Amit Chakrabarti Dartmouth College Main result joint with Oded - PowerPoint PPT Presentation

Optimal Lower Bound for GHD May 2011 Gap-Hamming-Distance: The Journey to an Optimal Lower Bound Amit Chakrabarti Dartmouth College Main result joint with Oded Regev, Tel Aviv University Sublinear Algorithms Workshop at Bertinoro, May 2011 Amit Chakrabarti 1

Optimal Lower Bound for GHD May 2011 The Gap-Hamming-Distance Problem Input: Alice gets x ∈ { 0 , 1 } n , Bob gets y ∈ { 0 , 1 } n . Output: 2 + √ n • ghd ( x, y ) = 1 if ∆( x, y ) > n 2 − √ n • ghd ( x, y ) = 0 if ∆( x, y ) < n Want: randomized, constant error protocol Cost: Worst case number of bits communicated 0 1 0 0 1 0 1 1 0 0 0 1 x = y = 0 0 0 0 0 0 1 1 1 0 0 1 √ √ n = 12; ∆( x, y ) = 3 ∈ [6 − 12 , 6 + 12] Amit Chakrabarti 2

Optimal Lower Bound for GHD May 2011 Implications Data stream lower bounds • Distinct elements • Frequency moments • Norms • Entropy • General form of bound: ps = Ω(1 /ε 2 ) Distributed functional monitoring lower bounds Connections to differential privacy Amit Chakrabarti 3

Optimal Lower Bound for GHD May 2011 The Reductions E.g., Distinct Elements (Other problems: similar) 0 1 0 0 1 0 1 1 0 0 0 1 x = ) ) ) 0 0 1 ) ) ) ) ) ) ) ) ) 0 1 0 0 1 0 1 1 0 , , , σ : 0 1 2 , , , , , , , , , 1 2 3 4 5 6 7 8 9 1 1 1 ( ( ( ( ( ( ( ( ( ( ( ( y = 0 0 0 0 0 0 1 1 1 0 0 1 ) ) ) 0 0 1 ) ) ) ) ) ) ) ) ) 0 0 0 0 0 0 1 1 1 , , , τ : 0 1 2 , , , , , , , , , 1 2 3 4 5 6 7 8 9 1 1 1 ( ( ( ( ( ( ( ( ( ( ( ( Alice: x �− → σ = � (1 , x 1 ) , (2 , x 2 ) , . . . , ( n, x n ) � Bob: y �− → τ = � (1 , y 1 ) , (2 , y 2 ) , . . . , ( n, y n ) �  2 − √ n, or  < 3 n 1 Notice: F 0 ( σ ◦ τ ) = n + ∆( x, y ) = Set ε = √ n . 2 + √ n.  > 3 n Amit Chakrabarti 4

Optimal Lower Bound for GHD May 2011 Ancient History Amit Chakrabarti 5

Optimal Lower Bound for GHD May 2011 One-Pass Bounds Indyk, Woodruff [FOCS 2003] • Considered one-pass lower bound for dist-elem • Recognized relevance of ghd , difficulty of lower-bounding • Defined “related” problem Π ℓ 2 , showed R → (Π ℓ 2 ) = Ω( n ) • Concluded Ω( ε − 2 ) bound for dist-elem m,ε with m = � Ω(1 /ε 9 ) Amit Chakrabarti 6

Optimal Lower Bound for GHD May 2011 One-Pass Bounds Indyk, Woodruff [FOCS 2003] • Considered one-pass lower bound for dist-elem • Recognized relevance of ghd , difficulty of lower-bounding • Defined “related” problem Π ℓ 2 , showed R → (Π ℓ 2 ) = Ω( n ) • Concluded Ω( ε − 2 ) bound for dist-elem m,ε with m = � Ω(1 /ε 9 ) Woodruff [SODA 2004] • Worked with ghd itself, showed R → ( ghd ) = Ω( n ) • Very intricate combinatorial proof, with hairy probability estimations • Conjectured R( ghd ) = Ω( n ) , implying multi-pass lower bounds Amit Chakrabarti 6-a

Optimal Lower Bound for GHD May 2011 The VC-Dimension Technique • Consider communication matrix of ghd as set system • The system has Ω( n ) VC-dimension 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 1 • Thus, R → ( ghd ) = Ω( n ) Amit Chakrabarti 7

Optimal Lower Bound for GHD May 2011 The VC-Dimension Technique • Consider communication matrix of ghd as set system • The system has Ω( n ) VC-dimension 1 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 Instance of INDEX Amit Chakrabarti 7

Optimal Lower Bound for GHD May 2011 The VC-Dimension Technique • Consider communication matrix of ghd as set system • The system has Ω( n ) VC-dimension 1 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 Instance of INDEX • Thus, R → ( ghd ) = Ω( n ) Amit Chakrabarti 7-a

