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

Communication Complexity

Lecture 23 Computing with remote inputs

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Communication Complexity

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Communication Complexity

Setting

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Communication Complexity

Setting Alice wants to compute f(x,y)

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Communication Complexity

Setting Alice wants to compute f(x,y) Alice is given only x. Her friend Bob gets y.

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Communication Complexity

Setting Alice wants to compute f(x,y) Alice is given only x. Her friend Bob gets y. Least amount of communication to achieve this

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Communication Complexity

Setting Alice wants to compute f(x,y) Alice is given only x. Her friend Bob gets y. Least amount of communication to achieve this Compare with decision tree complexity

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Communication Complexity

Setting Alice wants to compute f(x,y) Alice is given only x. Her friend Bob gets y. Least amount of communication to achieve this Compare with decision tree complexity Trivial upper-bound of |x|

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Communication Complexity

Setting Alice wants to compute f(x,y) Alice is given only x. Her friend Bob gets y. Least amount of communication to achieve this Compare with decision tree complexity Trivial upper-bound of |x| Interested in proving lower bounds for various f

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Examples

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Examples

PARITY(x,y) = ⊕i (xi⊕yi)

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Examples

PARITY(x,y) = ⊕i (xi⊕yi) CC(PARITY) = 1

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Examples

PARITY(x,y) = ⊕i (xi⊕yi) CC(PARITY) = 1 EQ(x,y) = 1 iff x=y

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Examples

PARITY(x,y) = ⊕i (xi⊕yi) CC(PARITY) = 1 EQ(x,y) = 1 iff x=y Lower-bound?

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Examples

PARITY(x,y) = ⊕i (xi⊕yi) CC(PARITY) = 1 EQ(x,y) = 1 iff x=y Lower-bound? DISJ(x,y)=1 if x∧y=0n

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Motivation

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Motivation

Distributed computing

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Motivation

Distributed computing Lower-bounds for Circuit complexity

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Motivation

Distributed computing Lower-bounds for Circuit complexity Amount of communication across a cut in the circuit

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Motivation

Distributed computing Lower-bounds for Circuit complexity Amount of communication across a cut in the circuit Proving optimality of algorithms and data-structures

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Protocol

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Protocol

We’ll consider deterministic protocols

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Protocol

We’ll consider deterministic protocols Fixed number of rounds (Alice to Bob, then Bob to Alice), each party sends a fixed number of bits in each round

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Protocol

We’ll consider deterministic protocols Fixed number of rounds (Alice to Bob, then Bob to Alice), each party sends a fixed number of bits in each round Can even consider protocol to have Alice and Bob alternately exchanging single bits (since not considering number of rounds)

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Protocol

We’ll consider deterministic protocols Fixed number of rounds (Alice to Bob, then Bob to Alice), each party sends a fixed number of bits in each round Can even consider protocol to have Alice and Bob alternately exchanging single bits (since not considering number of rounds) At most doubles the communication complexity

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Protocol Execution

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Protocol Execution

ith message from Alice is a function of her input and previous messages

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Protocol Execution

ith message from Alice is a function of her input and previous messages Her output is a function of the final “transcript” and her own input (her “view”)

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Protocol Execution

ith message from Alice is a function of her input and previous messages Her output is a function of the final “transcript” and her own input (her “view”) Similarly for Bob. His view = transcript + his input

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Protocol Execution

ith message from Alice is a function of her input and previous messages Her output is a function of the final “transcript” and her own input (her “view”) Similarly for Bob. His view = transcript + his input #transcripts ≤ 2CC. i.e. CC ≥ log(#transcripts)

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Transcript Table

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Transcript Table

Consider the transcript table

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Transcript Table

Consider the transcript table If on (a1,b1) and (a2,b2) same transcript

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Transcript Table

Consider the transcript table If on (a1,b1) and (a2,b2) same transcript Then same transcript

• n (a1,b2) also!

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Transcript Table

Consider the transcript table If on (a1,b1) and (a2,b2) same transcript Then same transcript

• n (a1,b2) also!

