Upper Bounds for Maximally Greedy Binary Search Trees

Transcription

1 Upper Bounds for Maximally Greedy Binary Search Trees Kyle Fox -- University of Illinois at Urbana-Champaign

2 Balanced Binary Search Trees Want data structures for dictionary, predecessor search, etc. Red-black trees and AVL trees do have O(log n) insert and delete, but

3 Balanced Binary Search Trees Want data structures for dictionary, predecessor search, etc. Red-black trees and AVL trees do have O(log n) insert and delete, but 1,,n, 1, 1, 1,

4 Splay Trees Splay trees have O(log n) amortized performance, and access commonly searched nodes far more quickly [Sleator, Tarjan 85] They are - balanced - statically optimal - statically fingerful - working set w x z w x z w x z x y z x y z x y z

5 Splay Trees

6 Splay Trees They are - dynamically optimal? Conjecture: For a given search sequence, splay trees access the smallest number of nodes possible, within a constant factor. They are

7 Splay Trees They are - dynamically optimal? Conjecture: For a given search sequence, splay trees access the smallest number of nodes possible, within a constant factor. They are - complicated

8 Splay Trees They are - dynamically optimal? Conjecture: For a given search sequence, splay trees access the smallest number of nodes possible, within a constant factor. They are - complicated - Proof of dynamic finger theorem spread over 80 pages and two articles [Cole, Mishra, Schmidt, Siegel 00; Cole 00]

9 GreedyFuture [Lucas 88; Munro 00] An offline candidate for dynamic optimality - Appears to be optimal within additive term for # node accesses Easier to analyze? If dynamically optimal, then there exists an online dynamically optimal algorithm! [Demaine, Harmon, Iacono, Kane, Pătraşcu 09]

10 GreedyFuture Only touches nodes on the search path Picks new root (and right child) to place next search as high as possible Places lessor (greater) nodes on search path using lessor (greater) upcoming searches We need an example

11 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

12 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

13 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

14 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

15 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

16 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

17 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

18 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

19 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

20 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

21 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

22 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

23 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

24 GreedyFuture Search sequence 2, 3, 5, 4, 7, 1,

25 Basic Questions

26 Basic Questions Nothing written about GREEDYFUTURE s performance Do searches take O(log n) amortized?

27 Basic Questions Nothing written about GREEDYFUTURE s performance Do searches take O(log n) amortized? Yes!

28 Main Results Balance Theorem: GREEDYFUTURE accesses amortized O(log n) nodes per search. Static Optimality Theorem: Let t(x) be the number of times x appears in a given search sequence of length m. GREEDYFUTURE accesses amortized log (m / t(x)) nodes per search. Static Finger Theorem: Fix some node f. GREEDYFUTURE accesses amortized log ( si - f + 1) nodes when searching for si. ( si - f is the difference in ranks.) Working Set Theorem: Let si be the ith search node and d(i) be the number of distinct nodes searched since the last search of si. GREEDYFUTURE accesses amortized log (d(i) + 1) nodes during search i.

29 Main Results Sequential Access Theorem: Starting from any arbitrary binary search tree, the number of nodes accessed by GREEDYFUTURE when searching for each node once in order is O(n).

30 Main Strategy First four theorems all rely on an access lemma as in [ST 85] But first, some formalism and a different view of the world

31 Formalism Only consider searches, no insertions or deletions n: number of nodes in tree m: number of searches Nodes numbered {1,, n} Must execute search sequence S = s1,, sm

32 More Formalism In each search access nodes in a subtree containing the root The ith subtree must contain si from the search sequence S = s1,, sm Can rearrange/perform rotations in accessed subtree for free Cost of a search is the number of nodes accessed

33 A Geometric Model For the access lemma, we focus entirely on a geometric view of BSTs [DHIKP 09] The model uses points in 2D. Each point p has integer coordinates (p.x, p.y) with 1 p.x n and 1 p.y m

