[PPT] - On the communication complexity of sparse set disjointness and PowerPoint Presentation

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On the communication complexity of sparse set disjointness and exists-equal problems

Mert Saglam University of Washington Joint work with Gábor Tardos

SLIDE 2

Communication complexity

Alice Bob X Y f(X,Y)=?

SLIDE 3

Communication complexity

Alice Bob X Y f(X,Y)=?

R

R: Shared random source

SLIDE 4

Communication complexity

Alice Bob X Y f(X,Y)=? m1(R,X) m2(R, Y, m1) m3(R, X, m1,m2)

R

R: Shared random source

mr(R, Y, m1,...,mr-1)

SLIDE 5

Communication complexity

Alice Bob X Y f(X,Y)=? m1(R,X) m2(R, Y, m1) m3(R, X, m1,m2) f(X,Y)

R

R: Shared random source

mr(R, Y, m1,...,mr-1)

SLIDE 6

Communication complexity

Alice Bob X Y f(X,Y)=? m1(R,X) m2(R, Y, m1) m3(R, X, m1,m2) f(X,Y)

R

R: Shared random source

}

# rounds mr(R, Y, m1,...,mr-1)

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Disjointness problem

S={3,7,8,11} T={2,5,8,14} S,T⊆[m]

SLIDE 8

Disjointness problem

S={3,7,8,11} T={2,5,8,14} |S ∩ T| ≟ 0 S,T⊆[m]

SLIDE 9

Disjointness problem

S={3,7,8,11} T={2,5,8,14} |S ∩ T| ≟ 0 S,T⊆[m]

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Disjointness problem

S={3,7,8,11} T={2,5,8,14} |S ∩ T| ≟ 0 S,T⊆[m] In Sparse Set Disjointness DISJk |T|, |S|≤k

m

SLIDE 11

Previous work

Total Bits Rounds Error 𝛁(k), m≥k2 Arbitrary 1/3 Babai, Frankl, Simon 86 𝛁(k) Arbitrary 1/3 Kalyanasundaram, Schnitger 92, Razborov 92, Bar-Yossef et al. 02 𝚷(k log k) 1 1/k Folklore 𝛁(k log k) 1 1/3 Folklore, Buhrman et al. 13, Woodruff 08 𝚷(k) 𝚷(log k) 0.01 Håstad, Wigderson 93

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Our contributions

Bits Rounds Error Best Previous

𝚷(k log(r) k) r 1/exp(r)(c log(r) k)

𝚷(k log k), for r=1

𝚷(k) log* k exp(-k1-𝜁)

𝚷(log k) rounds, 0.01 error 0.01 error

𝛁(k log(r) k) r 1/3 error

𝛁(k) 𝛁(k log k), for r=1

Defn: exp(r)(x) = 22

⋰2x

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Our contributions

Bits Rounds Error Best Previous

𝚷(k log(r) k) r 1/exp(r)(c log(r) k)

𝚷(k log k), for r=1

𝚷(k) log* k exp(-k1-𝜁)

𝚷(log k) rounds, 0.01 error 0.01 error

𝛁(k log(r) k) r 1/3 error

𝛁(k) 𝛁(k log k), for r=1

Holds for any r ≤ log* k

Defn: exp(r)(x) = 22

⋰2x

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The upper bound

SLIDE 15

S:

:T

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p

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S:

:T

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S

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S:

:T

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S

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S:

:T Zk*

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S

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S:

:T Zk*

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S
Pr[Zi⊇S] = p|S|, so

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S:

:T Zk*

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S

E[ k* ] = 1/p|S|

Pr[Zi⊇S] = p|S|, so

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S:

:T Zk*

Håstad-Wigderson protocol

Z1 Z2 Z3 ⋯ Zk ⋯

Zi: random, ∀a∈[m], Pr[a∈Zi]=p
Finds the first k*, Zk*⊇S

E[ k* ] = 1/p|S|

Pr[Zi⊇S] = p|S|, so
Send k* to Bob: |S|log 1/p bits

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S:

:T Zk*

Håstad-Wigderson protocol

If a ∈ S ∩ T ⇒ a ∈ Zk*,

Z1 Z2 Z3 ⋯ Zk ⋯

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S:

:T Zk*

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,

Z1 Z2 Z3 ⋯ Zk ⋯

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S:

:T Zk*

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,

Z1 Z2 Z3 ⋯ Zk ⋯

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S:

:T Zk*

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,
If S ∩T=∅, E[|T’|] = p|T|

Z1 Z2 Z3 ⋯ Zk ⋯

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S:

:T

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,
If S ∩T=∅, E[|T’|] = p|T|
Bob repeats for T’

Z1 Z2 Z3 ⋯ Zk ⋯

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S:

:T

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,
If S ∩T=∅, E[|T’|] = p|T|
Bob repeats for T’

