L ECTURE 6 Last time Limitations of sublinear-time algorithms - PowerPoint PPT Presentation

Sublinear Algorithms L ECTURE 6 Last time • Limitations of sublinear-time algorithms • Yao’s Minimax Principle • Example: testing monotonicity • Communication complexity Today • Communication complexity • Other models of computation 10/5/2020 Sofya Raskhodnikova;Boston University

Communication Complexity A Method for Proving Lower Bounds [Blais Brody Matulef 11 ] Use known lower bounds for other models of computation Partially based on slides by Eric Blais

(Randomized) Communication Complexity 𝑇ℎ𝑏𝑠𝑓𝑒 𝑠𝑏𝑜𝑒𝑝𝑛 𝑡𝑢𝑠𝑗𝑜𝑕 1101000101110101110101010110 … Alice Bob 0100 11 001 ⋯ 0011 𝐽𝑜𝑞𝑣𝑢: 𝑦 Input: 𝑧 Compute 𝐷 𝑦, 𝑧 Goal: minimize the number of bits exchanged. Communication complexity of a protocol is t he maximum number of bits • exchanged by the protocol. • Communication complexity of a function 𝐷 , denoted 𝑆(𝐷) , is the communication complexity of the best protocol for computing C. 3

Example: Set Disjointness 𝐸𝐽𝑇𝐾 𝒍 1101000101110101110101010110 … Alice Bob 𝐽𝑜𝑞𝑣𝑢: 𝑇 ⊆ [𝑜] , 𝑇 = 𝑙 . Input: 𝑈 ⊆ [𝑜] , 𝑈 = 𝑙 Compute 𝐸𝐽𝑇𝐾 𝑙 𝑇, 𝑈 = ቊ 𝒃𝒅𝒅𝒇𝒒𝒖 if 𝑇 ∩ 𝑈 = ∅ 𝒔𝒇𝒌𝒇𝒅𝒖 otherwise Theorem [Kalyanasundaram Schmitger 92, Razborov 92] 𝑜 𝑆 DISJ 𝑙 ≥ Ω 𝑙 for all 𝑙 ≤ 2 . 4

A lower bound using CC method Testing if a Boolean function is a k-parity

Linear Functions Over Finite Field 𝔾 2 A Boolean function 𝑔: 0,1 𝑜 → {0,1} is linear (also called parity ) if 𝑔 𝑦 1 , … , 𝑦 𝑜 = 𝑏 1 𝑦 1 + ⋯ + 𝑏 𝑜 𝑦 𝑜 for some 𝑏 1 , … , 𝑏 𝑜 ∈ {0,1} no free term • Work in finite field 𝔾 2 example – Other accepted notation for 𝔾 2 : 𝐻𝐺 2 and ℤ 2 – Addition and multiplication is mod 2 001001 + – 𝒚 = 𝑦 1 , … , 𝑦 𝑜 , 𝒛 = 𝑧 1 , … , 𝑧 𝑜 , that is, 𝒚, 𝒛 ∈ 0,1 𝑜 011001 𝒚 + 𝒛 = 𝑦 1 + 𝑧 1 , … , 𝑦 𝑜 + 𝑧 𝑜 010000 6

Linear Functions Over Finite Field 𝔾 2 A Boolean function 𝑔: 0,1 𝑜 → {0,1} is linear (also called parity ) if 𝑔 𝑦 1 , … , 𝑦 𝑜 = 𝑏 1 𝑦 1 + ⋯ + 𝑏 𝑜 𝑦 𝑜 for some 𝑏 1 , … , 𝑏 𝑜 ∈ {0,1} ⇕ [𝑜] is a shorthand for {1, … 𝑜} 𝑔 𝑦 1 , … , 𝑦 𝑜 = σ 𝑗∈S 𝑦 𝑗 for some 𝑇 ⊆ 𝑜 . Notation: 𝜓 𝑇 𝑦 = σ 𝑗∈𝑇 𝑦 𝑗 . 7

k-Parity Functions 𝑙 -Parity Functions A function 𝑔 ∶ 0,1 𝑜 → {0,1} is a 𝑙 -parity if 𝑔 𝑦 = 𝜓 𝑇 𝑦 = σ 𝑗∈𝑇 𝑦 𝑗 for some set 𝑇 ⊆ 𝑜 of size 𝑇 = 𝑙 . 9

