Optimality of Randomized Algorithms for the Intersection Problem

1. Optimality of Randomized Algorithms for the Intersection Problem Presenters : 李宜益 范家豪 王紹中 Advisor : 呂學一

2. Outline • Introduction • Definitions • Randomized Algorithm • Randomized Complexity Lower Bound

3. Introduction • Conjunctive query • Comparison model • Redundancy analysis • More natural assumptions • More precise

4. Definition (1) • Signature : (k, n1,…,nk) is a signature, where k is number of arrays, n1,…,nk are k sorted arrays U, U is a totally ordered space.

5. Example • This is a signature (7,1,4,4,4,4,4,4) A = 9 A : 9 B = 1 2 9 11 B : 1 2 9 11 3 12 13 C = 3 9 12 13 C : 9 9 14 15 16 D = 9 14 15 16 D : E = 4 10 17 18 E : 4 10 17 18 F = 5 6 7 10 F : 5 6 7 10 G = 8 10 19 20 G : 8 10 19 20

6. Definition(2) • Intersection The Intersection of an instance is the set A1… Akcomposed of the elements that are present in k distinct arrays

7. This is a signature (7,1,4,4,4,4,4,4) The intersection of this signature is 9 Example A = 9 A : 9 B = 1 2 9 11 B : 1 2 9 11 3 12 13 C = 3 9 12 13 C : 9 9 14 15 16 D = 9 14 15 16 D : E = 9 10 17 18 E : 9 10 17 18 F = 5 6 9 10 F : 5 6 9 10 G = 9 10 19 20 G : 9 10 19 20

8. Definition(3) • Partition-Certificate A partition- certificate is a partition (Ij) jδ (δis the minimal number of such partition of an instance) of U into intervals such that any singleton {x} corresponds to an element x of iAi, and each other interval I has an empty intersection I Ai with at least on array Ai

9. This is a signature (7,1,4,4,4,4,4,4) δ=3; (-, 9], [9, 10), [10, +) Example A = 9 A : 9 B = 1 2 9 11 B : 1 2 9 11 3 12 13 C = 3 9 12 13 C : 9 9 14 15 16 D = 9 14 15 16 D : E = 4 10 17 18 E : 4 10 17 18 F = 5 6 7 10 F : 5 6 7 10 G = 8 10 19 20 G : 8 10 19 20

10. Difficulty of Partition-Certificate • For each singleton {x} of the partition, any algorithm must find the position of x in all arrays Ai, which takes k searches

11. 9 9 1 2 9 11 9 A: B: C: D: E: F: G: 12 13 3 9 9 14 15 9 9 4 9 9 10 9 9 10 5 6 7 8 9 9 10

12. Difficulty of Partition-Certificate • For each interval Ij of the partition, any algorithm must find an array, or a set of arrays, such that the intersection of Ij with this array, or with the intersection of those arrays, is empty

13. 9 1 2 9 11 A: B: C: D: E: F: G: 12 13 3 9 10 14 15 9 4 9 10 9 10 10 5 6 7 8 9 10 10

14. Def(4) • Redundancy Let A1,…,Ak be k sorted arrays, and let (Ij)j be a partition-certificate for this instance • The redundancy (I) of an interval or singleton I is defined as equal to 1 if I is a singleton, and equal to 1/#{i, AiI = } otherwise.

15. Redundancy • The redundancy ((Ij)j) of a partition-certificate (Ij)j is the sum of j(Ij) the redundancies of the intervals composing it. • The redundancy ((Ij)jk) of an instance of the intersection problem is the minimal redundancy of a partition-certificate of the instance, min{((Ij)j), (Ij)j}

16. Unbounded search • looks for an element x in a sorted array A • unknown size • starting at position init • returns value p such that A[p-1]< x ≦A[p] the insertion point of x in A.

17. Unbounded search • It can be implemented using: • Doubling search • Binary search • Complexity 2

18. The algorithm for all i do pi 1 end for I ø ; s 1 repeat m As[ps] #NO 0; #YES 1; while #YES < k and #NO = 0 do Let As be a random array s.t. As[ps] ‡m. ps Unbounded Search (m,As, ps) if Ai[pi] ‡ m then #NO 1 else #YES #YES + 1 end if end while if #YES = k then II U{m} end if for all i such that Ai[pi] = m do pipi + 1 end for until m = +∞ return I

19. Theorem 1 • Thm 1: Algorithm rand intersection performs on average O(ρΣlog(ni/ρ)) comparisons on an instance of signature (k, n1,…,nk) and of redundancy ρ

