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Longest Increasing Subsequence and Distance to Monotonicity in Data Stream Model Hossein Jowhari Simon Fraser University Joint work with Funda Ergun Dagstuhl.

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Presentation on theme: "Longest Increasing Subsequence and Distance to Monotonicity in Data Stream Model Hossein Jowhari Simon Fraser University Joint work with Funda Ergun Dagstuhl."— Presentation transcript:

1 Longest Increasing Subsequence and Distance to Monotonicity in Data Stream Model Hossein Jowhari Simon Fraser University Joint work with Funda Ergun Dagstuhl August 2008

2 Problem Definitions Longest Increasing Subsequence (LIS)  LIS(A)= length of longest increasing subsequence of sequence A A = 5,3,0,7,10,8,2,13,15,9,2,20,2,3. LIS(A)=6  (ε-LIS) Approximate LIS(A) within 1- ε factor for ε<1 Distance to Monotonicity  DM(A)= minimum number of elements needed to be deleted from A to get a sorted sequence = |A|-LIS(A)  Approximate the length of DM(A) within 1+ ε factor for ε>0

3 LIS problem in data stream model An exact algorithm in space O(|LIS|). [LVZ05] Every exact algorithm needs Ω(n) space. [GJKK07],[WS07] There is a O(ε -1/2 n 1/2 ) space deterministic streaming algorithm [GJKK07] [GJKK07] Conjecture: Every deterministic streaming algorithm for ε-LIS needs Ω(n 1/2 ) space. This Talk: A proof for the above conjecture (constant ε ). A lower bound of Ω (ε -1/2 n 1/2 ) was discovered independently by Gal and Gopalan (FOCS 07). (Next talk!)

4 Distance to Monotonicity in Data Stream Model Complexity of exact computations is the same as the LIS computation. A deterministic (1+ε) approximation algorithm using O(ε -1/2 n 1/2 ) space [GJKK07] A randomized (4+ε) approximation algorithm using O(ε -2 log 2 n) space [GJKK07] Our Result: There is a deterministic (2+ε) approximation algorithm which uses O(ε -2 log 2 n) space. (A brief description in this talk)

5 Space Lower Bound for Approximating LIS an Algorithm for Approximating Distance to Monotonicity

6 Communication Complexity of ε-LIS [GJKK07]  There is an O(ε -1 logn) deterministic protocol for 2 Players. 2-player model does not help.  There is an extension of the protocol to √n - players, where each player sends O(√n) bits. Let’s consider O(√n) player setting.

7 Lower Bound using multiplayer communication complexity (general idea) We split the input equally among √n players. The players compute a function g which is reducible to ε-LIS. We decompose g into primitive functions h i with high communication complexity Finally using a direct-sum approach, we show a lower bound total communication complexity of g

8 √n Player Framework g: A  {0,1} Boolean function h defined over rows g(A) = h(R 1 ) V h(R 2 ) V … V h(R √n ) √n Rows √n players

9 Description of the primitive function h h(R) = 0 if no consecutive nonzero elements in R. 1 if there are at least β√n nonzero elements in R. (β > 0.5) g(A) = h(R 1 ) V h(R 2 ) V … V h(R √n )

10 Going from g to LIS √n × √n matrix Numbers are increasing in each column (upward) and each row (leftward)

11 A 0 no consecutive 1’s in a row. g(A 0 ) = 0 LIS(A 0 ) ≤ 3/2 √n A 1 There is one row with β√n number of 1’s g(A 1 )=1 LIS(A 1 ) ≥ (1+β) √n Description of function g in terms of binary matrices

12 Description of the fooling set for h A is a k-Fooling set for h if A is a collection of subsets of {1,..,√n}. For all u in A, no consecutive member of [√n] appear in u. The union of every k members of A has size at least β√n. Using the probabilistic method we can prove that there exists a fooling set for h of size c √n where c>1 If A is a k-fooling set for f then CC tot (f) ≥ log (|A|/k-1)

13 Let F 1, F 2, …, F √n be the fooling sets for h. F 1 × F 2 × … × F √n is a k √n -fooling set for g Each F i has size c √n CC tot (g) ≥ log c n /k √n = Ω (n) CC max (g) = Ω(√n) Fooling set for g CC max (ε-LIS) = Ω(√n)

14 Space Lower Bound for Approximating LIS an Algorithm for Approximating Distance to Monotonicity

15 An approximate characterization based on inversion High level idea: We detect a set of elements that highly violate the monotonicity of the sequence. (bad elements) These elements form a set of disjoint decreasing subsequences (lower bound) Deleting twice the number of these elements from the sequence results in a sorted sequence (upper bound)

16 An approximate characterization based on inversion σ(i) is red if there is a interval I=[j,i-1] such that number of inversions in I (with respect to σ(i)) is bigger than number of red elements in that interval Definition of a bad (red) element.

17 Number of red elements is smaller than DM(σ) |R| <= DM(σ) (Idea) We can decompose the red elements into disjoint decreasing subsequences of σ.

18 |R| > ½ DM(σ) We delete at most 2|R| elements and we get a sorted sequence

19 A streaming friendly characterization  σ(i) is red if most of the elements in the interval are inverted with respect to σ(i) and red elements in the interval are far from being the majority.  This gives a 2+ε approximation  We can do this using existing deterministic algorithms for quantile approximation in all intervals [LLXY, ICDE04].

20 Open Questions Is randomness useful in approximating LIS? Is there a polylog approximation scheme for distance to monotonicity?


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