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Estimating the Sortedness of a Data Stream

Estimating the Sortedness of a Data Stream. Storage. Data Stream Model of Computation. X 1 X 2 X 3 … X n. Input. Computing with Massive data sets. Sequential access. Small storage space, update time. [Alon-Matias-Szegedy, …]. Sorting on Data-Streams. Cannot sort efficiently.

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Estimating the Sortedness of a Data Stream

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  1. Estimating the Sortedness of a Data Stream

  2. Storage Data Stream Model of Computation X1 X2 X3 … Xn Input • Computing with Massive data sets. • Sequential access. • Small storage space, update time. [Alon-Matias-Szegedy, …]

  3. Sorting on Data-Streams Cannot sort efficiently. Can we tell if the data needs to be sorted? • [Ergun-Kannan-Kumar-Rubinfeld-Vishwanathan, • Ajtai-Jayram-Kumar-Sivakumar, Gupta-Zane, • Cormode-Muthukrishnan-Sahinalp, LibenNowell-Vee-Zhu, • Ailon-Chazelle-Commandur-Liu]

  4. Sorting on Data-Streams • Cannot sort efficiently on a data-stream. • Can we tell if the data needs to be sorted? • [Ergun-Kannan-Kumar-Rubinfeld-Vishwanathan, • Ajtai-Jayram-Kumar-Sivakumar, Gupta-Zane, • Cormode-Muthukrishnan-Sahinalp, LibenNowell-Vee-Zhu, • Ailon-Chazelle-Commandur-Liu] • Measuring distance from Sortedness: • Kendall Tau distance • Spearman Footrule distance • Ulam distance

  5. Candidate metrics 1.Spearman’s footrule [ℓ1 distance] :  3 5 7 9 10 4 1 2 6 8 e 1 2 3 4 5 6 7 8 9 10 Easy to compute.

  6. Candidate metrics 2.Kendall Tau distance [No. of Inversions] Inversions: Positions i < j where (i) > (j)  3 5 7 9 10 4 1 2 6 8

  7. Candidate metrics 2.Kendall Tau distance [No. of Inversions] Inversions: Positions i < j where (i) > (j)  3 5 7 9 10 4 1 2 6 8

  8. Candidate metrics 2.Kendall Tau distance [No. of Inversions] Within a factor-2 of Spearman’s footrule. [Diaconis-Graham] An O(log n) space, 1-pass (1 + ) algorithm. [Ajtai-Jayram-Kumar-Sivakumar]

  9. Candidate metrics 3. Ulam distance [Edit Distance]: Ed(): Number of deletions needed to sort. Ulam: Fastest way to sort a bridge hand.

  10. Edit Distance and the LIS Ed(): Number of deletions needed to sort. 5 7 8 1 10 4 2 3 6 9

  11. Edit Distance and the LIS Ed(): Number of deletions needed to sort. 5 7 8 1 10 4 2 3 6 9 Delete 5 7 8 10 Insert 1 2 3 4 5 6 7 8 9 10

  12. Edit Distance and the LIS Ed() : Number of deletions needed to sort . LIS() : Length of the longest increasing sequence. Ed() + LIS() = n • Studied in statistics, biology, computer science … • Bothtake a global view of the sequence. • Hard for models like streaming, sketching, property-testing. 151 … 190 51 … 80 81 … 100

  13. Prior Work • Exact Computation of Ed() and LIS() : • Patience Sorting [Ross,Mallows]

  14. Patience Sorting 5 7 8 1 10 4 2 3 6 9 5 7 8 1 10 4 2 3 6 9 0 5 7 8

  15. Patience Sorting 5 7 8 1 10 4 2 3 6 9 5 7 8 1 10 4 2 3 6 9 0 1 5 7 8 10

  16. Patience Sorting 5 7 8 1 10 4 2 3 6 9 5 7 8 1 10 4 2 3 6 9 0 1 4 5 7 8 10

  17. Patience Sorting 5 7 8 1 10 4 2 3 6 9 5 7 8 1 10 4 2 3 6 9 0 2 1 4 5 7 8 10 Number in place i: Earliest end to IS of length i.

