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Information Complexity:

A Paradigm for Proving Lower Bounds

Amit ChakrabartiDartmouth College

Hanover, NH, USA

STOC 2013, Palo Alto

1

History

The applications came first; theory built in service of applications

2

History


• Ablayev (Theor. Comp. Sci., 1996)

Proved lower bound for communication problem index

Used technique that we now recognize as “information complexity”

2-a

History





• Chakrabarti, Shi, Wirth, Yao (FOCS, 2001)

Proved direct sum results for simultaneous message complexity

Formally defined “information cost” and “information complexity”

Introduced notations icost, IC

Anticipated wider applicability of paradigm

2-b

History










!

"

#

$

“We introduce a new notion of informational complexity which is

related to SM complexity and has nice direct sum properties. This

notion is used as a tool to prove the above results; it appears to be

quite powerful and may be of independent interest.”

2-c

History










2

History










• Bar-Yossef, Jayram, Kumar, Sivakumar (FOCS, 2002)

Gave extension to interactive communication

Cleverly handled non-product distributions: “conditional icost”

Improved some communication (hence data stream) lower bounds

2-a

This Talk

Goals:

• Tutorial style

• Diversity of results

• Extract common patterns in applying IC

Not goals:

• Be comprehensive

• Present latest results (but see Woodru!’s talk next)

3

(Generalized) Direct Sum Theorems

Situation:

• Task A : a simple computational task

• Task B: combines N independent copies of task A

Direct sum theorem:

Complexity(B) = !(N) · Complexity(A )

4

The Information Complexity Paradigm

Situation:



The paradigm:

1. Define information cost

2. Simulation Argument

3. Basic IC lower bound

5


Situation:



The paradigm:

1. Define information cost, hence information complexity (IC)

Get Complexity(B) ! IC(B)



5-a


Situation:



The paradigm:



2. Simulation Argument: solving B " solving each copy of A

Get IC(B) ! N · IC(A )


5-b


Situation:



The paradigm:



2. Simulation Argument: solving B " solving each copy of A

Get IC(B) ! N · IC(A )

3. Basic IC lower bound: apply to simple task A

Get IC(A ) ! Complexity(A )

5-c

Part One:

No Interaction

6

The Problem

Definition:

Alice holds x # {0, 1}n, Bob holds i # [n]; find xi (error $ !)

iAlice

1 2 nx = x x ... xmsg( )x,R Bob

Correctness requirement:

%x, i Pr[output &= xi] $ !

7

The Problem

Definition:

Alice holds x # {0, 1}n, Bob holds i # [n]; find xi (error $ !)

iAlice

1 2 nx = x x ... xmsg( )x,R Bob

Correctness requirement:

%x, i Pr[output &= xi] $ !

Theorem: Alice needs to send !(n) bits. [Ablayev’96]

7-a

Information Complexity Paradigm Demo

The echo problem:

Alice holds b # {0, 1}, Bob to output b with error $ !

BobAliceb

msg( )b,R

8


The echo problem:


BobAliceb

msg( )b,R

Simple task (A ): echo

Complex task (B): index

8-a


The echo problem:


BobAliceb

msg( )b,R


Complex task (B): index ' combines n independent copies of echo

8-b


The echo problem:


BobAliceb

msg( )b,R


Complex task (B): index ' combines n independent copies of echo

The paradigm



3. Basic IC lower bound (for echo)

8-c

Step 1: Define Information Cost

Generic notion, for a communication protocol ":

icost(") = amount of info about (part of) the input to "

revealed by (some of) the messages in "

(possibly conditioned on some prior knowledge)

9

Step 1: Define Information Cost

Generic notion, for a communication protocol ":

icost(") = amount of info about (part of) the input to "

revealed by (some of) the messages in "

(possibly conditioned on some prior knowledge)

In this case...