Optimal Lower Bound for GHD May 2011 The Middle Ages Amit Chakrabarti 8

Optimal Lower Bound for GHD May 2011 A Nice Simplification Jayram, Kumar, Sivakumar [circa 2005] • Simpler proof of R → ( ghd ) = Ω( n ) • Much simpler: direct reduction from index • Geometric intuition: � � n √ n , − 1 1 x ∈ { 0 , 1 } n ∈ R n Alice: �− → x ∈ � √ n e j = (0 , . . . , 0 , 1 , 0 , . . . , 0) ∈ R n Bob: j ∈ [ n ] �− → • Observe: � � x, e j � �≈ 0 , and x j determined by sgn � � x, e j � • We’ve reduced index to “gap-inner-product”, or gip Amit Chakrabarti 9

Optimal Lower Bound for GHD May 2011 Inner Product ↔ Hamming Distance • Obviously, ghd → gip : y � = 1 − 2∆( x, y ) � � x, � n 2 ± √ n y � ≷ ∓ 2 √ n ⇒ ∆( x, y ) ≶ n � � x, � • Also, gip → ghd by “discretization transform”: Pick random Gaussians r 1 , . . . , r N , with N = 10 n x ∈ R n x, r N � ) ∈ {± 1 } N Alice: ¯ �− → x = (sgn � ¯ x, r 1 � , . . . , sgn � ¯ y ∈ R n y, r N � ) ∈ {± 1 } N Bob: ¯ �− → y = (sgn � ¯ y, r 1 � , . . . , sgn � ¯ √ whp y � ≷ ∓ 1 ∆( x, y ) ≶ N � ¯ x, ¯ = ⇒ 2 ± O ( N ) √ n Amit Chakrabarti 10

Optimal Lower Bound for GHD May 2011 The Renaissance Era Amit Chakrabarti 11

Optimal Lower Bound for GHD May 2011 Round Elimination Brody, Chakrabarti [CCC 2009] • Can we at least rule out a two-pass improvement for dist-elem ? • A cheap first message makes little progress? Then rinse, repeat • Tends to decimate problem [Miltersen-Nisan-Safra-Wigderson’98] [Sen’03] Input: Input: ⇒ (k rounds) (k−1 rounds) Padding: Amit Chakrabarti 12

Optimal Lower Bound for GHD May 2011 Another VC-Dimension Argument: Subcube Lifting First message constant on large set: 1 } 1 0 1 1 0 0 1 0 0 1 1 1 0 1 0.99 n 1 1 1 1 1 1 1 2 points 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 Alice, Bob lift their ( n/ 3) -dim inputs from inner coords to full n -dim space First message now redundant, so eliminate! Amit Chakrabarti 13

Optimal Lower Bound for GHD May 2011 Another VC-Dimension Argument: Subcube Lifting First message constant on large set: 1 } 1 0 0 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 0.99 n 1 1 1 1 1 1 1 1 1 2 points 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 S: inner coords, the real input (Rest: outer coords, padding) Alice, Bob lift their ( n/ 3) -dim inputs from inner coords to full n -dim space First message now redundant, so eliminate! Amit Chakrabarti 13

Optimal Lower Bound for GHD May 2011 Another VC-Dimension Argument: Subcube Lifting First message constant on large set: � � � � � � � � � � � � � � � � � � � � ���� � � � � � � � � � � � ������ � � � � � � � � � � � � � � � � � � � � � � � � � � �� ���������������������������� ����������������������������� Amit Chakrabarti 13

Optimal Lower Bound for GHD May 2011 Another VC-Dimension Argument: Subcube Lifting First message constant on large set: � � � � � � � � � � � � � � � � � � � � ���� � � � � � � � � � � � ������ � � � � � � � � � � � � � � � � � � � � � � � � � � �� ���������������������������� ����������������������������� Alice, Bob lift their ( n/ 3) -dim inputs from inner coords to full n -dim space First message now redundant, so eliminate! [Brody-C.’09] Amit Chakrabarti 13-a

Optimal Lower Bound for GHD May 2011 Better Round Elimination Brody, Chakrabarti, Regev, Vidick, de Wolf [RANDOM 2010] • Previous argument reduced dimension too rapidly • Gives R k ( ghd ) = n/ 2 O ( k 2 ) • Can improve to R k ( ghd ) = n/O ( k 2 ) Amit Chakrabarti 14

Optimal Lower Bound for GHD May 2011 Round Elimination V2.0: Geometric Perturbation First message constant over large set A {0,1} n x z A ERR y 1/2 n c Amit Chakrabarti 15

Optimal Lower Bound for GHD May 2011 Round Elimination V2.0: Geometric Perturbation First message constant over large set A {0,1} n x z A ERR y 1/2 n c Alice: replace x with z = NearestNeighbour ( x, A ) Amit Chakrabarti 15-a

Optimal Lower Bound for GHD May 2011 Modern History Amit Chakrabarti 16

Optimal Lower Bound for GHD May 2011 Main Theorem Chakrabarti, Regev [STOC 2011] And now, we show: R( ghd ) = Ω( n ) Amit Chakrabarti 17

Optimal Lower Bound for GHD May 2011 The Rectangle Property Input universe U = { 0 , 1 } n × { 0 , 1 } n Deterministic protocol P , communicating ≤ c bits partitions U into ≤ 2 c rectangles A i × B i , where A i , B i ⊆ { 0 , 1 } n Bob Alice If P computes f : U → { 0 , 1 } , then f − 1 (1) = R 1 ∪ R 2 ∪ · · · ∪ R 2 c Amit Chakrabarti 18