  18. Patience Sorting 5 7 8 1 10 4 2 3 6 9 5 7 8 1 10 4 2 3 6 9 0 2 1 3 4 5 7 8 10 Number in place i: Earliest end to IS of length i.

  19. Patience Sorting 5 7 8 1 10 4 2 3 6 9 0 2 1 3 4 6 5 7 8 10 9 Number in place i: Earliest end to IS of length i.

  20. Patience Sorting 5 7 8 1 10 4 2 3 6 9 0 2 0 3 1 4 6 5 7 8 10 9 Number in place i: Earliest end to IS of length i.

  21. Patience Sorting 5 7 8 1 10 4 2 3 6 9 0 LIS 2 0 3 1 4 6 5 7 8 10 9 Length of LIS

  22. Prior Work • Exact Computation of Ed() and LIS() : • Patience Sorting [Ross,Mallows] • O(n) space, 1-pass streaming algorithm. • (√n) space lower bound. [LibenNowell-Vee-Zhu] • Approximating Ed() and LIS() : • No sub-linear space algorithms, no lower bounds. [Ajtai et al, Cormode et al, LibenNowell et al] • LIS Algorithms parametrized by length of LIS : [LibenNowell-Vee-Zhu, Sun-Woodruff] • Computing Ed() in other models: • Property Testing[Ergun et al, Ailon et al] • Sketching[Charikar-Krauthgamer]

  23. Our Results • Approximating Ed() : • An O(log2 n) space, randomized 4-approximation for Ed(). • A O(√n) space, deterministic (1 + ε)-approximation for Ed(). • Approximating the LIS: • A O(√n) space, deterministic (1 + ε)-approximation for LIS(). • Exact Computation of Ed() and LIS(): • An (n) space lower bound for randomized algorithms. • Independently proved by [Sun-Woodruff]. • Lower bounds for approximating the LIS: • Conjecture: Deterministic algorithms require(√n) space for (1 + ε)-approximation

  24. Computing the Edit Distance Thm: For any ε > 0,there is a one-pass randomized algorithm using O(ε-2log2 n) space and update time, that gives a (4 + ε) approximation toEd(). • Combinatorial measure that approximates Ulam distance. Builds on [Ergun et al, Ailon et al]. • Sampling scheme to compute this measure in one pass.

  25. 1 3 7 8 6 5 9 2 A Voting Scheme [Ergun et al.] Combinatorial measure called Unpopularity. Neighborhoods of (i): Intervals starting or ending at i.

  26. A Voting Scheme [Ergun et al.] Combinatorial measure called Unpopularity. Neighborhoods of (i): Intervals starting or ending at i. • Deciding if (i) is unpopular: • For every neighborhood of (i) • Every number in the neighborhood votes on “Is (i) out of order?” • If majorityinsome neighborhood vote against (i), it is marked unpopular. Let U() denote no. of unpopular numbers. [Ergun et al]: Ed() ≤ U() [Ailon et al]: U() ≤ 2 Ed()

  27. A Voting Scheme [Ergun et al.] Can we estimate U() using a streaming algorithm? 4 5 3 71 2

  28. A Voting Scheme [Ergun et al.] Can we estimate U() using a streaming algorithm? 4 5 3 7 1 2 Impossible to decide if (i)is unpopular before seeing the entire input.

  29. A New Voting Scheme • Neighborhoods of (i): Intervals ending at i. • If majority in some neighborhood vote against (i), it is marked unpopular. • Unpopularity based only on past, not the future. Thm: Let V() denote no. of unpopular numbers. Then Ed()/2 ≤V() ≤ 2 Ed()

  30. A Voting Scheme • Let Ed() = k. Then V() ≤ 2k. • Fix an optimal Bad set of size k to delete. How many numbers can be Unpopular ? Partition Unpopular into Good and Bad. Good numbers form an increasing sequence. Good never votes against Good. Good +Unpopular≡Bad neighborhood !

