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Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence. Anna GalUT Austin Parikshit GopalanU. Washington & UT Austin

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Data Stream Model of Computation X 1 X 2 X 3 …X n Input Storage Single pass. Small storage space, update time. Surprisingly powerful [Alon-Matias-Szegedy, …]

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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

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LIS( ): Length of Longest Increasing Subsequence Longest Increasing Subsequence

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LIS( ): Length of Longest Increasing Subsequence Studied in statistics, biology, computer science … [Gusfeld, Pevzner, Aldous-Diaconis…] Longest Increasing Subsequence

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Prior Work Exact Computation of LIS( ) : –Patience Sorting [Ross,Mallows] O(n) space, 1-pass streaming algorithm. – n) space lower bound. [G.-Jayram-Krauthgamer- Kumar07, 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 n) space.

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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.

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A Communication Problem Consider the following problem: 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 – t players, t numbers each. Goal: Approximate length of the LIS. Enough to show a lower bound of (t) on maximum message size P1P2…PtP1P2…Pt

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A Communication Problem No Yes P1P2…PtP1P2…Pt [GJKK]: Consider the following decision problem –

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No Yes All columns non-increasing P1P2…PtP1P2…Pt A Communication Problem [GJKK]: Consider the following decision problem –

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No Yes P1P2…PtP1P2…Pt A Communication Problem [GJKK]: Consider the following decision problem – All columns non-increasing

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No Yes Some column increasing P1P2…PtP1P2…Pt A Communication Problem [GJKK]: Consider the following decision problem – All columns non-increasing

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No Yes Some column increasing P1P2…PtP1P2…Pt A Communication Problem [GJKK]: Consider the following decision problem – All columns non-increasing

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Direct Sum Paradigm x1x1 y1y1 p(x 1, y 1 ) Primitive Problem:

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Direct Sum Paradigm x 1,…,x n y 1,…,y n Ç i p(x i, y i ) 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…] Direct Sum Problem:

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Primitive Problem No Yes P1P2…PtP1P2…Pt

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Direct Sum of Primitive Problems No Yes P1P2…PtP1P2…Pt All No instances

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Direct Sum of Primitive Problems No Yes P1P2…PtP1P2…Pt All No instancesOne Yes instance

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Direct Sum of Primitive Problems No Yes P1P2…PtP1P2…Pt

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Direct Sum of Primitive Problems No Yes P1P2…PtP1P2…Pt Techniques for proving direct-sum theorems: [KN,CSWY,BJKS,SS,…]

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No Yes [GG] An Easier Problem 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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NoYes Problem is Easy in the BlackBoard model BlackBoard protocol with max. communication 2 log(m).

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NoYes Problem is Easy in the BlackBoard model 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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No Yes 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/ ). Private Messages Model 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 No Yes P1P2…PtP1P2…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).

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Lower Bound for Primitive Problem aaa a P i s message is specified by prefix x 1 …x i. M i (a): Prefixes where P i sends the same message as a…a. q i (a): Length of longest IS in M i (a) ending below a. a…a x 1 …x i a a…a

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Lower Bound for Primitive Problem M i (a): Inputs where P i sends the same message as a…a. q i (a): Length of longest IS in M i (a) ending below a. i q i (a) Monotone x 1 …x i 2 M i (a) ) x 1 …x i a 2 M i+1 (a) Bounded by t/2 Correctness. aaa a

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Lower Bound for Primitive Problem M i (a): Inputs where P i sends the same message as a…a. q i (a): Length of longest IS in M i (a) ending below a. i q i (a) Map a to first i s.t q i-1 (a) = q i (a). Some i occurs m/t times. aaa a

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Lower Bound for Primitive Problem P i-1 PiPi x 1 < … < x i-1 = a Claim: P i-1 must distinguish a…a from b…b from c…c. a…a x 1 …x i-1 b…b c…c y 1 …y i-1 z 1 …z i-1 m/t y 1 < … < y i-1 = b z 1 < … < z i-1 = c

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Lower Bound for Primitive Problem Hence P i-1 must distinguish a…a from b…b from c…c. Gives log(m/t) lower bound. a…a x 1 …x i-1 b…b y 1 …y i-1 a…ab x 1 …x i-1 b b…bb y 1 …y i-1 b x 1 · … · x i-1 = a · b But q i (b) = i-1. Contradiction. P i-1 PiPi

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Lower Bound for General Problem a 1 …a t M i (a 1 …a t ): i £ t prefixes where P i sends the same message as (a 1 …a t ) i. q i,j (a 1 …a t ): Length of longest IS in column j ending at/before a j. a 1 …a t x 1,1 x 1,2 …x 1,t ………… x i,1 x i,2 …x i,t a1a1 a2a2 …atat ………… a1a1 a2a2 …atat

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M i (a 1 …a t ): i £ t prefixes where P i sends the same message as (a 1 …a t ) i. q i,j (a 1 …a t ): Length of longest IS in column j ending at/before a j.... q i,1 (a) q i,t (a) Lower Bound for General Problem a 1 …a t

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M i (a 1 …a t ): i £ t prefixes where P i sends the same message as (a 1 …a t ) i. q i,j (a 1 …a t ): Length of longest IS in column j ending at/before a j. Lower Bound for General Problem a 1 …a t... q i,1 (a) q i,t (a)

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Part I: By pigeonhole, find 1.A good player P i 2.A good set S µ [t] of columns 3.A good set I µ [m] t of (m/t) t inputs where... q i,1 (a) q i,t (a) Lower Bound for General Problem a 1 …a t

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Part II: Show that P i-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 General Problem a 1 …a t

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Part I: Messages sent by P i in round 2 and beyond depend on entire input. Need to change defn. of M i (a 1 …a t ). Lower Bound for Many Rounds a 1 …a t

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Part I: Messages sent by P i in round 2 and beyond depend on entire input. Need to change defn. of M i (a 1 …a t ). Part II: Reduce to 2-player protocol involving P i-1 and P t. Thm: Any deterministic O(1)-pass algorithm that gives a (1 + ) approximation to the LIS requires space (n/ ). Lower Bound for Many Rounds a 1 …a t

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

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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. A[i]: Smallest number ending an IS of length i. Patience Sorting: Dynamic program to compute A[i].

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Approximate Patience Sorting [GJKK] Track best inc. seq. of length i, for all i. A[i]: Smallest number ending an IS of length i. Patience Sorting: Dynamic program to compute A[i]. Approx. Patience Sorting: Store A[i] for at most n values of i.

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Lower Bounds for approximating the LIS Conjecture [GJKK]: For some ε 0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε 0 ) approximation to LIS( ) requires n) space. Candidate Hard Instances: P1P2…PtP1P2…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 No Yes P1P2…PtP1P2…Pt Does the direct sum property hold for this problem?

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