1. A Simple Algorithm for Stable Minimum Storage Merging Pok-Son Kim Kookmin University, Department of Mathematics, Seoul 135-702, Korea Arne Kutzner Seokyeong University, Department of E-Business, Seoul 136-704, Korea

2. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 2 Merging Make one sorted array out of two consecutive sorted arrays 491392 3, 491, 92

3. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 3 Lower Bounds for Merging Number of comparisons –Argumentation over the decision tree (see Knuth) Number of assignments –Each element can change its position in the final sequence for

4. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 4 Notions An algorithm merges two adjacent sequences with “minimum storage” when it needs bits additional space at most. Stability: Merging algorithm preserves the initial ordering of elements with equal value.

5. We present..... …a simple, asymptotically optimal (regarding comparisons), stable, minimum storage merging algorithm called SPLITMERGE

6. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 6 SPLITMERGE: Step 1 - Splitting We compute boundaries in our inputs u and v so that we get the following property: u1u1 u2u2 v1v1 v2v2  

7. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 7 SPLITMERGE: Step 2 – Rotation We rotate u 2 v 1 to v 1 u 2 u1u1 u2u2 v1v1 v2v2 u1u1 u2u2 v1v1 v2v2

8. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 8 SPLITMERGE: Step 3 – Recursive Application We have: 1. u 1  u 2 and u 1  v 2 2. v 1  u 2 and v 1  v 2 u1u1 u2u2 v1v1 v2v2 left recursionright recursion

9. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 9 Stability Lemma: SPLITMERGE is stable. Proof: u1u1 u2u2 v1v1 v2v2 u1u1 u2u2 v1v1 v2v2 > rotation !!!!

10. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 10 How do we split? Repeated halving of two search spaces until these are reduced to single points (the borders of a rotation). For asymmetric inputs one element of the smaller side “freezes” after some time

11. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 11 Symmetric Splitting – Graphical Description u 21 xu 22 v 21 yv 22   search spaces x  y u 22 v 21   xy u 21 v 22   xy x  y

12. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 12 Splitting Example 98651655201880 98651655201880 98651655201880   98651655201880 ? ? ?

13. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 13 Symmetric Splitting – Asymmetric Inputs (1) x u1u1 u3u3 v1v1 v 21 yv 22 v3v3   single search space After some time one element “freezes” (here x) x can “belong” to u 1, u 3 or both x belongs to u 1 means u 1 x  v 3 x belongs to u 3 means xu 3  v 1

14. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 14 Symmetric Splitting – Asymmetric Inputs (2) x u1u1 u3u3 v1v1 v3v3   Rotation – we distinguish two different cases if x belongs to u 3 : x u1u1 u3u3 v1v1 v3v3 otherwise : x u1u1 u3u3 v1v1 v3v3

15. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 15 Symmetric Splitting – Asymmetric Inputs (3) Recursion – we distinguish several cases if x belongs to u 1 x u1u1 u3u3 v1v1 v3v3 otherwise x u1u1 u3u3 v1v1 v3v3 left Recursionright Recursion

16. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 16 Algorithm in Pseudo code

17. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 17 Some Remarks … The “belongs to” information can be followed from the variables l, r and m

18. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 18 Worst Case Complexity Lemma: The recursion depth is limited by min, where Corollary: Minimum Storage Algorithm

19. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 19 Worst Case Complexity (Comparisons) Special case m→ m/2 (1) m10m10 n32n32 m12m12 m22m22 m32m32 m42m42 m11m11 m21m21 n42n42 n22n22 n12n12 n11n11 n21n21 n10n10 m m/2 m/4 1111 recursion depth

20. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 20 Worst Case Complexity (Comparisons) Special case m→ m/2 (2) m10m10 n32n32 m12m12 m22m22 m32m32 m42m42 m11m11 m21m21 n42n42 n22n22 n12n12 n11n11 n21n21 n10n10 0 1 2 i k recursion level

21. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 21 Worst Case Complexity (Comparisons) Special case m→ m/2 (3) Overall number of comparisons: Mathematical Machinery

22. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 22 Worst Case Complexity How to treat the other cases? (1) Example: Maximum spanning case

23. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 23 Worst Case Complexity How to treat other cases? (2) Trick: Mapping into a corresponding “ m → m/2 case” by the introduction of recursion groups –A recursion group consists of one or several recursion levels –Grouping of recursion levels into recursion groups so that each recursion group holds at most 2 i-1 recursion levels. –Altogether k+1 recursion groups

24. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 24 Worst Case Complexity How to treat other cases? (3) Mapping for the maximum spanning case:

25. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 25 Assignments / Worst Case Complexity Theorem: The S PLITMERGE algorithm needs assignments. Sketch of Proof: At most k+1 recursion groups with O(m+n) many assignments for every group

26. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 26 Benchmarks n,m : lengths of input sequences (m=n) i : number of different input elements t e : execution time in ms #comp : number of comparisons

27. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 27 Summary Simply structured algorithm using a novel binary partition technique Doesn’t rely on any third algorithm as subpart Short definition in Pseudo Code Asymptotically optimal regarding the number of comparisons Nice Textbook algorithm ?

28. SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 28 Thank you very much for your attention