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Unformatted text preview: 6.896 Sublinear Time Algorithms April 19, 2007 Lecture 20 Lecturer: Ronitt Rubinfeld Scribe: Erdong Chen 1 Graph Isomorphism ∼ = We talked about a lower bound last time. Recall from last lecture (all are based on adjacent matrix model): Thm H known H unknown 1-sided error ˜ Θ( n ) ˜ Θ( n 3 2 ) 2-sided error ˜ Θ( √ n ) Ω( n ) , ˜ O ( n 5 4 ) Difference between the case that H is known and H is unknown: when H is known, query complexity to H is zero. We will discuss two upper bounds in today’s class. 1.1 1-Sided Tester In this subsection, we will discuss the upper bound of 1-sided testers when H is known. Lemma 1 When H is known, there is a 1-sided tester with ˜ Θ( n ) queries. Proof Algorithm: 1 Q ← pick each edge slot with probablity ln n εn i.i.d. (no queries yet!) 2 if | Q | ≤ 10 n ln n ε , ACCEPT, else ask queries in Q of G . 3 if there is a isomorphism from G to H consistent with Q , ACCEPT runtime: O ( n ! n ln n ε ) (because n ! different permutations exist) query complexity:...
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- Fall '04
- Algorithms, Isomorphism, Upper and lower bounds, special case, ln n εn