The Communication Complexity of Gap Hamming Distance

Transcript

1. 2016/9/30 1 The Communication Complexity of Gap Hamming Distance Himemiya Nanao TheoryGroup@NJU himemiya@smail.nju.edu.cn
2. The Communication Model 2016/9/30 2
3. The GHD Problem  2016/9/30 3
4. The Motivation  2016/9/30 4
5.  2016/9/30 5
6.  2016/9/30 6
7.  2016/9/30 7
8.  2016/9/30 8
9. Gap Orthogonality problem  2016/9/30 9
10. Near Orthogonal Vectors 2016/9/30 10 
11. Many +1  2016/9/30 11
12.  2016/9/30 12 ≥
13.  2016/9/30 13
14.  2016/9/30 14
15. Rectangle Must Small  2016/9/30 15 ≥
16.  Let = /10 . We can find 1 ,⋯ , ∈ ′ as Lemma 1.  Let ′ = ൛ ∈ |Pr∈[] , ≤ /4 ≥ 1 − 2016/9/30 16
17.  Thus Pr∈ −1,1 ቂ , ≤ 4 for at least 2016/9/30 17
18. ORT = Ω  Let be the uniform distribution over −1,1 × −1,1 .  Lemma 3 guarantees that ∩ ORT −1 1 > ≥ ∩ ORT −1 −1 for any rectangle with > 2−, where , > 0 are some small constants.  ORT −1 −1 = Θ 1 2016/9/30 18
19. ORT −1 −1 = Θ 1  Fix any ∈ −1,1 .  Recall that , = − 2H ,  When is close to /2, it is true that ≥ 2 , where is some constant.  # such that , ≤ 8 = σ = 2 − 8 2 + 8 ≥ 4 ∙ 2 = 2 4 . 2016/9/30 19
20. ORT = Ω  Let be the uniform distribution over −1,1 × −1,1 .  Lemma 3 guarantees that ∩ ORT −1 1 > ≥ ∩ ORT −1 −1 for any rectangle with > 2−, where , > 0 are some small constants.  ORT −1 −1 = Θ 1  By the corrupted bound [2], ORT ≥ log 2 ORT −1 −1 − 2016/9/30 20
21. Proof of Lemma 3  Let = for short.  σ=1 ≤ σ=1 2 1 2 ≤ .  σ=+1 ≤ rank − +1  Let = σ be a SVD of , by definition  , ෩ = σ , ෩ = σ ෩ ≤ 1 ෩ σ 2016/9/30 21
22. Proof of Lemma 3  σ=1 ≤ σ=1 2 1 2 ≤ .  σ=+1 ≤ rank − +1  Let = σ be a SVD of , by definition  , ෩ = σ , ෩ = σ ෩ ≤ 1 ෩ σ  σ ≥ , ෩ /1 ෩ 2016/9/30 22
23. Proof of Lemma 3  σ=1 ≤ σ=1 2 1 2 ≤ .  σ=+1 ≤ rank − +1  σ ≥ , ෩ /1 ෩  ⟹ + rank − +1 ≥ , ෩ /1 ෩  ⟹ +1 ≥ , ෩ 1 ෩ − / rank − 2016/9/30 23
24. Reference 1) Alon, Noga, and Joel H. Spencer. The probabilistic method. John Wiley & Sons, 2004. 2) Beame, Paul, et al. "A strong direct product theorem for corruption and the multiparty communication complexity of disjointness." Computational Complexity 15.4 (2006): 391-432. 3) Chakrabarti, Amit, and Oded Regev. "An optimal lower bound on the communication complexity of gap-hamming-distance." SIAM Journal on Computing 41.5 (2012): 1299-1317. 4) Sherstov, Alexander A. "The Communication Complexity of Gap Hamming Distance." Theory of Computing 8.1 (2012): 197-208. 5) Talagrand, Michel. "Concentration of measure and isoperimetric inequalities in product spaces." Publications Mathématiques de l'Institut des Hautes Etudes Scientifiques 81.1 (1995): 73-205. 6) Vidick, Thomas. "A concentration inequality for the overlap of a vector on a large set." (2011). 2016/9/30 24
25. Thank you 2016/9/30 25