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

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. Storage 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, • Woodruff-Sun,G.-Jayram-Kumar-Sivakumar] • Measuring Sortedness: • Length of Longest Increasing Subsequence. • Ulam/Edit distance • Inversion/Kendall Tau distance

4. Longest Increasing Subsequence LIS(): Length of Longest Increasing Subsequence. 5 7 8 1 4 2 10 3 6 9

5. Longest Increasing Subsequence LIS(): Length of Longest Increasing Subsequence. 5 7 8142103 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-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.

7. 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.

8. 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.

9. 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.

10. A Communication Problem [GJKK]: Consider the following decision problem – Yes No P1 P2 … Pt

11. A Communication Problem [GJKK]: Consider the following decision problem – Yes No P1 P2 … Pt All columns non-increasing

12. A Communication Problem [GJKK]: Consider the following decision problem – Yes No P1 P2 … Pt All columns non-increasing

13. A Communication Problem [GJKK]: Consider the following decision problem – Yes No P1 P2 … Pt All columns non-increasing Some column increasing

14. A Communication Problem [GJKK]: Consider the following decision problem – Yes No P1 P2 … Pt All columns non-increasing Some column increasing

15. Direct Sum Paradigm Primitive Problem: p(x1, y1) y1 x1

16. 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…]

17. Primitive Problem Yes No P1 P2 … Pt

18. Direct Sum of Primitive Problems Yes No P1 P2 … Pt All No instances

19. Direct Sum of Primitive Problems Yes No P1 P2 … Pt All No instances One Yes instance

20. Direct Sum of Primitive Problems Yes No P1 P2 … Pt

21. [GG] An Easier Problem Yes No Hope: Some player distinguishes between many No instances.

22. BlackBoard Model of One-Way Communication • Players speak in order. • Every message seen by all. • Last player outputs answer.

23. Problem is Easy in the BlackBoard model No Yes BlackBoard protocol with max. communication 2 log(m).

24. Problem is Easy in the BlackBoard model No Yes BlackBoard protocol with max. communication 2 log(m).

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

26. 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.

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

28. 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).

29. 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.

30. 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

31. 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

32. 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.

33. 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.

34. 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.

35. 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)

36. 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)

37. 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)

38. 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).

39. 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/).

40. 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.

41. 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!