• Let "B be a protocol for task B (i.e., index)

• Let X = random input (distrib µ) for Alice

R = random coins of Alice

M = msg(X,R); then

icostµ("B) := I(X : M)

• Notice:

icostµ("B) $ H(M) $ length(M) = cost("B)

9-a

Step 2: Simulation Argument

Take X = X1 . . .Xn ( µ1) · · ·)µn =: µ; then X1, . . . , Xn independent

cost("B) ! icostµ("B)

= I(X1X2 . . .Xn : M)

! I(X1 : M) + I(X2 : M) + · · ·+ I(Xn : M) [superadditivity]

10




= I(X1X2 . . .Xn : M)


= icostµ1("A,1) + icostµ2("A,2) + · · ·+ icostµn("A,n)

To make this work, want protocols "A,j s.t.

M * Alice’s message in "A,j on input Xj ( µj

10-a




= I(X1X2 . . .Xn : M)





xAlice inWB Bob inx,Rmsg( ) BW

i

Notice: each "A,j solves echo

10-b




= I(X1X2 . . .Xn : M)





Alice inWB Bob inx,Rmsg( ) BWi

W

i = 2b

Bob in

rand+x = b

Alice in A,2W,2A

10




= I(X1X2 . . .Xn : M)






W

i = 2b

Bob in

rand+x = b

Alice in A,2W,2A

Notice: each "A,j solves echo

10-a

Step 3: Basic IC Lower Bound

Comm complexity: R!! (B) = min {cost(") : " solves B with error !}

Info complexity: ICµ,!! (B) = inf {icostµ(") : " solves B with error !}

Pick µ = uniform distrib, "; so far

R!! (B) ! IC",!

! (B)

IC",!! (B) ! n · IC",!(A )

11





R!! (B) ! IC",!

! (B)

IC",!! (B) ! n · IC",!(A )

Intuitively clear that ICµ,!! (A ) &= 0

Implication: R!! (B) = !(n). QED

11-a





R!! (B) ! IC",!

! (B)

IC",!! (B) ! n · IC",!(A )

Intuitively clear that ICµ,!! (A ) &= 0

Implication: R!! (B) = !(n). QED

In fact we can work out the constant precisely...

11-b

Step 3: Basic IC Lower Bound: Details

• Let "A be a protocol for task A (i.e., echo)

• Let Z = random input (uniform distrib ") for Alice

M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}

• Basic information theory:

icost"("A) = I(Z : M)= 12 (DKL(M (0)+M) + DKL(M (1)+M))

12




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}


icost"("A) = I(Z : M)= 12 (DKL(M (0)+M) + DKL(M (1)+M))

DTV(P,Q) : total variation distance

DKL(P+Q) : Kullback-Leibler divergence

DJS(P,Q) : Jensen-Shannon divergence

Hb(x) : binary entropy function

= ,x log x, (1, x) log(1, x)

12-a




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}

icost"("A) = I(Z : M)

12




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}


icost"("A) = I(Z : M) = 12 (DKL(M (0)+M) + DKL(M (1)+M))

12-a




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}



= DJS(M(0),M (1))

! 1, Hb

!1,DTV(M (0),M (1))

2

"

12-b




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}



= DJS(M(0),M (1))

! 1, Hb

!1,DTV(M (0),M (1))

2

"

• Error $ ! implies DTV(M (0),M (1)) ! 1, 2!

12-c




M = msg(Z,R)

M (z) = msg(z, R) for z # {0, 1}



= DJS(M(0),M (1))

! 1, Hb

!1,DTV(M (0),M (1))

2

"

• Error $ ! implies DTV(M (0),M (1)) ! 1, 2!

• Thus icost"("A) ! 1,Hb(!)

and so R!! (index) ! (1,Hb(!))n ... a tight bound!

12-d

The Problem: Applications

A humble lower bound, but with many applications!

• Complexity of sampling procedures

• Lower bounds for succinct data structures

• Space lower bounds for (one-pass) data stream algorithms

– Median of n numbers: !(n)

– Mode of n numbers: !(n)

– Connectivity of n-vertex graph, given edges: !(n)

– Triangle-freeness of n-vertex graph: !(n2)...