Alice and Bob never realize the difference through out the protocol

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Fooling Set

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Fooling Set

If on (a1,b1) and (a2,b2) same transcript, then same transcript on (a1,b2) also

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Fooling Set

If on (a1,b1) and (a2,b2) same transcript, then same transcript on (a1,b2) also Showing a set S of input-pairs that must have distinct transcripts

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Fooling Set

If on (a1,b1) and (a2,b2) same transcript, then same transcript on (a1,b2) also Showing a set S of input-pairs that must have distinct transcripts All pairs have same output

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Fooling Set

If on (a1,b1) and (a2,b2) same transcript, then same transcript on (a1,b2) also Showing a set S of input-pairs that must have distinct transcripts All pairs have same output “Cross” of no two pairs has the same output

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Fooling Set

If on (a1,b1) and (a2,b2) same transcript, then same transcript on (a1,b2) also Showing a set S of input-pairs that must have distinct transcripts All pairs have same output “Cross” of no two pairs has the same output If S is a set of such pairs, CC ≥ log(|S|)

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Fooling Set for EQ

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Fooling Set for EQ

S = set of all pairs (x,x)

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Fooling Set for EQ

S = set of all pairs (x,x) CC(EQ) ≥ log(|S|) ≥ n

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Fooling Set for EQ

S = set of all pairs (x,x) CC(EQ) ≥ log(|S|) ≥ n True for any function in which each row and column has exactly one 1

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Fooling Set for EQ

S = set of all pairs (x,x) CC(EQ) ≥ log(|S|) ≥ n True for any function in which each row and column has exactly one 1 Other functions too

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Fooling Set for EQ

S = set of all pairs (x,x) CC(EQ) ≥ log(|S|) ≥ n True for any function in which each row and column has exactly one 1 Other functions too e.g.: DISJ(x,y) if x∧y=0n

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Fooling Set for EQ

S = set of all pairs (x,x) CC(EQ) ≥ log(|S|) ≥ n True for any function in which each row and column has exactly one 1 Other functions too e.g.: DISJ(x,y) if x∧y=0n S = set of complementary pairs, (x,¬x)

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Monochromatic Rectangles

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Monochromatic Rectangles

Rectangle: a subset of D1xD2

• f the form S1xS2

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Monochromatic Rectangles

Rectangle: a subset of D1xD2

• f the form S1xS2

Monochromatic: same f-value

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Monochromatic Rectangles

Rectangle: a subset of D1xD2

• f the form S1xS2

Monochromatic: same f-value Recall: for any protocol, set

• f all input-pairs with the

same transcript is a rectangle

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Monochromatic Rectangles

Rectangle: a subset of D1xD2

• f the form S1xS2

Monochromatic: same f-value Recall: for any protocol, set

• f all input-pairs with the

same transcript is a rectangle For protocol to be correct, the rectangles should be monochromatic

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Tiling Lower-Bound

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Tiling Lower-Bound

For protocol to be correct, same-transcript rectangles should be monochromatic

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Tiling Lower-Bound

For protocol to be correct, same-transcript rectangles should be monochromatic Find the least number of monochromatic rectangles that can tile the function, χ(f)

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Tiling Lower-Bound

For protocol to be correct, same-transcript rectangles should be monochromatic Find the least number of monochromatic rectangles that can tile the function, χ(f) #transcripts ≥ χ(f)

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Tiling Lower-Bound

For protocol to be correct, same-transcript rectangles should be monochromatic Find the least number of monochromatic rectangles that can tile the function, χ(f) #transcripts ≥ χ(f) CC(f) ≥ log(χ(f))

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Tiling Lower-Bound

For protocol to be correct, same-transcript rectangles should be monochromatic Find the least number of monochromatic rectangles that can tile the function, χ(f) #transcripts ≥ χ(f) CC(f) ≥ log(χ(f)) How to lower-bound χ(f)?

