1. Uniquely Represented Data Structures Advanced Algorithms & Data Structures Lecture Theme 10 Prof. Dr. Th. Ottmann Summer Semester 2006

2. 2 Motivation A structure storing a set of keys is called oblivious, if it is not possible to infer its generation history from its current shape. A structure is called persistent, if it supports access to multiple versions. Partially persistent: All versions can be accessed but only the newest version can be modified. Fully persistent: All versions can be accessed and modified. Confluently persistent: Two or more old versions can be combined into one new version.

3. 3 Methods for making structures oblivious Unique representation of the structure: Set/size uniqueness: For each set of n keys there is exactly one structure which can store such a set. The storage is order unique, i.e. the nodes of the strucure are ordered and the keys are stored in ascending order in nodes with ascending numbers. Randomise the structure: Assure that the expectation for the occurrence of a structure storing a set M of keys is independent of the way how M was generated. Observation: The address-assingment of pointers has to be subject under a randomised regime!

4. 4 Example of a randomised structure Z-stratified search tree On each stratum, randomly choose the distribution of trees from Z. Insertion? Deletion? … … …….... …..

5. 5 Uniquely represented structures (a) Generation history determines structure (b) Set-uniqueness:Set determines structure 1, 3, 5, 7 5, 1, 3, 7 1, 3, 5, 7 1 3 5 7 3 1 5 7 1 3 5 7

6. 6 Uniquely represented structures (c) Size-uniqueness:Size determines structure 1, 3, 5, 7 2, 4, 5, 8 Common structure Order-uniqueness: Fixed ordering of nodes determines where the keys are to be stored. 1 3 2 4 2 4 5 8 1 3 5 7

7. 7 Set- and order-unique structures Lower bounds? Assumptions: A dictionary of size n is represented by a graph of n nodes. Node degree finite (fixed), Fixed order of the nodes, i-th node stores i-largest key. Operations allowed to change a graph: Creation | Removal of a node Pointer change Exchange of keys Theorem: For each set- and order-unique representation of a dictionary with n keys, at least one of the operations access, insertion, or deletion must require time  (n 1/3 ).

8. 8 Uniquely represented dictionaries Problem: Find set-unique oder size-unique representations of the ADT „dictionary“ Known solutions: (1)set-unique, oder-unique Aragon/Seidel, FOCS 1989: Randomized Search Trees universal hash-function Update as for priority search trees! Search, insert, delete can be carried out in O(log n) expected time. (s, h(s)) priority s  X

9. 9 The Jelly Fish (2) L. Snyder, 1976, set-unique, oder-unique Upper Bound: Jelly Fish, search insert delete in time O(  n). body:  n nodes  n tentacles of length  n each 10 5 1 2 3 6 7 8 11 12

10. 10 Lower bound for tree-based structures set-unique, oder-unique Lower bound: For “ tree-based ” structures the following holds: Update-time · Search-time = Ω (n) Number of nodes n ≤ h  L + 1 L ≥ (n – 1)/h At least L-1 keys must have moved from leaves to internal nodes. Therefore, update requires time Ω(L). Delete x 1 Insert x n+1 > x n L leaves · x n x 1 h

11. 11 Cons-structures (3) Sunder/Tarjan, STOC 1990, Upper bound: (Nearly) full, binary search trees Einzige erlaubte Operation für Updates: Search time O(log n) Einfügen Entfernen in Zeit O(  n) möglich · · · · 3115353 L R x LR x Cons,,

12. 12 Jump-lists (Half-dynamic) 2-level jump-list 2-level jump-liste of size n Search:O(i) = O( ) time Insertion: Deletion: O( ) time tail 0i2in (n-1)/i·i 23578101112141719

13. 13 Jump-lists: Dynamization 2-level-jump-list of size n search:O(i) = O(  n) time insert delete : O(  n) time Can be made fully dynamic: (i-1) 2 i2i2 n(i+1) 2 (i+2) 2

14. 14 3-level jump-lists level 2 Search(x): locate x by following level-2-pointers identifying i 2 keys among which x may occur, level-1-pointers identifying i keys among which x may occur, level-0-pointers identifying x time: O(i) = O(n 1/3 ) 0i2ii 2 i 2 +i2·i 2

15. 15 3-level jump-lists level 2 Update requires Changing of 2 pointers on level 0 Changing of i pointers on level 1 Changing of all i pointers onlevel 2 Update time O(i) = O(n 1/3 ) 0i2ii 2 i 2 +i2·i 2

16. 16 c-level jump-lists Let Lower levels: level 0: all pointers of length 1:... level j: all pointers of legth i j-1 :... level c/2 :... Upper levels: level j: connect in a in list all nodes 1, 1·i j-1 +1, 2· i j-1 +1, 3· i j-1 +1,... level c:

17. 17 c-level jump-lists Theorem: For each c ≥ 3, the c-level jump-list is a size and order-unique representation of dictionaries with the following characteristics: Space requirement O(c·n) Access time O(c·n 1/c ) Update time, if n is even, if n is odd

18. 18 1 top-level tree with  n leaves All low-level trees for each sequence of  n consecutive keys Top-level tree direct search to the root of the currently active low-level trees Semi-dynamic structure: low-level-tree-size s+1 = top-level-tree-size Shared-search-trees Reduction of search time

19. 19 Shared-search-trees Pointers at: Level 0: (p-2 0 ) p (p+ 2 0 ) Level 1: (p-2 1 ) p (p+ 2 1 ) … Level k-2: (p-2 k-2 ) p (p+ 2 k-2 ) 2(k-1) Pointers per node p, k = O(log n) Search time O(log n) SpaceO(n log n) 01234567891011121314151617 1 0

20. 20 Insertion:Determine insertion position; Change all pointers jumping over the insertion position; Add 2 new pointers per level; Completely rebuild top-level tree.

21. 21 Number of pointerchanges:level 0: 2 · 2 0 +2 level 1: 2 · 2 1 +2 … level k-2:2 · 2 k-2 +2 2 · (2 k-1 -1)+2(k-1) =

22. 22 Shared search trees: Summary Theorem: Shared search trees are a size- und order-unique representation of dictionaries with the following characteristics: Space requirement: O(n log n) Search time: O(log n) Upadate time: O(  n ) Open problem: Is there a size- and order-unique representation of by graphs with bounded node degree, search time O(log n), and update time o(n) (e.g.. O(  n))?