This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6.896 Sublinear Time Algorithms September 11, 2004 Lecture 17 Lecturer: Eli BenSasson Scribe: Rafael Pass 1 Lower Bounds for Property Testing: Rest of the Proof This lecture is the continuation of the previous one, in which we prove the following Theorem, stated last time: Theorem 1 ([1]) There exists integers c, d , and values > , < < 1 / 2 , > such that for all suciently large n , with high probability, for a random low density parity check code (LDPC) every ( , , , q ) linear test requires q n queries. As mentioned in the previous lecture, this result also extends to any test, and not just linear tests. Towards the goal of proving the above theorem we rely on Yaos minmax principle, which (in our case) boils down to showing one bad distribution over inputs, such that every deterministic linear test fails with probability larger than . We continue to use the notation and definitions from previous lecture. 1.1 The bad distribution Recall that the code V defined by the constraint matrix A is V = ker( A ) = { x F n 2  Ax = 0 } . Let B i be the uniform distribution over strings that falsify constraint A i , but satify the rest, i.e., B i = ker( A i ) ker( A ), where A i is the matrix obtained by removing row i from A . Define the bad distribution to be the uniform convex combination of B i s. In other words, the bad distribution is obtained by uniformly chosing i [ m ] (recall m is the number of rows in A ) and then uniformly chosing an element in B i . Note that it follows from the definition of B that for every w B , w / V . In fact, every w B is far from V , as is shown in the following lemma (a proof sketch of the lemma is given in the next section)....
View
Full
Document
This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Spring '04 term at MIT.
 Spring '04
 RonittRubinfeld
 Algorithms

Click to edit the document details