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Lower-Bounding χ(f)

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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

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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

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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

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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(Mf)

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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(Mf) Discrepancy lower-bound

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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(Mf) Discrepancy lower-bound χ(f) ≥ Discrepancy(f)

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Rank(M)

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Rank(M)

Rank of a matrix

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Rank(M)

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

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Rank(M)

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

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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

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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

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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 = Σi≤r Dii Ui(mx1) Vi(1xn) = Σi≤r Bi, where Rank(Bi)=1

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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 = Σi≤r Dii Ui(mx1) Vi(1xn) = Σi≤r Bi, where Rank(Bi)=1 If M = Σi≤r Bi = UDV, Rank(M) ≤ min{Rank(U),Rank(D),Rank(V)} ≤ Rank(D) = r

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χ(f) ≥ Rank(Mf)

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χ(f) ≥ Rank(Mf)

If M = Σi≤r Bi with Rank(Bi)=1, then Rank(M) ≤ r

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χ(f) ≥ Rank(Mf)

If M = Σi≤r Bi with Rank(Bi)=1, then Rank(M) ≤ r Mf = Σi≤χ(f) Tilei, where Tilei has a monochromatic rectangle and 0’ s elsewhere

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χ(f) ≥ Rank(Mf)

If M = Σi≤r Bi with Rank(Bi)=1, then Rank(M) ≤ r Mf = Σi≤χ(f) Tilei, where Tilei has a monochromatic rectangle and 0’ s elsewhere Rank(Tilei)=1

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χ(f) ≥ Rank(Mf)

If M = Σi≤r Bi with Rank(Bi)=1, then Rank(M) ≤ r Mf = Σi≤χ(f) Tilei, where Tilei has a monochromatic rectangle and 0’ s elsewhere Rank(Tilei)=1 Rank(Mf) ≤ χ(f)

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χ(f) ≥ Rank(Mf)

If M = Σi≤r Bi with Rank(Bi)=1, then Rank(M) ≤ r Mf = Σi≤χ(f) Tilei, where Tilei has a monochromatic rectangle and 0’ s elsewhere Rank(Tilei)=1 Rank(Mf) ≤ χ(f) CC(f) ≥ log(χ(f)) ≥ log(Rank(Mf))

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Discrepancy

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Discrepancy

Discrepancy of a 0-1 matrix

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Discrepancy

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

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Discrepancy

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

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Discrepancy

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

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Discrepancy

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

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Discrepancy

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

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Discrepancy

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

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CC Lower-bounds Summary

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts)

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CC Lower-bounds Summary

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

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) )

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) ) To lower-bound χ(f): fooling-set, rank, 1/Disc

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) ) To lower-bound χ(f): fooling-set, rank, 1/Disc χ(f) ≥ |max fooling-set| ≥ (Rank(Mf))2

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) ) To lower-bound χ(f): fooling-set, rank, 1/Disc χ(f) ≥ |max fooling-set| ≥ (Rank(Mf))2 1/Discrepancy lower-bounds can be very loose

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) ) To lower-bound χ(f): fooling-set, rank, 1/Disc χ(f) ≥ |max fooling-set| ≥ (Rank(Mf))2 1/Discrepancy lower-bounds can be very loose Conjecture: Rank(Mf) (and hence fooling set) is fairly tight

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CC Lower-bounds Summary

CC(f) ≥ log(#transcripts) Tiling Lower-bound: #transcripts ≥ χ(f) Both fairly tight: CC(f) = O( log2(χ(f)) ) To lower-bound χ(f): fooling-set, rank, 1/Disc χ(f) ≥ |max fooling-set| ≥ (Rank(Mf))2 1/Discrepancy lower-bounds can be very loose Conjecture: Rank(Mf) (and hence fooling set) is fairly tight i.e., CC(f) = O(polylog(Rank(Mf))

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Many Variants

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Many Variants

Randomized protocols: significant savings in expectation

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Many Variants

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

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Many Variants

Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess Multi-party: Input split across multiple parties. Broadcast channels for communication.

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Many Variants

Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess Multi-party: Input split across multiple parties. Broadcast channels for communication. Number on the forehead version

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Many Variants

Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess Multi-party: Input split across multiple parties. Broadcast channels for communication. Number on the forehead version Non-boolean output

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Many Variants

Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess Multi-party: Input split across multiple parties. Broadcast channels for communication. Number on the forehead version Non-boolean output Multi-valued functions: agree on one value

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Many Variants

Randomized protocols: significant savings in expectation Non-deterministic: Alice and Bob are non-deterministic. “Communication” now includes shared guess Multi-party: Input split across multiple parties. Broadcast channels for communication. Number on the forehead version Non-boolean output Multi-valued functions: agree on one value Different costs: asymmetric communication, average-case complexity

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