13

The Problem: Applications

A humble lower bound, but with many applications!

• Complexity of sampling procedures

• Lower bounds for succinct data structures

• Space lower bounds for (one-pass) data stream algorithms

– Median of n numbers: !(n)

– Mode of n numbers: !(n)

– Connectivity of n-vertex graph, given edges: !(n)

– Triangle-freeness of n-vertex graph: !(n2)...

– Diameter of n-vertex graph, k-approx: !(n1+1/k)

A very sophisticated reduction [Feigenbaum-K-M-S-Z’05]

13-a

The Problem: Extensions

• Generalize to augmented-index

kï11x ... x

x = x x ... x21 n

W

k

xkBobAlice

• Still !(n); replace superadditivity step with chain rule:

I(X1X2 . . .Xn : M) =#n

i=1 I(Xi : M | X1 . . .Xi"1)

14

The Problem: Extensions

• Generalize to augmented-index

kï11x ... x

x = x x ... x21 n

W

k

xkBobAlice

• Still !(n); replace superadditivity step with chain rule:

I(X1X2 . . .Xn : M) =#n

i=1 I(Xi : M | X1 . . .Xi"1)

• Generalize to interactive communication

– Communication complexity drops to O(log n)

– Seek tradeo!s [Magniez-Mathieu-Nayak’10], [C.-Kondapally’11]

14-a

Simultaneous Message Communication

= f(x,y) w.h.p.

Refereesketch(x) sketch(y)

Alice Bobx y

Lower bound method: R#(f) := R#1/3(f) = !(

$D#(f)) [Babai-Kimmel’97]

15


= f(x,y) w.h.p.


Alice Bobx y



Equality function eqn(x, y) = 1 -" x = y ... x, y # {0, 1}n

A neat result: R#(eqn) = #(.n) [Ambainis’96]

15-a


= f(x,y) w.h.p.


Alice Bobx y





Direct sum: oreqn,m(x1 . . . xm, y1 . . . ym) =%m

i=1 eqn(xi, yi)

What is R#(oreqn,m)? Above method only shows !(.mn)

15-b


= f(x,y) w.h.p.


Alice Bobx y





Direct sum: oreqn,m(x1 . . . xm, y1 . . . ym) =%m

i=1 eqn(xi, yi)

What is R#(oreqn,m)? Above method only shows !(.mn)

Theorem: (via IC) R#(oreqn,m) = !(m.n) [C.-Shi-Wirth-Yao’01]

15-c

IC Paradigm: Second Demo

Alice: input X ( ", message U ; Bob: input Y ( ", message V




16




icost"(") = I(X : U) + I(Y : V )


To solve eqn by simulating protocol for oreqn,m

Alice, Bob plug input into ith position, fill rest at random ( "

May change answer from 0 to 1 w.p. $ m/2n = o(1)


16-a




icost"(") = I(X : U) + I(Y : V )





3. Basic IC lower bound (for eq)

So far: R#(oreqn,m) ! IC",#(oreqn,m) ! m · IC",#(eqn)

Must show IC",#(eqn) ! R#(eqn)

16-b




icost"(") = I(X : U) + I(Y : V )








For last step: compress Alice’s/Bob’s messages down to their info content

Main idea: Whittle down message space via rejection sampling [CSWY’01]

16-c




icost"(") = I(X : U) + I(Y : V )








For last step: compress Alice’s/Bob’s messages down to their info content

Main idea: Whittle down message space via rejection sampling [CSWY’01]

Deeper version of idea: comm complexity of correlation [Harsha-J-M-R’07]

16-d

Part Two:

Interaction, But Not Really

17

Lower Bounds for Data Structures

Preprocess data Y / data structure T = T (Y ) ... low storage space

Query x / algorithm A(x, T ) / output z ... low query time

Satisfying some relation R(x, Y, z).

Examples: take x # {0, 1}d, Y 0 {0, 1}d with |Y | = n

• Predecessor Search

Treat data as d-bit integers

R(x, Y, z) i! z # Y is the predecessor of x in Y .

