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Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence. Anna Gal UT Austin Parikshit Gopalan U. Washington & UT Austin. Storage. Data Stream Model of Computation. X 1 X 2 X 3 … X n. Input. Single pass.

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slide1

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

Anna Gal UT Austin

Parikshit Gopalan U. Washington & UT Austin

data stream model of computation

Storage

Data Stream Model of Computation

X1 X2 X3 … Xn

Input

  • Single pass.
  • Small storage space, update time.
  • Surprisingly powerful [Alon-Matias-Szegedy, …]
estimated sortedness on data streams
Estimated Sortedness on Data-Streams

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
slide4

Longest Increasing Subsequence

LIS(): Length of Longest Increasing Subsequence.

5 7 8 1 4 2 10 3 6 9

slide5

Longest Increasing Subsequence

LIS(): Length of Longest Increasing Subsequence.

5 7 8142103 6 9

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

prior work
Prior Work
  • 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.

our results
Our Results

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.
a communication problem
A Communication Problem

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.
a communication problem1
A Communication Problem

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.
a communication problem2
A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

Pt

slide11

A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

Pt

All columns non-increasing

a communication problem3
A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

Pt

All columns non-increasing

a communication problem4
A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

Pt

All columns non-increasing

Some column increasing

a communication problem5
A Communication Problem

[GJKK]: Consider the following decision problem –

Yes

No

P1

P2

Pt

All columns non-increasing

Some column increasing

direct sum paradigm
Direct Sum Paradigm

Primitive Problem:

p(x1, y1)

y1

x1

direct sum paradigm1
Direct Sum Paradigm

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…]

primitive problem
Primitive Problem

Yes

No

P1

P2

Pt

direct sum of primitive problems
Direct Sum of Primitive Problems

Yes

No

P1

P2

Pt

All No instances

direct sum of primitive problems1
Direct Sum of Primitive Problems

Yes

No

P1

P2

Pt

All No instances

One Yes instance

slide21

[GG] An Easier Problem

Yes

No

Hope: Some player distinguishes between many No instances.

blackboard model of one way communication
BlackBoard Model of One-Way Communication
  • Players speak in order.
  • Every message seen by all.
  • Last player outputs answer.
slide23

Problem is Easy in the BlackBoard model

No

Yes

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

slide24

Problem is Easy in the BlackBoard model

No

Yes

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

private messages model
Private Messages Model
  • Messages seen by next player only.
  • Suffices for streaming lower bound.
  • Requires non-standard techniques.
slide26

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.

proof outline
Proof Outline
  • Step 1: Primitive Problem (one round).
  • Step 2: Direct-sum Problem (one-round).
  • Multi-round Protocols.
primitive problem1
Primitive Problem

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).

lower bound for primitive problem
Lower Bound for Primitive Problem

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.

lower bound for primitive problem1
Lower Bound for Primitive Problem

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

lower bound for primitive problem2
Lower Bound for Primitive Problem

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

lower bound for primitive problem3
Lower Bound for Primitive Problem

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.

lower bound for primitive problem4
Lower Bound for Primitive Problem

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.

lower bound for general problem
Lower Bound for General Problem

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.

lower bound for general problem1
Lower Bound for General Problem

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)

lower bound for general problem2
Lower Bound for General Problem

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)

lower bound for general problem3
Lower Bound for General Problem

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)

lower bound for many rounds
Lower Bound for Many Rounds

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).

lower bound for many rounds1
Lower Bound for Many Rounds

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/).

conclusions
Conclusions
  • 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.
randomized complexity of lis
Randomized Complexity of LIS

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!