WS 2011 / 2012 Lecture 8 : Construction of Expanders

In this lecture we study the explicit constructions of expander graphs. Although we can construct expanders probabilistically, this does not suffice for many applications. • One applications of expander graphs is for reducing the randomness complexity of algorithms (cmp. Lecture 6), thus constructing the graph itself randomly does not serve this purpose. • Sometimes we may even need expanders of exponential size. In this case, we cannot store the whole description of the graph. Hence we need a more explicit version of expanders. We call a family of expander graphs explicitly constructible if the construction satisfies the following properties: 1. We can construct the whole graph in time poly(n), where n is the number of vertices in a graph. 2. For any vertex v and integer i ∈ {1, · · · , d}, we can find the i-th neighbor of v in time poly(log n, log d). 3. For any vertices v and u, we can determine if they are adjacent in time poly(log n). Let G = (V,E) be a d-regular graph. For each vertex v ∈ V , we label the edges adjacent to v and let v[i] be the i-th edge of v. Define a rotation map RotG : V × [d] → V × [d] by RotG(u, i) = (v, j) where v is the i-th neighbor of u and u is the j-th neighbor of v. Example: For the graph shown in Figure 1, we have Rot(a, 3) = (h, 1).