# Expanders, Property Testing and the PCP theorem

## Presentation guidelines

Posted in presentations, spr09 by atri on April 6, 2009

I wrote down a guidelines/advice post for the presentations in my coding theory course. Please follow the guidelines while preparing for your talk in the seminar.

## Theory Seminar Talks

Posted in annoucements, presentations, spr09 by atri on February 6, 2009

We have two confirmed talks in the theory seminar for this semester. The first one is on March 2nd and the next one is on May 4th. For the latter please use the comments section to let me know what times work for you to attend the talk (it is the finals week).

Due to the March 2nd theory seminar, the first four student presentation dates have been moved up. I have updated the dates in the presentation schedule accordingly.

## Spr09 1st presentation schedule

Posted in presentations, spr09 by atri on February 3, 2009

Here is the schedule of the first set of presentations:

1. Thanh (Feb 16 ): Lecture notes on Parallel repetition from Venkat and Ryan’s course.
2. Steve (Feb 18 ) Ben-Sasson, Sudan: Short PCPs wth polylog query complexity.
3. Nathan (Feb 23  ): Continue with lectures notes on the Parallel repetition from Venkat and Ryan’s course.
4. Swapnoneel (Feb 25 )
5. Yang (Mar 4 )

Please let us know once you have chosen your paper so that we can make a note of it above. For your reference here is a link to the suggested list of papers.

## List of PCP-related papers

Posted in annoucements, presentations, spr09 by atri on January 14, 2009

The list of suggested papers for the first set of presentations is now up on the webpage. Some of them have some caveats, so read them carefully. These papers are probably much much harder than any you presented last semester, so we highly encourage you to start early on picking/preparing for your presentations.

## Student Presentation #8

Posted in presentations by atri on January 13, 2009

(Guest post by Swapnoneel Roy)

I presented the paper titled Bounds on 2-Query Codeword Testing by Eli Ben-Sasson, Oded Goldreich, and Madhu Sudan. In the paper, the authors study  $2$-query codeword testers. The main results in the paper are the upper bounds on the size of linear (respectively binary) codes that admit such testers (respectively testers of perfect completeness).

In other words, it was showed, if $C\subseteq F^n$ be a $(2, c, s)$-locally testable linear code with minimal relative distance $\delta>0$, for $c> s$, we have $|C| \le |F|^{3/\delta}$.

## Student Presentation # 9

Posted in presentations by atri on January 13, 2009

(Guest post by Steve Uurtamo)

I presented the result (by Noga Alon, Eldar Fischer, Ilan Newman and Asaf Shapira) that using the dense graph model and allowing for two-sided error, the set of graph properties that can be tested for using a constant number of queries to the adjacency matrix of a graph (constant for any fixed error distance $\epsilon$) exactly correspond with those that can be determined using a set of Szemeredi regularity constraints.

Three examples of such reductions are given in the paper; vertex $k$-colorability (testable), co-subgraph isomorphism (testable) and graph isomorphism (not testable).

## Student presentation # 2

Posted in presentations by atri on January 13, 2009

(Guest post by Steve Uurtamo)

I presented the result by Omer Reingold that the decision problem of determining whether two given vertices ($s$ and $t$) are connected in a given graph can be determined in space logarithmic in the size of the graph.

The approach given in the paper was to transform each connected component of the graph into an subgraph with good expansion properties (using an iterated transformation that respects the original connectivity properties), and then simply to walk the vertices of the expander starting at $s$, either arriving at $t$ or proving that no such path exists.  Because the new graph has small diameter, it is possible to enumerate and check all such possible paths in logarithmic space.

## Student Presentation #3

Posted in presentations by atri on November 17, 2008

(Guest post by Nathan Russell)

I presented the first half of the paper On-Line Algorithms for Path Selection in a Nonblocking Network, by Arora, Leighton and Maggs.

Broadly speaking, nonblocking networks are networks of switches (represented by vertices) such that it is possible for any set of inputs to be pairwise connected to any set of outputs without conflict.  The authors, in my part of the paper, set up background to present a network in which $N$ inputs can be connected to $N$ outputs by $O(N \log N)$ switches of bounded degree, using only $O(\log N)$ bit steps.  All of these bounds are known to be optimal. Additionally, the network can do this for multiple  connections simultaneously.

I presented the butterfly and Benes networks, as well as the multi-butterfly and multi-Benes networks constructed by combining them, and gave intuition and prepatory proofs to set up for the authors’ eventual proof of how the networks are nonblocking when a series of splitters is combined with a series of mergers, each built out of multi-Benes networks.

## Student Presentation #4

Posted in presentations by atri on November 17, 2008

(Guest post by Swapnoneel Roy)

I presented the first algorithmic result in the paper Online Algorithms for Path Selection in a Nonblocking Network by Sanjeev Arora, Tom Leighton and Bruce Maggs. The algorithm was applied on a Multi-Butterfly network with $N$ inputs and outputs with the $(\alpha, \delta)$ unshared neighbor property. In the algorithm, all the paths were extended from level $\ell$ to level $\ell+1$ (for any level $\ell$) of the network by performing the following steps:

1. every path that waits to be extended sends out a “proposal” to each of its output level $\ell+1$  neighbors in the desired direction (up or down),
2. every output node that receives one proposal sends back its acceptance to that proposal,
3. every path that receives an acceptance advances to one of its accepting outputs on level $\ell+1$.

Going by this way, the algorithm extends the paths from level 0 to level $\log N$. The time taken to extend the paths from one level to the next was shown to be $O(\log N)$. This makes the total running time of the algorithm $O(\log^2{N})$.

## Student Presentations

Posted in annoucements, presentations by atri on November 9, 2008

Here is the schedule of the student presentation (all of them start at 10:30am in Bell 224):

1. Yang (Nov 17): Towards 3-query locally decodable codes of subexponential length (Yekhanin)
2. Nathan+Swapnoneel (Nov 21+24): Bounds on 2-query codeword testing (Ben-Sasson, Goldreich, Sudan).
3. Steve (Dec 1)
4. Than (Dec 5): Approximating the Minimum Spanning Tree Weight in Sublinear Time (Chazelle, Rubinfeld, Trevisan).

Please let us know once you choose your paper and we will update the list above.