Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subseque

1
Lower Bounds on Streaming Algorithms for
Approximating the Length of theLongest
Increasing Subsequence.
Anna Gal UT Austin Parikshit Gopalan U.
Washington UT Austin
2
Data Stream Model of Computation
X1 X2 X3 Xn
Input

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

3
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, Wood
ruff-Sun, G.-Jayram-Kumar-Sivakumar

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

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Longest Increasing Subsequence
LIS(?) Length of Longest Increasing Subsequence.
5 7 8 1 4 2 10 3 6 9
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Longest Increasing Subsequence
LIS(?) Length of Longest Increasing Subsequence.
5 7 8 1 4 2 10 3 6 9
Studied in statistics, biology, computer science
Gusfeld, Pevzner, Aldous-Diaconis
6
Prior Work

• Exact Computation of LIS(?)
• Patience Sorting Ross,Mallows
• O(n) space, 1-pass streaming algorithm.
• ?(n) space lower bound. G.-Jayram-Krauthgamer-Kum
  ar07, Woodruff-Sun07
• Approximating LIS(?)
• Deterministic, O(n/?)1/2 space, (1 ?)-approx.
• G.-Jayram-Krauthgamer-Kumar07

Conjecture GJKK Every 1-pass deterministic
algorithm that gives a 1.1-approximation to
LIS(?) requires ?(vn) space.
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Our Results
Thm Any det. O(1)-pass algorithm that gives a (1
?) approximation to the LIS requires space
v(n/?).

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

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A Communication Problem

• Consider the following problem

1 2 3.2 4.2
1.6 2.8 3.5 4.6
1.8 2.9 3.7 4.9

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

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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.

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A Communication Problem
GJKK Consider the following decision problem
Yes
No
P1 P2 Pt
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A Communication Problem
GJKK Consider the following decision problem
Yes
No
P1 P2 Pt
All columns non-increasing
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A Communication Problem
GJKK Consider the following decision problem
Yes
No
P1 P2 Pt
All columns non-increasing
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A Communication Problem
GJKK Consider the following decision problem
Yes
No
P1 P2 Pt
Some column increasing
All columns non-increasing
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A Communication Problem
GJKK Consider the following decision problem
Yes
No
P1 P2 Pt
Some column increasing
All columns non-increasing
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Direct Sum Paradigm
Primitive Problem
p(x1, y1)
y1
x1
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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-Disjoint
ness, Inner Product Techniques for proving
direct-sum theorems KN,CKSW,BJKS,SS
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Primitive Problem
Yes
No
P1 P2 Pt
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Direct Sum of Primitive Problems
Yes
No
P1 P2 Pt
All No instances
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Direct Sum of Primitive Problems
Yes
No
P1 P2 Pt
All No instances
One Yes instance
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Direct Sum of Primitive Problems
Yes
No
P1 P2 Pt
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Direct Sum of Primitive Problems
Yes
No
P1 P2 Pt
Techniques for proving direct-sum
theorems KN,CSWY,BJKS,SS,
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GG An Easier Problem
Yes
No
Hope Some player distinguishes between many No
instances.
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BlackBoard Model of One-Way Communication

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

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Problem is Easy in the BlackBoard model
No
Yes
BlackBoard protocol with max. communication 2
log(m).
25
Problem is Easy in the BlackBoard model
No
Yes
BlackBoard protocol with max. communication 2
log(m).
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Private Messages Model

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

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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
v(n/?).
Separation between blackboard and private
messages.
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Proof Outline

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

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Primitive Problem
Yes
No
P1 P2 Pt
Alphabet of size m gt t. Yes Case LIS(?) gt
t/2. Easy Bound of (log m)/t on max
communication. Thm Max communication is at least
log (m/t).
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Lower Bound for Primitive Problem
a
a
a
a
a
aa
aa
aa
x1xi
Pis message is specified by prefix x1xi. Mi(a)
Prefixes where Pi sends the same message as
aa. qi(a) Length of longest IS in Mi(a) ending
below a.
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Lower Bound for Primitive Problem
a
a
a
a
Mi(a) Inputs where Pi sends the same message as
aa. qi(a) Length of longest IS in Mi(a) ending
below a.

