Lecture 9 : Expanders Part 2 , Extractors
نویسندگان
چکیده
In the previous lecture we saw the definitions of eigenvalue expanders, edge expanders, and vertex expanders for d-regular graphs. To recap, in an eigenvalue expander, all except the first eigenvalue of the graph’s adjacency matrix are bounded below d. In an edge expander, every subset of vertices of size below a certain threshold has a large number of edges “going out” of the subset. And in a vertex expander, every such subset of vertices has a large neighborhood.
منابع مشابه
Randomness Conductors and Constant-Degree Lossless Expanders [Extended Abstract]
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1− ǫ)D, where D is the degree and ǫ is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the...
متن کاملQuantum expanders and the quantum entropy difference problem
Classical expanders and extractors have numerous applications in computer science. However, it seems these classical objects have no meaningful quantum generalization. This is because it is easy to generate entropy in quantum computation simply by tracing out registers. In this paper we define quantum expanders and extractors in a natural way. We show that this definition is exactly what is nee...
متن کاملRandomness Conductors and Constant-Degree Expansion Beyond the Degree / 2 Barrier PRELIMINARY VERSION
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: , where is the degree and is an arbitrarily small constant. Such graphs are essential components in networks that can implement fast distributed, routing algorithms e.g. [PU89, ALM96, BFU99]. They are useful in exp...
متن کاملLecture 7 : Expanders
In this lecture we give basic background about expander graphs. Expanders are graphs with strong connectivity properties: every two large subsets of vertices in an expander have many edges connecting them. Surprisingly, one can construct very sparse graphs that are expanders, and this is what makes them so useful. Expanders have a huge number of applications in theoretical computer science: in ...
متن کاملExistence of Extractors Figure 1: an Extractor Viewed as a Bipartite Graph
A vertex u on the left is connected to v on the right, if Ext(u, d) = v for some seed d. A (k, ǫ)-extractor on a flat k-source gives a distribution which is ǫ-close to uniform. In particular, at most ǫ fraction of vertices on the right side, may not have any incoming edges. In other words, for any set on the left side of size at list K, we will have |N(K)| ≥ (1 − ǫ)M . This suggests that a good...
متن کامل