Communication Complexity

Rahul Jain

Centre for Quantum Technologies andDepartment of Computer ScienceNational University of Singapore.

Communication Protocols

x 2 X y 2 Y

² Qent² (f ) : Quantum protocols with entanglement, tolerate error ² on anyinput (x;y)

jÃi½

f : X £ Y ! Z

Why do we care ?

Generic tool for results in several areas of complexity theory

• Circuit Lower Bounds, Formula Size Bounds• Time-Space trade-offs for Data Structure problems• Space lower bounds in the streaming model• Direct Sum leading to possible class separations like NC1 and NC2

• Showing lower bounds for Locally Decodable Codes

x 2 X y 2 Y

f : X £ Y ! Z

Two main topics we will look at

Lower bound methods

i. the rectangle bound, the smooth rectangle boundii. the discrepancy bound, the smooth discrepancy bound (equivalently the bound

equivalently the approximate rank bound)iii. the partition bound

Direct Sum and Direct Product results

iv. Two way modelv. One way modelvi. Simultaneous Message Passing model

Lower Bound Methods

The Rectangle Bound [Yao 83; Babai, Frankl, Simon 86; Razborov 92]

f : X £ Y ! Z

x1x2x3...

y1 y2 y3 ¢¢¢R1R3

R2

R7 R6

R4R5

(xi ;yj )

(xk;yl)

(xi ;yl )

(xk;yj )

A deterministic protocol with c bits of communication divides the inputs into at most 2c rectangles.

The Rectangle Bound

The Smooth Rectangle Bound [J, Klauck 10]

x1x2x3...

y1 y2 y3 ¢¢¢R1R3

R7 R6

R4R5

(xi ;yj )

(xk;yl)

(xi ;yl)

(xk;yj )

[Yao 79]

The Discrepancy Bound [Yao 83; Babai, Frankl, Simon 86]

The Smooth Discrepancy Bound [Klauck 07, Linail Shraibman 07]

The smooth discrepancy bound is equivalent to the 2 bound. [Linial, Shraibman 07], [Lee, Shraibman 08].

Quantum World: The 2 Bound [Linial, Shraibman 07]

• • The master lower bound : beats all other generic lower bounds

Incomparable with information theoretic lower bound methods which are not generic and not known to beat the 2 bound either

°2(A) = minX ;Y :X Y =A

r(X )c(Y )°®2 (A) = min

B :8(i ;j ) 1· A (i ;j )B (i ;j )· ®°2(B)

r(X ) = the largest 2̀ normof the rows of Xc(X ) = the largest 2̀ normof thecolumns of Y

The Partition Bound [J, Klauck 10]

min: P zPR wz;R

8(x;y) : P R :(x;y)2R wf (x;y);R ¸ 1¡ ²8(x;y) : P z

PR :(x;y)2R wz;R =1

8z;8R : wz;R ¸ 0

Initially all w0z;R = 0w0z1 ;R 11 =w

0z1 ;R 11

+p(r1)w0z2 ;R 12 =w

0z2 ;R 12

+p(r1)

For any (x;y;z) : Pr[Protocol outputs z on input(x;y)]= P R :(x;y)2R w0z;R

r1r2...rm

p(r1)p(r2)...p(rm)

R11 R12 R13 : : : R12cR21 R22 R23 : : : R22c

...: : : : : : Rm2c

: : :

Public-coin protocol with communication c

f : X £ Y ! Z

...