Testing if a Boolean Function is a k-Parity Input: Boolean function 𝑔: 0,1 𝑜 → {0,1} and an integer 𝑙 Is the function a 𝑙 -parity or 𝜁 -far from a 𝑙 -parity Question: ( ≥ 𝜁2 𝑜 values need to be changed to make it a 𝑙 -parity)? Time: O 𝑙 log 𝑙 [Buhrman, Carcia−Soriano, Matsliah, de Wolf 13] W min(𝑙, 𝑜 − 𝑙 ) [Blais Brody Matulef 12] • Today: Ω(𝑙) for 𝑙 ≤ 𝑜/2 • Today’s bound implies W min(𝑙, 𝑜 − 𝑙 ) 10

Important Fact About Linear Functions Two different linear functions disagree on half of the values. Fact. • Consider functions 𝜓 𝑇 and 𝜓 𝑈 where 𝑇 ≠ 𝑈 . 0 0 – Let 𝑗 be an element on which 𝑇 and 𝑈 differ 1 1 (w.l.o.g. 𝑗 ∈ 𝑇 ∖ 𝑈 ) 1 0 – Pair up all 𝑜 -bit strings: (𝒚, 𝒚 𝑗 ) 𝒚 𝑏 𝑐 where 𝒚 𝑗 is 𝒚 with the 𝑗 th bit flipped. 0 1 ⋅ ⋅ – For each such pair, 𝜓 𝑇 (𝒚) ≠ 𝜓 𝑇 (𝒚 𝑗 ) ⋅ ⋅ but 𝜓 𝑈 (𝒚) = 𝜓 𝑈 (𝒚 𝑗 ) ⋅ ⋅ So, 𝜓 𝑇 and 𝜓 𝑈 differ on exactly one of 𝒚, 𝒚 𝑗 . 𝒚 𝑗 1 − 𝑏 𝑐 0 0 – Since all 𝒚 ’s are paired up, 1 0 𝜓 𝑇 and 𝜓 𝑈 differ on half of the values. 0 1 𝜓 𝑇 ( x ) 𝜓 𝑈 ( x ) A 𝑙 ′ −parity function, where 𝑙 ′ ≠ 𝑙, is ½-far from any k-parity. Corollary. 11

Reduction from 𝐸𝐽𝑇𝐾 𝒍/𝟑 to Testing k-Parity • Let 𝑈 be the best tester for the 𝑙 -parity property for 𝜁 = 1/2 – query complexity of T is 𝑟 testing 𝑙−parity . • We will construct a communication protocol for 𝐸𝐽𝑇𝐾 𝒍/𝟑 that runs 𝑈 and has communication complexity 2 ⋅ 𝑟 (testing 𝑙 −parity) . holds for CC of every protocol for 𝐸𝐽𝑇𝐾 𝒍 [Hastad Wigderson 07] • T hen 2 ⋅ 𝑟 (testing 𝑙 −parity) ≥ 𝑆 DISJ 𝑙/2 ≥ Ω 𝑙/2 for 𝑙 ≤ 𝑜/2 ⇓ 𝑟 (testing 𝑙 -parity) ≥ Ω 𝑙 for 𝑙 ≤ 𝑜/2 12

Reduction from 𝐸𝐽𝑇𝐾 𝒍/𝟑 to Testing k-Parity 1101000101110101110101010110 … ℎ = 𝑔 + 𝑕 (𝑛𝑝𝑒 2) ℎ 𝑦 ? 𝑔 𝑦 + 𝑕 𝑦 𝑛𝑝𝑒 2 Alice Bob T 𝒃𝒅𝒅𝒇𝒒𝒖/𝒔𝒇𝒌𝒇𝒅𝒖 𝑔(𝑦) 𝑕(𝑦) 𝐽𝑜𝑞𝑣𝑢: 𝑇 ⊆ [𝑜] , 𝑇 = 𝑙/2 . Input: 𝑈 ⊆ [𝑜] , 𝑈 = 𝑙/2 Compute: 𝑔 = 𝜓 𝑇 Compute: 𝑕 = 𝜓 𝑈 Output T’s answer • 𝑈 receives its random bits from the shared random string. 13

