1. ENCODING NEAREST LARGER VALUES Pat Nicholson* and Rajeev Raman** * MPII ** University of Leicester

2. Input Data (Relatively Big) DÉJÀ VU: THE ENCODING APPROACH

3. Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller)

4. Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller) DÉJÀ VU: THE ENCODING APPROACH

5. Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) DÉJÀ VU: THE ENCODING APPROACH

6. Succinct Data Structure: Minimum Space Possible Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) Input Data (Relatively Big) Preprocess w.r.t. Some Query DÉJÀ VU: THE ENCODING APPROACH

7. Succinct Data Structure: Minimum Space Possible Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) Query (Hope: as fast as non- succinct counterpart) Input Data (Relatively Big) Preprocess w.r.t. Some Query DÉJÀ VU: THE ENCODING APPROACH

8. NEAREST LARGER VALUES 102319871154

9. OVERVIEW: ENCODING NLV For all these results: space bound is optimal to within lower order terms DistinctProblemSpaceQNotes Yes Unidirectional Cartesian Tree Bidirectional Cartesian Tree Nondirectional NoUnidirectional [Fischer et al. 2009] Cartesian Tree Bidirectional [Fischer 2011] Schröder Trees (Navigate CSA) Nondirectional

10. OVERVIEW: ENCODING NLV DistinctProblemSpaceQNotes Yes Unidirectional Cartesian Tree Bidirectional Cartesian Tree NondirectionalThis paper: NLV Tree NoUnidirectional [Fischer et al. 2009] Cartesian Tree Bidirectional [Fischer 2011] Schröder Trees (Navigate CSA) Nondirectional

11. BIGGER PICTURE

13. CARTESIAN TREES REVIEW We can rebuild him. We have the technology.

14. NONDIRECTIONAL NLV TREE Tie breaking rule: break ties to by choosing the one to the right.

15. TIEBREAKING MATTERS? 12345678910 Rule To the right1251440116341101030099012 To the smaller12514421263831178364011316 To the larger12512328824870219985696

16. IDEA: COMPRESS RUNS

20. DIGRESSION: PATH (OR CHAIN) COMPRESSION Degree two Degree one Terminal Subtree

21. COMPRESSING CARTESIAN TREES W.R.T. NLVS Forget about whether it zigs or zags, just store # in prefix…

22. THE ENCODING

23. SUB-OPTIMALITY EXAMPLES

28. LOWER BOUND SKETCH

29. 102319871154

30. CONCLUSIONS AND OPEN PROBLEMS

31. THANK YOU