prt²(f ) =

The Partition Bound

Primalmin: P z

PR wz;R

8(x;y) : P R :(x;y)2R wf (x;y);R ¸ 1¡ ²8(x;y) : P z

PR :(x;y)2R wz;R =1

8z;8R : wz;R ¸ 0

Dualmax: (1¡ ²) P (x;y) ¹ x;y ¡

P(x;y) Áx;y

8z;8R : P (x;y)2 f ¡ 1(z)\ R ¹ x;y ¡P(x;y)2R Áx;y · 1

8(x;y) : ¹ x;y ¸ 0;Áx;y 2 R

prt²(f ) =

f : X £ Y ! Z

The Partition Bound for Relations

f µ X £ Y £ Z

x 2 X y 2 Y

Task: Output z 2 Z such that (x;y;z) 2 f .Primalmin: P z

PR wz;R

8(x;y) : P z:(x;y;z)2 fPR :(x;y)2R wz;R ¸ 1¡ ²

8(x;y) : P zPR :(x;y)2R wz;R = 1

8z;8R : wz;R ¸ 0

Dualmax: P (x;y)(1¡ ²)¹ x;y ¡ Áx;y

8z;8R : P (x;y)2R :(x;y;z)2 f ¹ x;y ¡P(x;y)2R Áx;y · 1

8(x;y) : ¹ x;y ¸ 0;Áx;y 2 R

prt²(f ) =

All these bounds can be captured by linear programs

Primal

min:PzPR wz;R

8(x;y) :PR :(x;y)2R wf (x;y);R ¸ 1¡ ²

8(x;y) :PzPR :(x;y)2 R wz;R = 1

8z;8R : wz;R ¸ 0

Partition Bound

Primal

min:PR wR + vR

8(x;y) 2 f ¡ 1(1) :PR :(x;y)2 R wR ¡ vR ¸ 1

8(x;y) 2 f ¡ 1(0) :PR :(x;y)2 R vR ¡ wR ¸ 1

8R : wR ; vR ¸ 0

Discrepancy Bound

Primal

min:PR wR

8(x;y) 2 f ¡ 1(z) :PR :(x;y)2 R wR ¸ 1¡ ²

8(x;y) =2 f ¡ 1(z) :PR :(x;y)2 R wR · ²

8R : wR ¸ 0

Rectangle Bound [Lovász 90]

recz² (f ) =

rec²(f ) =maxz recz² (f )

Primal

min:PR wR

8(x;y) 2 f ¡ 1(z) :PR :(x;y)2 R wR ¸ 1¡ ²

8(x;y) 2 f ¡ 1(z) :PR :(x;y)2 R wR · 1

8(x;y) =2 f ¡ 1(z) :PR :(x;y)2 R wR · ²

8R : wR ¸ 0

Smooth Rectangle Bound

f : X £ Y ! Z

Applications

[Klauck 10]

[Chakrabarti, Regev 11]

Simpler proofs [Vidick 11, Sherstov 11]

1. Rpub² (f ) ¸ logprt²(f )2. prt²(f ) ¸ srec²(f ) ¸ rec²(f ) = ­ (disc(f ))3. srec²(f ) = ­ (sdisc²(f )) = £(°®2 (f ))4. Qent² (f ) = ­ (sdisc²(f )) = £(°®2 (f ))

Summary f : X £ Y ! Z

Questions

1. Is Rpub1=3(f ) · poly(logprt1=3(f )) for all functions/ relations f ?

2. Is logprt1=3(f ) · poly(logsrec1=3(f )) for all functions/ relations f ?3. Lower bounds for spec̄ c functions/ relations using thepartition bound?

Direct Sum and Direct Product

Direct Sum

x1; : : :;xk y1; : : : ;ykTask: Output z1; : : : ;zk such that for all i 2 [k];(xi ;yi ;zi ) 2 f

f µ X £ Y £ Z

Direct SumQuestion : Rpub² (f k) = ­ (k ¢Rpub² (f )) ?

1. Rpvt1=3(EQn) = £(logn) but Rpvt1=3(EQkn) = O(k logk+ logn)

2. Open D(f k) = ­ (k ¢D(f )) ?

Can we solve k copies faster ?

Direct Product

x1; : : :;xk y1; : : : ;yk

f µ X £ Y £ Z

Let­there­be­ERROORR!

Direct Product : Is the trivial protocol essentially optimal ?

Can we solve k copies faster ?

Open: Rpub1¡ 2¡ ­ ( k ) (f k) = ­ (k ¢Rpub² (f )) ?

