Communication Complexity Lecture 23 Computing with remote inputs - PowerPoint PPT Presentation

Lower-Bounding χ (f) If a fooling set of size S, no two input-pairs from S can be on the same tile in a monochromatic tiling χ (f) ≥ |S| for every fooling set S Rank lower-bound χ (f) ≥ Rank(M f ) Discrepancy lower-bound χ (f) ≥ Discrepancy(f) 12

Rank(M) 13

Rank(M) Rank of a matrix 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) Linear independence: operations in a field 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) Linear independence: operations in a field Rank-r matrix: after row & column reductions D (mxn) diagonal, with r 1’ s, rest 0’ s. M = UDV 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) Linear independence: operations in a field Rank-r matrix: after row & column reductions D (mxn) diagonal, with r 1’ s, rest 0’ s. M = UDV Rank(M) ≤ r, iff M can be written as sum of ≤ r rank 1 matrices 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) Linear independence: operations in a field Rank-r matrix: after row & column reductions D (mxn) diagonal, with r 1’ s, rest 0’ s. M = UDV Rank(M) ≤ r, iff M can be written as sum of ≤ r rank 1 matrices M = UDV = Σ i ≤ r D ii U i(mx1) V i(1xn) = Σ i ≤ r B i , where Rank(B i )=1 13

Rank(M) Rank of a matrix Maximum number of linearly independent rows (or equivalently, columns) Linear independence: operations in a field Rank-r matrix: after row & column reductions D (mxn) diagonal, with r 1’ s, rest 0’ s. M = UDV Rank(M) ≤ r, iff M can be written as sum of ≤ r rank 1 matrices M = UDV = Σ i ≤ r D ii U i(mx1) V i(1xn) = Σ i ≤ r B i , where Rank(B i )=1 If M = Σ i ≤ r B i = UDV, Rank(M) ≤ min{Rank(U),Rank(D),Rank(V)} ≤ Rank(D) = r 13

χ (f) ≥ Rank(M f ) 14

χ (f) ≥ Rank(M f ) If M = Σ i ≤ r B i with Rank(B i )=1, then Rank(M) ≤ r 14

χ (f) ≥ Rank(M f ) If M = Σ i ≤ r B i with Rank(B i )=1, then Rank(M) ≤ r M f = Σ i ≤ χ (f) Tile i , where Tile i has a monochromatic rectangle and 0’ s elsewhere 14

χ (f) ≥ Rank(M f ) If M = Σ i ≤ r B i with Rank(B i )=1, then Rank(M) ≤ r M f = Σ i ≤ χ (f) Tile i , where Tile i has a monochromatic rectangle and 0’ s elsewhere Rank(Tile i )=1 14

χ (f) ≥ Rank(M f ) If M = Σ i ≤ r B i with Rank(B i )=1, then Rank(M) ≤ r M f = Σ i ≤ χ (f) Tile i , where Tile i has a monochromatic rectangle and 0’ s elsewhere Rank(Tile i )=1 Rank(M f ) ≤ χ (f) 14

χ (f) ≥ Rank(M f ) If M = Σ i ≤ r B i with Rank(B i )=1, then Rank(M) ≤ r M f = Σ i ≤ χ (f) Tile i , where Tile i has a monochromatic rectangle and 0’ s elsewhere Rank(Tile i )=1 Rank(M f ) ≤ χ (f) CC(f) ≥ log( χ (f)) ≥ log(Rank(M f )) 14

Discrepancy 15

Discrepancy Discrepancy of a 0-1 matrix 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle Imbalance = | #1’ s - #0’ s | 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle Imbalance = | #1’ s - #0’ s | Disc(M) = 1/(mn) max rect imbalance(rect) 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle Imbalance = | #1’ s - #0’ s | Disc(M) = 1/(mn) max rect imbalance(rect) χ (f) ≥ 1/Disc(M f ) 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle Imbalance = | #1’ s - #0’ s | Disc(M) = 1/(mn) max rect imbalance(rect) χ (f) ≥ 1/Disc(M f ) Disc(M f ) ≥ 1/(mn) (size of largest monochromatic tile) 15

Discrepancy Discrepancy of a 0-1 matrix max “imbalance” in any rectangle Imbalance = | #1’ s - #0’ s | Disc(M) = 1/(mn) max rect imbalance(rect) χ (f) ≥ 1/Disc(M f ) Disc(M f ) ≥ 1/(mn) (size of largest monochromatic tile) χ (f) ≥ (mn)/(size of largest monochromatic tile) 15

CC Lower-bounds Summary 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) To lower-bound χ (f): fooling-set, rank, 1/Disc 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) To lower-bound χ (f): fooling-set, rank, 1/Disc χ (f) ≥ |max fooling-set| ≥ (Rank(M f )) 2 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) To lower-bound χ (f): fooling-set, rank, 1/Disc χ (f) ≥ |max fooling-set| ≥ (Rank(M f )) 2 1/Discrepancy lower-bounds can be very loose 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) To lower-bound χ (f): fooling-set, rank, 1/Disc χ (f) ≥ |max fooling-set| ≥ (Rank(M f )) 2 1/Discrepancy lower-bounds can be very loose Conjecture: Rank(M f ) (and hence fooling set) is fairly tight 16

CC Lower-bounds Summary CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ (f) Both fairly tight: CC(f) = O( log 2 ( χ (f)) ) To lower-bound χ (f): fooling-set, rank, 1/Disc χ (f) ≥ |max fooling-set| ≥ (Rank(M f )) 2 1/Discrepancy lower-bounds can be very loose Conjecture: Rank(M f ) (and hence fooling set) is fairly tight i.e., CC(f) = O(polylog(Rank(M f )) 16

Many Variants 17

Many Variants Randomized protocols: significant savings in expectation 17

Many Variants Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess 17