Randomized Algorithms (Spring 2010)/Expander graphs and rapid mixing random walks: Difference between revisions
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=== Expander graphs === | === Expander graphs === | ||
We | We consider undirected graphs <math>G(V,E)</math>. In addition, we allow multiple edges between two vertices. | ||
Some notations: | Some notations: | ||
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'''Expander graphs''' are <math>d</math>-regular graphs with | '''Expander graphs''' are '''<math>d</math>-regular''' (multi)graphs with <math>d=O(1)</math> and <math>\phi(G)=\Omega(1)</math>. | ||
=== Existence === | === Existence === |
Revision as of 07:36, 6 May 2010
Mixing Time
Graph Expansion
Expander graphs
We consider undirected graphs [math]\displaystyle{ G(V,E) }[/math]. In addition, we allow multiple edges between two vertices.
Some notations:
- For [math]\displaystyle{ S,T\subset V }[/math], let [math]\displaystyle{ E(S,T)=\{uv\in E\mid u\in S,v\in T\} }[/math].
- The Edge Boundary of a set [math]\displaystyle{ S\subset V }[/math], denoted [math]\displaystyle{ \partial S\, }[/math], is [math]\displaystyle{ \partial S = E(S, \bar{S}) }[/math].
Definition (Graph expansion)
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Expander graphs are [math]\displaystyle{ d }[/math]-regular (multi)graphs with [math]\displaystyle{ d=O(1) }[/math] and [math]\displaystyle{ \phi(G)=\Omega(1) }[/math].
Existence
Suppose that [math]\displaystyle{ d }[/math] is even. We can generate a random [math]\displaystyle{ d }[/math]-regular graph [math]\displaystyle{ G(V,E) }[/math] as follows:
- Let [math]\displaystyle{ V }[/math] be the vertex set. Uniformly and independently choose [math]\displaystyle{ \frac{d}{2} }[/math] cycles of [math]\displaystyle{ V }[/math].
- For each vertex [math]\displaystyle{ v }[/math], for every cycle, assuming that the two neighbors of [math]\displaystyle{ v }[/math] in that cycle is [math]\displaystyle{ w }[/math] and [math]\displaystyle{ u }[/math], add two edges [math]\displaystyle{ wv }[/math] and [math]\displaystyle{ uv }[/math] to [math]\displaystyle{ E }[/math].
The resulting [math]\displaystyle{ G(V,E) }[/math] is a multigraph. That is, it may have multiple edges between two vertices. We will show that [math]\displaystyle{ G(V,E) }[/math] is an expander graph with high probability. Formally, for some constant [math]\displaystyle{ d }[/math] and constant [math]\displaystyle{ \alpha }[/math],
- [math]\displaystyle{ \Pr[\phi(G)\ge \alpha]=1-o(1) }[/math].