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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 August 2008

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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

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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!)

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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)

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

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Communication Complexity of ε-LIS [GJKK07] 1096702153 48783 19 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.

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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

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√n Player Framework 1 7 13 3 5 31 8 7 6 4 26 1 3 9 15 10 9 33 18 4 22 28 3 5 11 9 56 42 0 45 33 18 24 27 10 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

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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 )

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Going from g to LIS 130300052 02802900 00025031 015220027 120200210 3080100 √n × √n matrix 101001 010100 000101 011001 101010 101010 Numbers are increasing in each column (upward) and each row (leftward)

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1010001 1001001 1001001 1010101 1010001 1001001 1000101 1010001 1001001 1111111 1010001 1010101 1000001 1010101 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

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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)

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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)

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

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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)

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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. 715670289345670 Definition of a bad (red) element.

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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.

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|R| > ½ DM(σ) We delete at most 2|R| elements and we get a sorted sequence. 438956

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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].

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

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