Z1 Z2 Z3 ⋯ Zk ⋯

Zh*

Zh*⊇T’

SLIDE 28

S:

:T

Håstad-Wigderson protocol

so set T’ = T ∩ Zk*

If a ∈ S ∩ T ⇒ a ∈ Zk*,
If S ∩T=∅, E[|T’|] = p|T|
For p=1/2, if S ∩T=∅, in O(log k) rounds S’=T’=∅
Bob repeats for T’

Z1 Z2 Z3 ⋯ Zk ⋯

Zh*

Zh*⊇T’

SLIDE 29

•
S:

:T

Håstad-Wigderson protocol

Run O(log k) rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

SLIDE 30

•
S:

:T

Håstad-Wigderson protocol

Cost: |S’| bits per round Run O(log k) rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

SLIDE 31

•
S:

:T

Håstad-Wigderson protocol

Cost: |S’| bits per round Run O(log k) rounds Total = k + k/2 + k/4 + ⋯ = O(k)

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

SLIDE 32

Our bound: Rr(DISJk)≤O(k log(r) k)

•
S:

:T Zi: random, ∀a∈[m], Pr[a∈Zi]=p

Z1 Z2 Z3 ⋯ Zk ⋯

SLIDE 33

Our bound: Rr(DISJk)≤O(k log(r) k)

•
S:

:T Zi: random, ∀a∈[m], Pr[a∈Zi]=p

Z1 Z2 Z3 ⋯ Zk ⋯

Bits per round: |S’|log 1/p

SLIDE 34

Our bound: Rr(DISJk)≤O(k log(r) k)

•
S:

:T Observation: Sets get smaller ⇒ can afford smaller p each round Zi: random, ∀a∈[m], Pr[a∈Zi]=p

Z1 Z2 Z3 ⋯ Zk ⋯

Bits per round: |S’|log 1/p

SLIDE 35

Our bound: Rr(DISJk)≤O(k log(r) k)

•
S:

:T Observation: Sets get smaller ⇒ can afford smaller p each round Zi: random, ∀a∈[m], Pr[a∈Zi]=p

Z1 Z2 Z3 ⋯ Zk ⋯

Bits per round: |S’|log 1/p Defn: exp(r)(x) = 22

⋰2x

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Our bound: Rr(DISJk)≤O(k log(r) k)

•
S:

:T Observation: Sets get smaller ⇒ can afford smaller p each round Zi: random, ∀a∈[m], Pr[a∈Zi]=p

Z1 Z2 Z3 ⋯ Zk ⋯

Bits per round: |S’|log 1/p Defn: exp(r)(x) = 22

⋰2x

pi = 1/exp(i)(5 log(r) k)

SLIDE 37

Our bound: Rr(DISJk)≤O(k log(r) k)

Run r rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

SLIDE 38

Our bound: Rr(DISJk)≤O(k log(r) k)

Run r rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

pi = 1/exp(i)(5 log(r) k)
if S ∩T=∅, in r rounds S’=T’=∅

Fact 1

SLIDE 39

Our bound: Rr(DISJk)≤O(k log(r) k)

Run r rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

pi = 1/exp(i)(5 log(r) k)
if S ∩T=∅, in r rounds S’=T’=∅

Fact 1 Fact 2 |messagei| ≤ 5k log(r) k / 2i

SLIDE 40

Our bound: Rr(DISJk)≤O(k log(r) k)

Run r rounds

➡ If S’=T’=∅, declare DISJOINT ➡ Otherwise, declare INTERSECT

pi = 1/exp(i)(5 log(r) k)
if S ∩T=∅, in r rounds S’=T’=∅

Fact 1 Fact 2 |messagei| ≤ 5k log(r) k / 2i |messagei| ≤ ∏

pi-2t+1 |S| log 1/pi

k exp(i-1) exp(i-3) ≤ log exp(i) For i>3, ≤ k / 2i

t=1 i/2

SLIDE 41

The lower bound

SLIDE 42

Exists-equal problem

Stronger lower bound: easier problem

exists-equal (EE)

SLIDE 43

Exists-equal problem

Stronger lower bound: easier problem

exists-equal (EE)

Let x,y∈[t]n

SLIDE 44

Exists-equal problem

Stronger lower bound: easier problem

exists-equal (EE)

Let x,y∈[t]n
EEn(x, y) = 1 iff ∃i, xi=yi

t

SLIDE 45

Exists-equal problem

Stronger lower bound: easier problem

exists-equal (EE)

Let x,y∈[t]n

3 4 4 5 1 2 3 4 2 4

x: y:

EEn(x, y) = 1 iff ∃i, xi=yi

t

SLIDE 46

Exists-equal problem

Stronger lower bound: easier problem

exists-equal (EE)