18

Lower Bounds for Data Structures

Preprocess data Y / data structure T = T (Y ) ... low storage space

Query x / algorithm A(x, T ) / output z ... low query time

Satisfying some relation R(x, Y, z).

Examples: take x # {0, 1}d, Y 0 {0, 1}d with |Y | = n

• Predecessor Search

Treat data as d-bit integers

R(x, Y, z) i! z # Y is the predecessor of x in Y .

• Approx Nearest Neighbor (ANN) Search

Treat data as points in Hamming cube

R(x, Y, z) i! z # Y is a #-ANN of x w.r.t. Y .

18-a

Cell-Probe Model

AlgorithmQuery

Preprocessed DatabaseQuery xT= Table

11110001000011001011001000

poly(d)bits per cell

cellspoly(n)

[Yao’81]

19

Cell-Probe Model

Index i1

1T[i ]

AlgorithmQuery


11110001000011001011001000


cellspoly(n)

[Yao’81]

19

Cell-Probe Model

i2

2T[i ]

Index i1

1T[i ]

AlgorithmQuery


11110001000011001011001000


cellspoly(n)

[Yao’81]

19

Cell-Probe Model

probest

i2

2T[i ]

Index i1

1T[i ]

AlgorithmQuery


11110001000011001011001000


cellspoly(n)

[Yao’81]

19

Cell-Probe Model

probest

i2

2T[i ]

Index i1

1T[i ]

AlgorithmQuery


11110001000011001011001000


cellspoly(n)

ALIC

E

BOB

rounds2t

Alice: O(log n)-bit messages; Bob: poly(d)-bit messages

19

Round Elimination: Predecessor Search

Repeatedly remove first round, shrink instance size

[Miltersen-Nisan-Safra-Wigderson’95], [Sen’03]

ALICE

d bits

d bits

npo

ints

BOB

Eventually: zero communication protocol for instance size (d/kt, n/$t)

Implying... Theorem: Query time t = !

!log d

log log d

"

20




ALICE

d/k

bitsd/k

npo

ints

BOB

bits



!log d

log log d

"

20




ALICE

d/k

bitsd/k

poin

tsn/

l

BOB

bits



!log d

log log d

"

20

Predecessor Search: Embeddability Property

Input (x, Y ) solvable using input (% 1 x 1 rand, % 1 Y 1 rand).

x

ALICE BOB

yy

y

3

n

1

2

y

Simple task (A ): instance size (n/k, d), using 2t rounds

Complex task (B): instance size (n, d), using 2t rounds ' k times A

21



x

ALICE BOB

yy

y

3

n

1

2

y

m

m

m

m

m

m

m

21



x

ALICE BOB

yy

y

3

n

1

2

y

m

m

m

m

m

m

m

Simple task (A ): instance size (n/k, d), using 2t rounds

Complex task (B): instance size (n, d), using 2t rounds ' k times A

21-a

IC Paradigm For Round Elimination


icostµ1 (") = I(X : M1) Alice’s input X ( µ, M1 = msg1(X,R)


3. Compression Argument

22





Put X = X1X2 . . .Xk, each Xi: a (d/k)-bit chunk

Protocol "A,#: pad instance using prefix % of length (i, 1)d/k

O(log n) ! icostµ!k

1 (!B) = I(X1X2 . . . Xk : M1)

=!k

i=1 I(Xi : M1 | X1 . . . Xi!1)

=k"

i=1

E![I(Xi : M1 | X1 . . .Xi!1 = !)] =k"

i=1

E![icostµ1 (!A,!)]


22-a








1 (!B) = I(X1X2 . . . Xk : M1)

=!k

i=1 I(Xi : M1 | X1 . . . Xi!1)

=k"

i=1

E![I(Xi : M1 | X1 . . .Xi!1 = !)] =k"

i=1

E![icostµ1 (!A,!)]


So far: exists "A,# with icostµ1 ("A,#) $ O((log n)/k) = o(1)

Pretend msg1(X$, R) sent as first message, X $ * X but indep.

22-b








1 (!B) = I(X1X2 . . . Xk : M1)

=!k

i=1 I(Xi : M1 | X1 . . . Xi!1)

=k"

i=1

E![I(Xi : M1 | X1 . . .Xi!1 = !)] =k"

i=1

E![icostµ1 (!A,!)]


So far: exists "A,# with icostµ1 ("A,#) $ O((log n)/k) = o(1)

Pretend msg1(X$, R) sent as first message, X $ * X but indep.

Error += O(&icostµ1 ("A,#)) = o(1) [Pinsker’s inequality]

22-c

Round Elimination: ANN Search


As before, icost1(") = I(X : M1)


ANN does not have embeddability property


23






Reduce from Longest Prefix Match (LPM), which does

Get icostµ1 ("A,#) $ O((log n)/k) but this bound = &(1)


23-a






Reduce from Longest Prefix Match (LPM), which does

Get icostµ1 ("A,#) $ O((log n)/k) but this bound = &(1)


Use comm complexity of correlation [Harsha-J-M-R’07]

Compress first message down to its info content

Now it’s short enough: easy (combinatorial) round eliminiation

Eventually... Theorem: Query time t = !

!log log d

log log log d

"[C.-Regev’10]

23-b

Pointer Jumping Problems

Input: One pointer per level in layered graph; plus one bit per leaf

Task: output bit at leaf reached by following pointers from root

Multilayer Ptr Jumping, n,p

Full layered DAG, n nodes/layer

Number-on-Forehead (NOF)

Speaking order P1, . . . , Pp

0

1

0

1

1

0

1P1 P2 P3 P4 P5

Tree Pointer Jumping, n,p+1

Complete (p+ 1)-level n-ary tree

Number-In-Hand (NIH)

Use p rounds, going up tree: """

1 0 0 1 1 1 00 1

Level

Level

2

3

Level 1

Theorems: R!(mpjn,p) = !(n/p)! Rp(tpjn,p+1) = !(n/p2)

24

The Importance of a Careful Definition

For the NOF problem mpjn,p [Chakrabarti’07]

Input value

Random value

Fixed value

XXX X321 4

Protocol P , input (X1, . . . , Xp) ( µ, M1 = message of player P1:

icostµ(") := I(M1 : X2 | X3, . . . , Xk)

Simulation argument works only under one of these protocol restrictions:

• Myopic: each player sees only one layer ahead ... !(n/p)

• Conservative: don’t see behind except know current node ... !(n/p2)

25

Pointer Jumping: Applications

• Layered DAG version, NOF

– Strong NOF lower bounds imply circuit lower bounds [Yao’90]

– If myopic/conservative restriction removed, ACC0 &= LOGSPACE

26

Pointer Jumping: Applications

• Layered DAG version, NOF

– Strong NOF lower bounds imply circuit lower bounds [Yao’90]

– If myopic/conservative restriction removed, ACC0 &= LOGSPACE

• Tree version, NIH

– Multi-pass data stream lower bounds

– Classic example: median of n numbers, p passes: !(n1/p) space

– Modern: median, randomly-ordered, p passes: !(n2"p

) space

A sophisticated reduction [C.-Cormode-McGregor’08]

26-a

Part Three:

Full Interaction

27

The problem

• Input: 22 n Boolean matrix

• Task: distinguish between the following two cases

– Case 0: Every column has weight $ 1

0 1 0 0 0 1 1 0

1 0 1 1 0 0 0 0

– Case 1: One column has weight 2, rest have weight $ 1

(i.e., the subsets of [n] represented by the rows intersect)

0 1 1 0 0 1 1 0

1 0 1 1 0 0 0 0

28

The problem

• Input: 22 n Boolean matrix

• Task: distinguish between the following two cases

– Case 0: Every column has weight $ 1

0 1 0 0 0 1 1 0

1 0 1 1 0 0 0 0

3,Alice

3,Bob

– Case 1: One column has weight 2, rest have weight $ 1

(i.e., the subsets of [n] represented by the rows intersect)

0 1 1 0 0 1 1 0

1 0 1 1 0 0 0 0

3,Alice

3,Bob

• This problem is called disjn,2

• Later: t-player generalization disjn,t

28-a

New Twist in Applying IC Paradigm

Problem and2:

• Alice holds a # {0, 1}, Bob holds b # {0, 1}

• Bob to output a 4 b

Simple task (A ): and2

Complex task (B): disjn,2 ' combines n copies of and2

29

New Twist in Applying IC Paradigm

Problem and2:

• Alice holds a # {0, 1}, Bob holds b # {0, 1}

• Bob to output a 4 b

Simple task (A ): and2

Complex task (B): disjn,2 ' combines n copies of and2

The key twist:

• Step 2 (Simulation): pad and2 instance to get disjn,2 instance

• Careless random padding will drown out the answer!

• Must ensure “padding” columns j have and2(Xj , Yj) = 0

• Need shared coins to do this ... so must condition on these coins

29-a

IC Paradigm Applied to (1/3)

Inputs: 'X = X1 . . .Xn, 'Y = Y1 . . . Yn; Auxiliary coins: D1 . . .Dn

Each (Xi, Yi, Di) ( µ; M = transcript!(X,Y,Rpub, Rpriv)

µ =

XY ! 00 01 10 11

D = 0 1/2 0 1/2 0

D = 1 1/2 1/2 0 0


icostµ(") = I( 'X 'Y : M | 'D,Rpub) (Note: external icost)


3. Basic IC lower bound (for and2)

30




µ =

XY ! 00 01 10 11

D = 0 1/2 0 1/2 0

D = 1 1/2 1/2 0 0




Protocols "A,i for and2(X,Y ) simulating "B for disjn,2

Public coins for 'D, private for 'X, 'Y s.t. each (Xj , Yj , Dj) ( µ


30-a




µ =

XY ! 00 01 10 11

D = 0 1/2 0 1/2 0

D = 1 1/2 1/2 0 0





Public coins for 'D, private for 'X, 'Y s.t. each (Xj , Yj , Dj) ( µ

Crucial property: Dj factorizes (Xj , Yj)

Plug in Xi 3 X and Yi 3 Y


30-b



µ =

XY ! 00 01 10 11

D = 0 1/2 0 1/2 0

D = 1 1/2 1/2 0 0



icostµ!n

("B) = I( 'X 'Y : M | 'D) !#n

i=1 I(XiYi : M | 'D)

=#n

i=1 I(XiYi : M | Di, 'D"i) =#n

i=1 icostµ("A,i)


31



µ =

XY ! 00 01 10 11

D = 0 1/2 0 1/2 0

D = 1 1/2 1/2 0 0



icostµ!n

("B) = I( 'X 'Y : M | 'D) !#n

i=1 I(XiYi : M | 'D)

=#n

i=1 I(XiYi : M | Di, 'D"i) =#n

i=1 icostµ("A,i)


To prove: % "A solving and2, have icostµ("A) = !(1)

Twist: distrib µ not hard for and2: E(X,Y )%µ[and2(X,Y )] = 0

31-b


• Protocol "A for and2; (X,Y,D) ( µ; M = trans!(X,Y,Rpriv)

• Let M (wz) = transcript!(w, z,Rpriv) for w, z # {0, 1}; then

icostµ("A) = I(XY : M | D) = 12 (I(X : M | D = 0) + I(Y : M | D = 1))

= 12 (DJS(M (00),M (10)) + DJS(M (00),M (01)))

! 12 (h

2(M (00),M (10)) + h2(M (00),M (01)))

! 14h

2(M (10),M (01))

32





= 12 (DJS(M (00),M (10)) + DJS(M (00),M (01)))

! 12 (h

2(M (00),M (10)) + h2(M (00),M (01)))

! 14h

2(M (10),M (01)) = 14h

2(M (00),M (11))

by cut-and-paste property, conseq of rectangle property for protocols

• Error $ ! implies

DTV(M(00),M (11)) ! 1, 2! =" h2(M (00),M (11)) ! 1, 2

.!

• Overall: R!(disjn,2) ! 14 (1, 2

.!)n [BarYossef-J-K-S’04]

32-a

Digression: Cut-And-Paste

• Protocol "; inputs w, z

• Let M (wz) = transcript!(w, z,Rpriv) for w, z # {0, 1}

• Rectangle property:

Can “factorize” distrib of M (wz) as f (w) 5 g(z)

This means % a : Pr[M (wz) = a] = f (w)(a)g(z)(a)

• Hellinger distance: h2((, P ), Q) = 1,#

a

&P (a)Q(a)

• Put these together:

h2(M (bc),M (wz)) = h2(M (bz),M (cw))

33





= 12 (DJS(M (00),M (10)) + DJS(M (00),M (01)))

! 12 (h

2(M (00),M (10)) + h2(M (00),M (01)))

! 14h

2(M (10),M (01)) = 14h

2(M (00),M (11))


34





= 12 (DJS(M (00),M (10)) + DJS(M (00),M (01)))

! 12 (h

2(M (00),M (10)) + h2(M (00),M (01)))

! 14h

2(M (10),M (01)) = 14h

2(M (00),M (11))


• Error $ ! implies

DTV(M(00),M (11)) ! 1, 2! =" h2(M (00),M (11)) ! 1, 2

.!

• Overall: R!(disjn,2) ! 14 (1, 2

.!)n [BarYossef-J-K-S’04]

34-b

Applications of

Multi-pass lower bounds for many data stream problems

• Connectivity of n-vertex graphs: !(n) space

Generalization (t players, NIH)

Theorem: R(disjn,t) = !(n/t) [C.-Khot-Sun’03], [Gronemeier’09]

t

Pad

Input to DISJInput to AND n,t

35

Applications of

Multi-pass lower bounds for many data stream problems

• Connectivity of n-vertex graphs: !(n) space

Generalization (t players, NIH)

Theorem: R(disjn,t) = !(n/t) [C.-Khot-Sun’03], [Gronemeier’09]

t

Pad

Input to DISJInput to AND n,t

• Classic data stream problem: frequency moments Fk =#

j fkj

where fj := number of occurrences of ‘j’ in stream

• Approximating Fk: space '#(n1"2/k)

lower bound via disjn,t [Alon-Matias-Szegedy’96]

35-a

Yet More Applications of IC

Sadly, left on cutting room floor ...

• Separation of nondet and randomized CC [Jayram-Kumar-Sivakumar’03]

• CC of read-once-formula problems [Saks-Leonardos’09]

[Jayram-Kopparty-Raghavendra’09]

• Det vs rand decision trees [Jayram-Kumar-Sivakumar’03]

• Increasingly complex data stream lower bounds

[Jayram-Woodru!’09], [Magniez-Mathieu-Nayak’10]

[C.-Cormode-Kondapally-McGregor’10], [Magniez’13]

• Data structure query/update time lower bounds [Patrascu’10]

• Quantum communication... [Jain-Radhakrishnan-Sen]

36

probest

i2

2T[i ]

Index i1

1T[i ]

AlgorithmQuery


11110001000011001011001000


cellspoly(n)

ALIC

E

BOB

rounds2t

kï11x ... x

x = x x ... x21 n

W

k

xkBobAlice 1 0 0 1 1 1 00 1

Level

Level

2

3

Level 1

x

ALICE BOB

yy

y

3

n

1

2

y

m

m

m

m

m

m

m THANKS!ALICE

d bits

d bits

np

oin

ts

BOB

Input value

Random value

Fixed value

XXX X321 4


W

i = 2b

Bob in

rand+x = b

Alice in A,2W,2A

= f(x,y) w.h.p.


Alice Bobx y

37

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