Van Emde Boas Trees

1. Van Emde Boas Trees Elif Tosun (252a-aa) CSC 252 – Final Project

2. Outline • Introduction • “Silly” Priority Queue • Stratified Tree Structure • Operations on the Stratified Tree Structure • Conclusions • My Contribution • References

3. Introduction • Main problem in operations for Set Manipulation • Random Access (insert/delete/membership) • Ordered Representation (predecessor/successor) • Some data structures: binary heap, AVL trees, 2-3 trees, etc. • Lower Bounds for a Restricted Universe • Motivation: Enumeration Algorithms

4. “Silly” Priority Queue • Structure • n = 2k leaves, height = k • Leaves with priorities from {1, 2,…, n} • k-bit representation (root to leaf) • 0, for left • 1, for right

5. “Silly” PQ – Structure (Cont’d) • Subset S  {1,..,n} Representation • Mark leaves in S and all nodes on the paths from root to the marked leaves.

6. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

7. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

8. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

9. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

10. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

11. “Silly” PQ – (Cont’d)Sketches of Algorithms • Insert(i) • Delete(i) • Member(i) • Min • Predecessor

12. Improvements? • Divide and Conquer Strategy. • Start with a tree of height 2k, n = 2^(2k) • Divide the universe {1, …, n} into n^(1/2) blocks of size n^(1/2). Insert/Delete locally • Run time recurrence: T(n) = T(n^(1/2)) + 1 • Result : T(n) = O(loglogn)

13. Stratified Tree Structure • Based on Divide and Conquer Approach mentioned before. • Two types of information available for each node: • Static Information • Dynamic Information

14. Static Information • Level: Length of path from a node to its leaves • Rank: x, for greatest divisor 2^x of the level of a node. For leaves, rank = k+1 where height= 2^k • Position: • Leaves: number in {1, …, n} representing the leaf • Internal nodes:position of rightmost leaf of its lefthand son. • Canonical Subtree of rank d (CS): Binary subtree having as a root node of rank >= d, as leaves descendants at the nearest level of rank >=d.

15. Static Information(Cont’d) • Left Canonical Subtree(LCS): Includes all left hand side descendants • Upper Canonical Subtree(UC): Subtree of rank d, having the node as a leaf. • Lower Canonical Subtree(LC): Subtree of rank d, having the node as a root. • Reach: LC+UC

16. Static Information Visualized

17. Dynamic Information • Internal Nodes: • lmin: leftmost present leaf in LLC(v) • lmax: rightmost present leaf in LLC(v) • rmin: leftmost present leaf in RLC(v) • rmax: rightmost present leaf in RLC(v) • ub: +, if there exists a branchpoint on the path between the top of LC and v. -, otherwise

18. Some definitions… • Branch Point: An internal node with 2 present sons • Active Node: A present node that has at least one dynamic field that doesn’t equal null. Note: For some operations to be done in O(1) nodes are prevented from being active unless needed.

19. Properness Condition Let v be a present internal node. V is active there exists a branchpoint in the interior of the reach of v. Note: a leaf is active it’s present root is active set  null

20. Representation of S • Leaves corresponding to elements of S and their ancestors are marked to be present. • Information on which present nodes are active is computed. – using Properness Condition • Dynamic fields of all active and non-active nodes are given their proper values.

21. An Example:

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30. Operations on the Stratified Tree • Insert:Let l to be the leaf to be inserted. If (no present leaves exists on l’s side of the tree) insert as before Else compute the fathers of l(=hl) and the present leaf p (=hp). if no branchpoint exists on l’s side of CS hp is present without being active. hl is present iff hp = hl

31. Operations- Insert (Cont’d) if hl is present insert l under hp = hl if hl is not present insert l as a leaf to a node above hp.

32. Operations – Insert (Cont’d) If a branch point does exist on l’s side of CS hl is present iff it’s active if hl is active insert leaf as a child to hl if hl is not active insert leaf as a child to a node above hp

33. Operations on the Stratified Tree • Delete: Makes a leaf non-present as well as its ancestors up to the lowest branch point. • Compute hl • Proceed to the upper branch point of hl • Consider the other son of the branchpoint • Change path from that son to the root of tree so that hl is not included • Mark as non-present the leaf and the path above until the “former” upper brachpoint of hl

34. Operations – Delete(Visualized)

35. Summary • What is important in these operations is that when inserting or deleting you only deal with the canonical subtree of the father of a leaf. • Basically the operations are done on the subtree of height k, which affects the whole tree of height 2^k, without touching the rest of the tree. And that’s why the time complexity of operations is O(loglogn) instead of O(logn).

36. Conclusions • Comparisons

37. Conclusions(Cont’d) • Advantages • Efficient PQ to work with VERY large amounts of data • Disadvantages • VERY difficult to implement • Can’t be used for Real-Valued Priority Queues. • Space – uses a full tree just for leaves

38. My Contributions • Research on Subject • Web • Periodicals • Books • LEDA • Reading Papers • UNDERSTANDING the Data Structure

39. References • P.van Emde Boas, R.Kaas, E.Zulstra. Design and Implementation of an Efficient Priority Queue. Mathematical Systems Theory. 1977:vol10, pgs 99-127 • P.van Emde Boas. An O(nloglogn) Online Algorithm for the Insert/Extract Min Problem. 1974 • http://www.cs.mcgill.ca/~mwozni/cs251/welcome.html • LEDA • eb_tree.h • _eb_tree.c