• Monotone
• x1xi 2 Mi(a) ) x1xia 2 Mi1(a)
• Bounded by t/2
• Correctness.

qi(a)
i
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Lower Bound for Primitive Problem
a
a
a
a
Mi(a) Inputs where Pi sends the same message as
aa. 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
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Lower Bound for Primitive Problem
Pi-1
Pi
aa
x1 lt lt xi-1 a
x1xi-1
bb
m/t
y1 lt lt yi-1 b
y1yi-1
cc
z1 lt lt zi-1 c
z1zi-1
Claim Pi-1 must distinguish aa from bb from
cc.
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Lower Bound for Primitive Problem
Pi-1
Pi
aab
aa
x1xi-1b
x1xi-1
y1yi-1b
y1yi-1
bbb
bb
x1 xi-1 a b But qi(b) i-1.
Contradiction.
Hence Pi-1 must distinguish aa from bb from
cc. Gives log(m/t) lower bound.
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Lower Bound for General Problem
a1at
a1at
a1at
a1at
Mi(a1at) i t prefixes where Pi sends the same
message as (a1at)i. qi,j(a1at) Length of
longest IS in column j ending at/before aj.
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Lower Bound for General Problem
a1at
a1at
a1at
a1at
Mi(a1at) i t prefixes where Pi sends the same
message as (a1at)i. qi,j(a1at) Length of
longest IS in column j ending at/before aj.
...
qi,t(a)
qi,1(a)
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Lower Bound for General Problem
a1at
a1at
a1at
a1at
Mi(a1at) i t prefixes where Pi sends the same
message as (a1at)i. qi,j(a1at) Length of
longest IS in column j ending at/before aj.
...
qi,t(a)
qi,1(a)
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Lower Bound for General Problem
a1at
a1at
a1at
a1at

• Part I By pigeonhole, find
• A good player Pi
• A good set S µ t of columns
• A good set I µ mt of (m/t)t inputs where

...
qi,t(a)
qi,1(a)
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Lower Bound for General Problem
a1at
a1at
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)
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Lower Bound for Many Rounds
a1at
a1at
a1at
a1at
Part I Messages sent by Pi in round 2 and beyond
depend on entire input. Need to change defn. of
Mi(a1at).
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Lower Bound for Many Rounds
a1at
a1at
Part I Messages sent by Pi in round 2 and beyond
depend on entire input. Need to change defn. of
Mi(a1at). 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 v(n/?).
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Conclusions

• Exact Computation of LIS(?)
• Patience Sorting Ross,Mallows
• O(n) space, 1-pass streaming algorithm.
• ?(n) space lower bound. G.-Jayram-Krauthgamer-Ku
  mar, Woodruff-Sun
• Approximating LIS(?)
• O(n/?)1/2 space, deterministic 1-pass algorithm.
  G.-Jayram-Krauthgamer-Kumar
• This paper The bound is tight for deterministic,
  O(1)-pass algorithms.
• Ergun-Jowhari08 Different proof.

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Randomized Complexity of LIS

• Problem Is the a randomized streaming algorithm
  to approximate the LIS using space o(vn) ?
• Woodruff-Sun O(log m) lower bound
• Chakrabarti Randomized private-messages
  protocol for the direct-sum problem.

Thank You!
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Prior Work

• Exact Computation of LIS(?)
• Patience Sorting Ross,Mallows

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Patience Sorting Ross,Mallows

• Track best inc. seq. of length i, for all i.
• Ai Smallest number ending an IS of length i.
• Patience Sorting Dynamic program to compute
  Ai.

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Approximate Patience Sorting GJKK

• Track best inc. seq. of length i, for all i.
• Ai Smallest number ending an IS of length i.
• Patience Sorting Dynamic program to compute
  Ai.
• Approx. Patience Sorting Store Ai for at most
  vn values of i.

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Lower Bounds for approximating the LIS
Conjecture GJKK For some e0 gt 0, every 1-pass
deterministic algorithm that gives a (1 e0)
approximation to LIS(?) requires ?(vn) space.
Candidate Hard Instances
P1 P2 Pt
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Protocol for BlackBoard model
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Protocol for BlackBoard model
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Protocol for BlackBoard model
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Primitive Problem
Yes
No
P1 P2 Pt
Does the direct sum property hold for this
problem?