Task: Output z1; : : :zk 2 Z k such that for all i;(xi ;yi ;zi ) 2 f

Why do we care ?

1. Direct Sum for Deterministic communication complexity for some relations can show strong complexity class separations

2. Direct Sum arguments lead to results in Data Structure model

3. Direct Product argument used for Privacy Amplification for Bounded Storage Model in Cryptography

4. Communication-Entanglement trade-offs for quantum protocols using Direct Product for a classical relation !

5. Communication-Space trade-offs using Direct Product

6. Direct Product in other models well studied e.g Raz’s Parallel Repetition, Yao XOR Lemma

Direct Sum – Two way model

[Karchmer, Kushilevitz, Nisan 92]

[Chen, Barak, Braverman, Rao 10]

[Braverman, Rao 10]

[Harsha, J, McAllester, Radhakrishnan 07]

x1; : : : ;xk y1; : : : ;yk

f µ X £ Y £ Z

No,­My­Dear!

Can we solve k copies faster ?

Rpub² (f k) = ­ (p k ¢Rpub² (f ))

D¹ k² (f k) = ­ (k ¢(D ¹² (f ) ¡ O(r)))r : rounds for theprotocol for f k achievingD ¹ k² (f k)

D¹ k² (f k) = ­ (k ¢(D ¹² (f ) ¡ O(r)))r : rounds for theprotocol for f k achievingD ¹ k² (f k)¹ : a product distribution across X £ Y

D(f k) = ­ (k ¢(pD(f ) ¡ O(logn)))

Direct Sum – One Way and SMP models

x1;: : : ;xk y1; :: :;yk

R1;pub² (f k) = ­ (k ¢R1;pub² (f ))[J, Radhakrishnan, Sen 05]

Q1;ent² (f k) = ­ (k ¢Q1;ent² (f ))

Rsmp;pub² (f k) = ­ (k ¢Rsmp;pub² (f ))Qsmp;pub=ent² (f k) = ­ (k ¢Qsmp;pub=ent² (f ))

x1;: : : ;xk y1; :: : ;yk

[J, Klauck 09]

Rsmp;pvt² (f k) = ­ (k ¢(Rsmp;pvt² (f ) ¡ O(logn))Qsmp;pvt=ent² (f k) = ­ (k ¢Qsmp;pvt=ent² (f ))

[Chakrabarti, Shi, Yirth, Yao 00]

Rsmp;pvt² (EQk) = ­ (k ¢(Rsmp;pvt² (EQ) ¡ O(logn)))

f µ X £ Y £ Z

f µ X £ Y £ Z

Direct Product

[Shaltiel 03]

This implies direct product for

[Lee, Shraibman, Špalek 08]

[Sherstov 10]

Implies direct product for first shown by [Klauck, Špalek, de Wolf 07]

[Parnafes, Raz, Wigderson 97]

x1; : : :;xk y1; : : : ;yk

Direct Product

[Beame, Pitassi, Segerlind, Wigderson 05]

[J, Klauck, Nayak 08]

[Klauck 10]

[J 10]

Implies result of [J, Klauck, Nayak 08] and [Shaltiel 03]

f µ X £ Y £ Z

x1; : : :;xk y1; : : : ;yk

f µ X £ Y £ Z

Conditional relative entropy bound (crent)

¸ has error at most ²

rec¹² (f ) =minf log 1¹ (R )g

x1x2x3...

y1 y2 y3 ¢¢¢

...

¢¢¢x1x2x3...

y1 y2 y3 ¢¢¢

...

¢¢¢

¸ ismessage-like for ¹

[J, Klauck, Nayak 08]

[Ben-Aroya, Regev, de Wolf 08]

[Gavinsky 08]

Implied communication entanglement trade-off

[J 10]

Direct Product x1;: : : ;xk y1; :: :;yk

f µ X £ Y £ Z

f µ X £ Y £ Z

Thanks­!

Questions ?Holy Grail: Rpub1¡ 2¡ ­ ( k ) (f k) = ­ (k ¢Rpub² (f )) ?En route: