Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence. - PowerPoint PPT Presentation

Lower Bounds on Streaming Algorithms for Approximating the Length of theLongest Increasing Subsequence.

Anna Gal UT Austin

Parikshit Gopalan U. Washington & UT Austin

Storage

X1 X2 X3 … Xn

Input

• Single pass.
• Small storage space, update time.
• Surprisingly powerful [Alon-Matias-Szegedy, …]

Cannot sort efficiently.

Can we tell if the data needs to be sorted?

• [Ajtai-Jayram-Kumar-Sivakumar, Gupta-Zane,
• Cormode-Muthukrishnan-Sahinalp, LibenNowell-Vee-Zhu,
• Woodruff-Sun,G.-Jayram-Kumar-Sivakumar]

• Measuring Sortedness:
• Length of Longest Increasing Subsequence.
• Ulam/Edit distance
• Inversion/Kendall Tau distance

Longest Increasing Subsequence

LIS(): Length of Longest Increasing Subsequence.

5 7 8 1 4 2 10 3 6 9

Longest Increasing Subsequence

LIS(): Length of Longest Increasing Subsequence.

5 7 8142103 6 9

Studied in statistics, biology, computer science … [Gusfeld, Pevzner, Aldous-Diaconis…]

• Exact Computation of LIS() :
• Patience Sorting [Ross,Mallows]

O(n) space, 1-pass streaming algorithm.

• (n) space lower bound. [G.-Jayram-Krauthgamer-Kumar’07, Woodruff-Sun’07]
• Approximating LIS() :
• Deterministic, O(n/)1/2space, (1 + )-approx.

[G.-Jayram-Krauthgamer-Kumar’07]

Conjecture [GJKK]: Every 1-pass deterministic algorithm that gives a 1.1-approximation toLIS() requires (√n) space.

Thm: Any det. O(1)-pass algorithm that gives a (1 + ) approximation to the LIS requires space √(n/).

• Tight bounds in n, .
• Proof via direct sum approach.
• Direct sum for maximum communication in the private messages model.
• Separation between communication models.

Consider the following problem:

1 2 3.2 4.2

1.8 2.9 3.7 4.9

1.6 2.8 3.5 4.6

• t players, t numbers each.
• Goal: Approximate length of the LIS.
• Enough to show a lower bound of (t) on maximum message size.

Consider the following problem –

P1

P2

…

Pt

• t players, t numbers each.
• Goal: Approximate length of the LIS.
• Enough to show a lower bound of (t) on maximum message size.

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

…

Pt

A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

…

Pt

All columns non-increasing

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

…

Pt

All columns non-increasing

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

…

Pt

All columns non-increasing

Some column increasing

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

…

Pt

All columns non-increasing

Some column increasing

Primitive Problem:

p(x1, y1)

y1

x1

Direct Sum Problem:

Çi p(xi,yi)

y1,…,yn

x1,…,xn

Can run n copies of protocol for p.

Direct-Sum Question: Is this the best possible?

Set-Disjointness, Inner Product…

Techniques for proving direct-sum theorems:

[KN,CKSW,BJKS,SS…]

Yes

No

P1

P2

…

Pt

Yes

No

P1

P2

…

Pt

All No instances

Yes

No

P1

P2

…

Pt

All No instances

One Yes instance

Yes

No

P1

P2

…

Pt

[GG] An Easier Problem

Yes

No

Hope: Some player distinguishes between many No instances.

• Players speak in order.
• Every message seen by all.
• Last player outputs answer.

Problem is Easy in the BlackBoard model

No

Yes

BlackBoard protocol with max. communication 2 log(m).

Problem is Easy in the BlackBoard model

No

Yes

BlackBoard protocol with max. communication 2 log(m).

• Messages seen by next player only.
• Suffices for streaming lower bound.
• Requires non-standard techniques.

Private Messages Model

Yes

No

Strong lower bound for maximum communication in the private messages model.

Thm: Any det. O(1)-pass algorithm that gives a (1 + ) approximation to the LIS requires space √(n/).

Separation between blackboard and private messages.

• Step 1: Primitive Problem (one round).
• Step 2: Direct-sum Problem (one-round).
• Multi-round Protocols.

Yes

No

P1

P2

…

Pt

Alphabet of size m > t. Yes Case: LIS() > t/2.

Easy: Bound of ≈ (log m)/t on max communication.

Thm:Max communication is at least log (m/t).

a

a

a

a

a

a…a

a…a

a…a

x1…xi

Pis message is specified by prefix x1…xi.

Mi(a): Prefixes where Pi sends the same message as a…a.

qi(a): Length of longest IS in Mi(a)ending below a.

a

a

a

a

Mi(a): Inputs where Pi sends the same message as a…a.

qi(a): Length of longest IS in Mi(a)ending below a.

• Monotone
• x1…xi2 Mi(a) ) x1…xia 2 Mi+1(a)
• Bounded by t/2
• Correctness.

qi(a)

i

a

a

a

a

Mi(a): Inputs where Pi sends the same message as a…a.

qi(a): Length of longest IS in Mi(a)ending below a.

Map a to first i s.t

qi-1(a) = qi(a).

Some i occurs m/t times.

qi(a)

i

Pi-1

Pi

a…a

x1< … < xi-1 = a

x1…xi-1

b…b

m/t

y1< … < yi-1 = b

y1…yi-1

c…c

z1< … < zi-1 = c

z1…zi-1

Claim:Pi-1 must distinguish a…a from b…b from c…c.

Pi-1

Pi

a…ab

a…a

x1…xi-1b

x1…xi-1

y1…yi-1b

y1…yi-1

b…bb

b…b

x1· … · xi-1 = a · b

But qi(b) = i-1. Contradiction.

HencePi-1 must distinguish a…a from b…b from c…c.

Gives log(m/t) lower bound.

a1…at

a1…at

a1…at

a1…at

Mi(a1…at):i £ t prefixes where Pi sends the same message as (a1…at)i.

qi,j(a1…at): Length of longest IS in column jending at/before aj.

a1…at

a1…at

a1…at

a1…at

Mi(a1…at):i £ t prefixes where Pi sends the same message as (a1…at)i.

qi,j(a1…at): Length of longest IS in column jending at/before aj.

...

qi,t(a)

qi,1(a)

a1…at

a1…at

a1…at

a1…at

Mi(a1…at):i £ t prefixes where Pi sends the same message as (a1…at)i.

qi,j(a1…at): Length of longest IS in column jending at/before aj.

...

qi,t(a)

qi,1(a)

a1…at

a1…at

Part II:

Show that Pi-1 distinguishes between inputs in I of ≈(m/t)t inputs.

Gives a lower bound of log(|I|) ≈ t log (m/t)

a1…at

a1…at

a1…at

a1…at

Part I: Messages sent by Pi in round 2 and beyond depend on entire input.

Need to change defn. of Mi(a1…at).

a1…at

a1…at

Part I: Messages sent by Pi in round 2 and beyond depend on entire input.

Need to change defn. of Mi(a1…at).

Part II: Reduce to 2-player protocol involving Pi-1 and Pt.

Thm: Any deterministic O(1)-pass algorithm that gives a (1 + ) approximation to the LIS requires space √(n/).

• Exact Computation of LIS() :
• Patience Sorting [Ross,Mallows]
• O(n) space, 1-pass streaming algorithm.
• (n) space lower bound. [G.-Jayram-Krauthgamer-Kumar, Woodruff-Sun]
• Approximating LIS() :
• O(n/)1/2space, deterministic 1-pass algorithm. [G.-Jayram-Krauthgamer-Kumar]
• This paper: The bound is tight for deterministic, O(1)-pass algorithms.
• [Ergun-Jowhari’08]: Different proof.

Problem: Is the a randomized streaming algorithm to approximate the LIS using space o(√n)?

• [Woodruff-Sun] O(log m) lower bound
• [Chakrabarti]: Randomized private-messages protocol for the direct-sum problem.

Thank You!