laplacian of a graph. This is not fully-general—Laplacians are SPSD matrices of a particular form—but equations of this form arise in many applications, certain other SPSD problems such as those based on SDD matrices can be reduced to this, and there has been a lot of work recently on this topic since it is a primitive for many other problems. The solver from last time is very simple, and it highlights the key ideas used in fast solvers, but it is very slow. Today, we will describe how to take those basic ideas and, by coupling them with certain graph theoretic tools in various ways, obtain a “fast” nearly linear time solver for Laplacian-based systems of linear equations.