34 Arborral Satisfaction A pair of points a and b in point set P is arborally satisfied if they lie on a horizontal or vertical line or if a point in P \ {a, b} lies in the closed axisaligned rectangle with corners a and b A set of points P is arborally satisfied if all pairs of points are arborally satisfied Unsatisfied Satisfied

35 Searches and Accesses There is a one-to-one correspondence between BST executions and arborally satisfied point sets The geometric view of BST execution E is P(E) = {(x, y) x is accessed during search y} The geometric view of BST search sequence S is P(S) = {(s1, 1), (s2, 2),, (sm, m)} An optimal BST algorithm for S is a minimum cardinality superset of P(S)

36 Online Arborally Satisfied Superset P(S) is revealed to us row by row (si, i) appears at time i. We must place points Pi on line y = i so that {(s1, 1), (s2, 2),, (si, i)} P1 P2 Pi is arborally satisfied Cost is total number of points in final superset ( + = ) Example

37 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

38 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

39 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

40 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

41 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

42 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

43 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

44 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

45 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

46 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

47 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

48 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

49 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

50 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

51 Online Arborally Satisfied Superset Search sequence 2, 3, 5, 4, 7, 1, 6

52 GreedyASS Just saw GREEDYFUTURE as the arborally satisfied superset algorithm GREEDYASS During search i, GREEDYASS places the minimum set of points needed to arborally satisfy {(s1, 1), (s2, 2),, (si, i)} P1 P2 Pi GREEDYASS is an online algorithm - Which implies an online BST algorithm competitive with GREEDYFUTURE [DHIKP 09]

53 Toward an Access Lemma The access lemma directly describes the behavior of GREEDYASS Fix a search sequence S and let X = P(S) Consider the execution of GREEDYASS on X

54 Neighborhoods ρ(x,i) is the last access of x at or before search i (highest point on ray from (x, i) to (x, - )) Left neighborhood of x at time i is Γl(x, i), the set of values {a+1, a+2,, x-1} where a is the smallest pos. integer < x such that ρ(a,i) ρ(x,i) Right neighborhood is Γr(x, i) Inclusive neighborhood is Γl(x, i) Γr(x, i) {x} Example

55 Neighborhoods Left neighborhood of x at time i is Γl(x, i), the set of values {a+1, a+2,, x-1} where a is the smallest pos. integer < x such that ρ(a,i) ρ(x,i) Right neighborhood is Γr(x, i) Inclusive neighborhood is Γl(x, i) Γr(x, i) {x} neighborhood at time 7 neighborhood at time 6

56 Intuition Behind Neighborhoods Intuitively, the nodes in x s neighborhood are in x s subtree in the BST model Formally, the inclusive neighborhood of x at time i contains each node e that will force an access of x if e is searched at time i+1

57 Weights and Potentials Assign each x {1,, n} a positive real weight w(x) Size of x at time i is σ(x, i), the sum of weights in x s inclusive neighborhood (σ(x, i) = ) (, ) ( ) Rank of x at time i is r(x, i) = lg σ(x, i) Potential function ( ) = [ ] (, )

58 The Access Lemma Amortized cost of search i is + + ( ) ( ) Access Lemma: Let = [ ] ( ). The amortized cost of a search at time i is at most + lg (, ) Plugging in appropriate weights implies main theorems [ST 85]

59 Main Proof Ideas The neighborhood of a node does not change if a node is not accessed - Can ignore these nodes when we examine the change in the potential function

60 Main Proof Ideas Upper bounding changes in accessed nodes ranks yields a telescoping sum that cancels out the cost of most accesses Essentially, the sizes for the remaining nodes we must count doubles as we examine them in order away from si x2 x2 x2

61 Sequential Access Theorem From any arbitrary BST, accessing nodes in order takes linear time Proven directly on BSTs Each node can only be accessed once as anything other than the root or root s right child

62 Future Directions Is GREEDYFUTURE/GREEDYASS dynamically optimal? Additional weaker results? - Dynamic Finger Theorem in fewer than 80 pages? - Iacono s unified bound [ 01]? Modify GREEDYFUTURE to support insertions/deletions?

63 Thank you!