Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes Kenji Obata.

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Transcript of Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes Kenji Obata.

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Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes Kenji Obata Slide 2 Codes Error correcting code C : {0,1} n {0,1} m with decoding procedure A s.t. for y {0,1} m with d(y,C(x)) m, A(y) = x Slide 3 Locally Decodable Codes Weaken power of A: Can only look at a constant number q of input bits Weaken requirements: A need only recover a single given bit of x Can fail with some probability bounded away from Study initiated by Katz and Trevisan [KT00] Slide 4 Locally Decodable Codes Define a (q, , )-locally decodable code: A can make q queries (w.l.o.g. exactly q queries) For all x {0,1} n, all y {0,1} m with d(y, C(x)) m, all inputs bits i 1,, n A(y, i) = x i w/ probability + Slide 5 LDC Applications Direct: Scalable fault-tolerant information storage Indirect: Lower bounds for certain classes of private information retrieval schemes (more on this later) Slide 6 Lower Bounds for LDCs [KT00] proved a general lower bound m n q/(q-1) (at best n 2, but known codes exponential) For 2-query linear LDCs Goldreich, Karloff, Schulman, Trevisan [GKST02] proved an exponential bound m 2 (n) Slide 7 Lower Bounds for LDCs Restriction to linear codes interesting, since known LDC constructions are linear But 2 (n) not quite right: Lower bound should increase arbitrarily as decoding probability 1 ( ) No matching construction Slide 8 Lower Bounds for LDCs In this work, we prove that for 2-query linear LDCs, m 2 (/(1-2)n) Optimal: There is an LDC construction matching this within a constant factor in the exponent Slide 9 Techniques from [KT00] Fact: An LDC is also a smooth code (A queries each position w/ roughly the same probability) so can study smooth codes Connects LDCs to information-theoretic PIR schemes: q queries q servers smoothness statistical indistinguishability Slide 10 Techniques from [KT00] For i 1,,n, define the recovery graph G i associated with C: Vertex set {1,,m} (bits of the codeword) Edges are pairs (q 1, q 2 ) such that, conditioned on A querying q 1, q 2, A(C(x),i) outputs x i with prob > Call these edges good edges (endpoints contain information about x i ) Slide 11 Techniques from [KT00]/[GKST02] Theorem: If C is (2, c, )-smooth, then G i contains a matching of size m/c. Better to work with non-degenerate codes Each bit of the encoding depends on more than one bit of the message For linear codes, good edges are non-trivial linear combinations Fact: Any smooth code can be made non-degenerate (with constant loss in parameters). Slide 12 Core Lemma [GKST02] Let q 1,,q m be linear functions on {0,1} n s.t. for every i 1,,n there is a set M i of at least m disjoint pairs of indices j 1, j 2 such that x i = q j 1 (x) + q j 2 (x). Then m 2 n. Slide 13 Putting it all together If C is a (2, c, )-smooth linear code, then (by reduction to non-degenerate code + existence of large matchings + core lemma), m 2 n/4c. If C is a (2, , )-locally decodable linear code, then (by LDC smooth reduction), m 2 n/8. Slide 14 Putting it all together Summary: locally decodable smooth big matchings exponential size This work: locally decodable big matchings (skip smoothness reduction, argue directly about LDCs) Slide 15 The Blocking Game Let G(V,E) be a graph on n vertices, w a prob distribution on E, X w an edge sampled according to w, S a subset of V Define the blocking probability (G) as min w ( max |S|n Pr (X w intersects S) ) Slide 16 The Blocking Game Want to characterize (G) in terms of size of a maximum matching M(G), equivalently defect d(G) = n 2M(G) Theorem: Let G be a graph with defect n. Then (G) min (/(1-), 1). Slide 17 The Blocking Game clique nn (1-)n Define K(n,) to be the edge- maximal graph on n vertices with defect n: K1K1 K2K2 Slide 18 The Blocking Game Optimization on K(n,) is a relaxation of optimization on any graph with defect n If d(G) n then (G) (K(n,)) So, enough to think about K(n,). Slide 19 The Blocking Game Intuitively, best strategy for player 1 is to spread distribution as uniformly as possible A ( 1, 2 )-symmetric dist: all edges in (K 1,K 2 ) have weight 1 all edges in (K 2,K 2 ) have weight 2 Lemma: ( 1, 2 )-symmetric dist w s.t. (K(n,)) = max |S|n Pr (X w intersects S). Slide 20 The Blocking Game Claim: Let w 1,,w k be dists s.t. max |S|n Pr (X w i intersects S) = (G). Then for any convex comb w = i w i max |S|n Pr (X w intersects S) = (G). Proof: For S V, |S| n, intersection prob is i (G) = (G). So max |S| n Pr (X w intersects S) (G). But by defn of (G), this must be (G). Slide 21 The Blocking Game Proof: Let w be any distribution optimizing (G). If w does, then so does (w) for Aut(G) = . By prior claim, so does w = (1/||) (w). For e E, , w(e) = (1/||) w((e)) = (1/||) w((e)) = w((e)).. So, if e, e are in the same -orbit, they have the same weight in w w is ( 1, 2 )-symmetric. Slide 22 The Blocking Game Claim: If w is ( 1, 2 )-sym then S V, |S| n s.t. Pr (X w intersects S) min (/(1-), 1). Proof: If 1 then can cover every edge. Otherwise, set S = any n vertices of K 2. Then Pr = ( 1/(1 - ) + n 2 (1 - ) 2 ) which, for < 1 - , is at least /(1 - ) (optimized when 2 = 0). Slide 23 The Blocking Game Theorem: Let G be a graph with defect n. Then (G) min (/(1-), 1). Proof: (G) (K(n,)). Blocking prob on K(n,) is optimized by some ( 1, 2 )-sym dist. For any such dist w, n vertices blocking w with Pr min (/(1-), 1). Slide 24 Lower Bound for LDLCs Still need a degenerate non-degenerate reduction (this time, for LDCs instead of smooth codes) Theorem: Let C be a (2, , )-locally decodable linear code. Then, for large enough n, there exists a non-degenerate (2, /2.01, )-locally decodable linear code C : {0,1} n {0,1} 2m. Slide 25 Lower Bound for LDLCs Theorem: Let C be a (2, , )-LDLC. Then, for large enough n, m 2 1/4.03 /(1-2) n. Proof: Make C non-degenerate Local decodability low blocking probability (at most - ) low defect ( 1 (/2.01)/(1-2)) big matching ( (/2.01)/(1-2) (2m) ) exponentially long encoding (m 2 (1/4.02) /(1-2)n 1 ) Slide 26 Matching Upper Bound Hadamard code on {0,1} n y i = a i x (a i runs through {0,1} n ) 2-query locally decodable Recovery graphs are perfect matchings on n-dim hypercube Success parameter = - 2 Can use concatenated Hadamard codes (Trevisan): Slide 27 Matching Upper Bound Set c = (1-2)/4 (can be shown that for feasible values of , , c 1). Divide input into c blocks of n/c bits, encode each block with Hadamard code on {0,1} n/c. Each block has a fraction c corrupt entries, so code has recovery parameter - 2 (1-2)/4 = Code has length (1-2)/4 2 4/(1-2)n Slide 28 Conclusions There is a matching upper bound (concatenated Hadamard code) New results for 2-query non-linear codes (but using apparently completely different techniques) q > 2? No analog to the core lemma for more queries But blocking game analysis might generalize to useful properties other than matching size