Let x,y∈[t]n

3 4 4 5 1 2 3 4 2 4

x: y: So EE(x,y)=1

EEn(x, y) = 1 iff ∃i, xi=yi

t

SLIDE 47

Exists-equal problem

EEn is OR of n equality problems over [t].
EEn = DISJn

t tn t

SLIDE 48

Exists-equal problem

3 4 4 5 1 2 3 4 2 4

EEn is OR of n equality problems over [t].
EEn = DISJn

t tn t

SLIDE 49

Exists-equal problem

3 4 4 5 1 2 3 4 2 4

EEn is OR of n equality problems over [t].
EEn = DISJn
=

t tn t

SLIDE 50

Exists-equal problem

3 4 4 5 1 2 3 4 2 4

EEn is OR of n equality problems over [t].
EEn = DISJn
=

t tn t

SLIDE 51

Exists-equal problem

3 4 4 5 1 2 3 4 2 4

EEn is OR of n equality problems over [t].
EEn = DISJn
=

|S| = |T| = n S, T ⊂ [nt]

t tn t

SLIDE 52

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

SLIDE 53

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

EE is OR of n equality problems, so decompose to

subproblems

SLIDE 54

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

EE is OR of n equality problems, so decompose to

subproblems

Show Ω(log(r) n) lower bound per subproblem

SLIDE 55

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

EE is OR of n equality problems, so decompose to

subproblems

Show Ω(log(r) n) lower bound per subproblem
Equality has O(1) bit communication protocol

SLIDE 56

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

EE is OR of n equality problems, so decompose to

subproblems

Show Ω(log(r) n) lower bound per subproblem
Equality has O(1) bit communication protocol

SLIDE 57

The lower bound

Show: any r-round EE protocol communicates

Ω(n log(r) n) bits.

EE is OR of n equality problems, so decompose to

subproblems

Show Ω(log(r) n) lower bound per subproblem
Equality has O(1) bit communication protocol
We get super-linear increase in complexity!

SLIDE 58

The lower bound

By Yao’s lemma, sufficient to consider deterministic protocols with random input

SLIDE 59

The lower bound

By Yao’s lemma, sufficient to consider deterministic protocols with random input

๏ Set t=4n

SLIDE 60

The lower bound

By Yao’s lemma, sufficient to consider deterministic protocols with random input

๏ Set t=4n ๏ Hard distribution ν: (x,y)∈ [t]n x [t]n, uniform random

SLIDE 61

The lower bound

By Yao’s lemma, sufficient to consider deterministic protocols with random input

๏ Set t=4n ๏ Hard distribution ν: (x,y)∈ [t]n x [t]n, uniform random ๏ 3/4 ⩽ Pr(x,y)~ν[EE(x,y)=0] = (1-1/(4n))n ⩽ e-1/4 < 0.78

SLIDE 62

The lower bound

By Yao’s lemma, sufficient to consider deterministic protocols with random input

๏ Set t=4n ๏ Hard distribution ν: (x,y)∈ [t]n x [t]n, uniform random ๏ 3/4 ⩽ Pr(x,y)~ν[EE(x,y)=0] = (1-1/(4n))n ⩽ e-1/4 < 0.78 ๏ ⇒ Any 0-round protocol has 0.22 error

SLIDE 63

Round elimination

Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r:

SLIDE 64

Round elimination

Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r: r-round C=𝛇n log(r) n-bits protocol for EEn

t

SLIDE 65

Round elimination

Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r:

⇒

construct r-round C=𝛇n log(r) n-bits protocol for EEn

t

SLIDE 66

Round elimination

Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r:

⇒

construct (r-1)-round 10C-bits protocol for EEn’ where n’=n/B and t’=t/B

B=2C/n

t’

r-round C=𝛇n log(r) n-bits protocol for EEn

t

SLIDE 67

Round elimination

Observe 10C=𝛑(n’ log(r-1) n’) Contradicts induction hypothesis! Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r:

⇒

construct (r-1)-round 10C-bits protocol for EEn’ where n’=n/B and t’=t/B

B=2C/n

t’

r-round C=𝛇n log(r) n-bits protocol for EEn

t

SLIDE 68

Round elimination

Observe 10C=𝛑(n’ log(r-1) n’) Contradicts induction hypothesis! Thm: No r-round C = O(n log(r) n)-bits protocol for EEn

t

Induction on r:

⇒

construct Note: t’=4n’ (r-1)-round 10C-bits protocol for EEn’ where n’=n/B and t’=t/B

B=2C/n

t’

r-round C=𝛇n log(r) n-bits protocol for EEn

t

SLIDE 69

Fixing the first message

x1 x2 x3 x4 x5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ xtn

SLIDE 70

Fixing the first message

x1 x2 x3 x4 x5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ xtn

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Fixing the first message

Throw x0, Pry[P(x0,y) ≠EE(x0,y)] ≥2δ

x1 x2 x3 x4 x5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ xtn

SLIDE 72