  31. A Voting Scheme • Let Ed() = k. Then V() ≤ 2k. • Fix an optimal Bad set of size k to delete. Good +Unpopular≡Bad neighborhood ! If k numbers are Bad, At most k are Good + Unpopular. Bad numbers might all be Unpopular. Hence V() ≤ 2k.

  32. A Voting Scheme • Let Ed() = k. Then V() ≤ 2k. • Bound can be tight. 100 99 98 … 91 1 2 3 … 10 11 12 … 90 100 99 98 … 91 1 2 3 … 10 11 12 … 90 100 99 98 … 91 1 2 3 … 10 11 12 … 90

  33. A Voting Scheme • Let V() = k. Then Ed() ≤ 2k. • Fix the set of k Unpopular elements. • Algorithm to produce an increasing sequence: • Scan right to left. • Delete Unpopular elements + Inversions w.r.t last number in sequence. • At least half of deletions are Unpopular numbers. • What remains is an increasing sequence.

  34. A Voting Scheme • Let V() = k. Then Ed() ≤ 2k. • Bound can be tight. 11 … 50 91 92 93 … 100 1 2 3 … 10 51 … 90 11 … 50 91 92 93 … 100 1 2 3 … 10 51 … 90 11 … 50 91 92 93 … 100 1 2 3 … 10 51 … 90

  35. A New Voting Scheme • Neighborhoods of (i): Intervals ending at i. • If majority in some neighborhood vote against (i), it is marked unpopular. • Unpopularity based only on past, not the future. Thm: Let V() denote no. of unpopular numbers. Then Ed()/2 ≤V() ≤ 2 Ed() Can we estimate V() efficiently?

  36. Outline of Sampling Scheme Taking a vote in one neighborhood: • Take O(log n) samples, take the (approx) majority. Reservoir Sampling [Vitter]. 1 3 7 8 6 5 9 2 Computing V() : Need O(log n) samples from every neighborhood. 1 3 7 8 6 5 9 2

  37. 1 3 7 8 6 5 9 2 Outline of Sampling Scheme Computing V() : Need O(log n) samples from every neighborhood. Key observation: Don’t need samples across intervals to be independent! Roughly O(log2 n) samples suffice.

  38. Deterministic Algorithm for LIS Thm: For any ε > 0,there is a one-pass deterministic algorithm using O(n/ε)1/2 space and update time, that gives a (1 - ε) approximation toLIS(). Based on multiplayer communication protocol for LIS: 10 51 … 19 32 … 80 15 … 50 • Algorithm simulates protocol for √n players.

  39. Two-Player Protocol for LIS 1000 5123 … 1319 3245 4582 … 8021 n/2 Patience Sorting 6 24 … 1000 k Multiples of εk 6…1000 1/ε

  40. Approximating the LIS Consider k-player communication protocol for LIS: 10 51 … 19 32 … 80 15 … 50 • As k increases, maximum message size increases. Conjecture: For some ε0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε0) approximation toLIS() requires (√n) space. Proving the conjecture requires analyzing k ≥ √n

  41. Lower Bounds for approximating the LIS Conjecture: For some ε0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε0) approximation toLIS() requires (√n) space. Candidate Hard Instances?

  42. Lower Bounds for approximating the LIS Conjecture: For some ε0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε0) approximation toLIS() requires (√n) space. Candidate Hard Instances? Yes No

  43. Lower Bounds for approximating the LIS Conjecture: For some ε0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε0) approximation toLIS() requires (√n) space. Candidate Hard Instances? Yes No

  44. Lower Bounds for approximating the LIS Conjecture: For some ε0 > 0, every 1-pass deterministic algorithm that gives a (1 + ε0) approximation toLIS() requires (√n) space. Candidate Hard Instances? Yes No

  45. Open Problems • Estimate the Edit distance between two permutations. • Tight bounds for approximation: • Show (√n) lower bound for deterministic algorithms. • Randomized algorithm for LIS ? Thank You!

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