20. Proof(1) • : #(binary searches) during phase i in array Aj • :#(binary searches) in array Aj over whole execution • 1 , if I is a singleton #{i, Ai∩ Ii= }, otherwise • If m  I then =1 • If m  I then is a random variable

21. Proof(2)

22. Proof(3)

23. Randomized Complexity Lower Bound • Yao’s Minimax Principle

24. Lemma 1 • For any k2, and 0<n1…nk, there is a distribution on instances of the Intersection problem with signature at most (k,n1,…,nk), and redundancy at most 4, such that any deterministic algorithm performs at least comparisons on average

25. At most  = 4, and output size at most 1 P N A1 Aw Ak

26. Input distribution • Let Fi = log2 (2ni + 1) • F = •  i each pi is chosen uniformly at random in {1,…,ni}. • An index w ∈ {2,…,k} equal to i with probability .

27. Case P: Aw[pw] = A1[1] • The redundancy of such instances is no more than 4.

28. Case N: Aw[pw] > A1[1] • The redundancy of such instances is no more than 4.

29. x-comps vs. comps between any element & x • Any algorithm performing C comparisons between arbitrary elements can be expressed as an algorithm performing no more than 2C x-comparisons.

30. x-comps vs. comps between any element & x • Any lower bound L on the complexity of algorithms using only x-comparisons is a L/2 lower bound on the complexity of algorithm using comparisons between arbitrary elements

31. Random variable Xi • Let Xi The number of x-comparisons performed by algorithms in array Ai for both P or N

32. Random variable Yi • Let Yi The number of x-comparisons performed by algorithms in array Ai for N

33. ζi • Let ζi be the indicator variable which equals 1 exactly if pi has been determined by algorithm on instance P.

34. C = • C   • E (Yiζi) = Σh Pr {Yiζi  h} • E (Yiζi)  Pr {Yiζi  h}

35. Pr {a ∨ b}  Pr {a} + Pr {b} • Pr {Yiζi h} = Pr {Yi h ∧ζi = 1} = 1 – Pr {Yi < h ∨ζi = 0}  1 - Pr {Yi < h} – Pr {ζi = 0} = Pr {ζi = 1} - Pr {Yi < h}

36. Pr {Yi < h}  (2h) / (2ni + 1)  (2h-Fi) • Pr {Yi < h} = (all of the positions could be investigated after h times of binary searches (Searchi(h))) / (all of the positions the x could be present in Ai list (Presenti))

37. Searchi (h) • Searchi (h)  2h

38. Presenti • Presenti = 2ni + 1

39. E(C)  • E(C)  E(Yiζi )  Fi Pr{ζi = 1} - 2(1 – 2-Fi)  Fi Pr{ζi = 1} + 2 2-Fi– 2 (k - 2)

40. Pr{ζi = 1 | p} =j:j  i Fj/F • Let’s fix p = (p2,…,pk). There are only k – 1 possible choices for w. Algorithm A can only differentiate between P and N when it finds w. Let  denote the order in which these instances are dealt with by A for p fixed. Thenζi = 1 iff i  w .

41. Pr{ζi = 1} • Pr{ζi = 1} = Pr{{ζi = 1 | p} Pr{p}} = Pr{p} • FiPr{ζi = 1} = Pr{p} = Pr{p}

42. ( Fi)2 • ( Fi)2 = 2 FiFj - Fi2, • FiFj =

43. FiPr{ζi = 1} = ( Fi) / 2

44. E(C)  • Lemma 1: proved

45. Lemma 2 • For any k ≧ 2, 0 < n1≦ . . . ≦ nk and ρ  {4, . . . ,4n1}, there is a distribution on instances of the Intersection problem of signature at most (k,n1, . . . ,nk), and redundancy at most ρ, such that any deterministic algorithm performs on average Ω(ρ log(ni/ρ)) comparisons.

46. p sub-instances • For p= • p sub-instances,(Pj,Nj)j {1,,,,p} ,of signature (k, ,…, )from the distribution of lemma 1 • ρ≦4n1 ,p ≦ n1 and > 0 • All the arrays are positive

47. Random choosing • Let’s choose uniformly at random each sub-instance Ij between the positive sub-instance Pj and the negative sub-instance Nj. • They form a larger instance I by unifying the arrays of same index from each sub-instance. • The elements from two different sub-instances never interleave.

48. p elementary instances unified to form a single large instance

49. United Instances • Redundancy at most 4p≦ρ • Singnature at most (k,n1,…,nk) • Solving this instance implies to solve all the p sub-instances. • From Lemma 1

50. p sub problems • A lower bound of • which is