Analysis of the Reduction Queries: Alice and Bob exchange 2 bits for every bit queried by 𝑈 Correctness: • ℎ = 𝑔 + 𝑕 𝑛𝑝𝑒 2 = 𝜓 𝑇 + 𝜓 𝑈 𝑛𝑝𝑒 2 = 𝜓 𝑇Δ𝑈 • 𝑇Δ𝑈 = 𝑇 + 𝑈 − 2 𝑇 ∩ 𝑈 𝑙 if S∩T = ∅ • SΔ𝑈 = ቊ ≤ 𝑙 − 2 if S∩T ≠ ∅ ℎ is ቊ 𝑙−parity if S∩T = ∅ 𝑙 ′ −parity where 𝑙 ′ ≠ 𝑙 if S∩T ≠ ∅ 1/2-far from every 𝑙 -parity Summary: 𝑟 (testing 𝑙 -parity) ≥ Ω 𝑙 for 𝑙 ≤ 𝑜/2 14

Testing Lipschitz Property on Hypercube Lower Bound

Lipschitz Property of Functions f : 0,1 𝑜 → R 1 • A function 𝑔 ∶ 0,1 𝑜 → R is Lipschitz 1 2 2 if changing a bit of 𝑦 changes 𝑔(𝑦) by at most 1. 1 1 0 0 • Is 𝑔 Lipschitz or 𝜁 -far from Lipschitz Lipschitz ( 𝑔 has to change on many points to become Lipschitz)? 0 – Edge 𝑦 − 𝑧 is violated by 𝑔 if 𝑔 𝑦 − 𝑔(𝑧) > 1 . 0 0 2 Time: 2 0 2 – 𝑃(𝑜/𝜁) , logarithmic in the size of the input, 2 𝑜 [Chakrabarty Seshadhri] 2 1 2 -far from Lipschitz – Ω(𝑜) [Jha Raskhodnikova] 16

Testing Lipschitz Property Theorem Testing Lipschitz property of functions f: 0,1 𝑜 → {0,1,2} requires Ω(𝑜) queries. Prove it. 17

Summary of Lower Bound Methods • Yao’s Principle – testing membership in 1*, sortedness of a list and monotonicity of Boolean functions • Reductions from communication complexity problems – testing if a Boolean function is a 𝑙 -parity 18

Other Models of Sublinear Computation

Algorithms Resilient to Erasures (or Errors) ⊥ ⊥ A - B L ⊥ ⊥ B L A - ⊥ L A - B L ⊥ - B L A - B L A - B L A ? L ? B ? L ? randomized algorithm • ≤ 𝜷 fraction of the input is erased (or modified) adversarially before algorithm runs • Algorithm does not know in advance what’s erased (or modified) • Can we still perform computational tasks? 20

Property Testing Property Tester [Rubinfeld Sudan 96, Goldreich Goldwasser Ron 98] randomized far from 𝜁 YES NO algorithm YES Accept with Reject with Don’t probability probability care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 21

Property Testing with Erasures Erasure-Resilient Property Tester [Dixit Property Tester [Rubinfeld Sudan 96, Raskhodnikova Thakurta Varma 16] Goldreich Goldwasser Ron 98] • ≤ 𝛽 fraction of the input is erased adversarially Any completion Can be randomized far from 𝜁 𝜁 is far from YES completed NO NO algorithm YES to YES YES Accept with Accept with Reject with Reject with Don’t Don’t probability probability probability probability care care ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 ≥ 𝟑/𝟒 Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 22

Can We Make Testers α -Erasure-Resilient? It is easy if a tester makes only uniform queries (and the property is extendable ). • Use the original tester as black box and ignore erasures: 1 O 1−α factor query complexity overhead for all α ∈ 0,1 . • Applies to many properties – Monotonicity over poset domains [Fischer Lehman Newman Raskhodnikova Rubinfeld Samorodnitsky 02] – Convexity of black and white images [Berman Murzabulatov Raskhodnikova 16] – Boolean arrays having at most 𝑙 alternations in values – … 23

Erasure-Resilient Sortedness Tester? Example: Testing sortedness of 𝑜 -element arrays • Every uniform tester requires Ω 𝑜 queries. [EKKRV00] (optimal) tester that makes 𝑃(log 𝑜) queries • ⊥ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 𝑜 random search element 1 • Can we make it erasure-resilient O 1−α factor overhead? • All known optimal sortedness testers [EKKRV00, BGJRW09, CS13a] break with just one erasure. Known optimal testers for monotonicity, Lipschitz property and convexity of functions [GGLRS00, DGLRRS99, EKKRV00, F04, CS13a, CS13b, CST14, BRY14, BRY14, CDST15, KMS15, BB16, JR13, CS13a, BRY14, BRY14, CDJS15, PRR03, BRY14] break on a constant number of erasures . 24