1. Pawe ł Gawrychowski* and Pat Nicholson** *University of Warsaw **Max-Planck-Institut für Informatik

2. Range Queries in Arrays

3. Encoding Range Queries in Arrays

4. Typical Data Structure Input Data (Relatively Big) Input Data (Relatively Big)

5. Typical Data Structure Input Data (Relatively Big) Input Data (Relatively Big) Data Structure Preprocess

6. Encoding Approach Input Data (Relatively Big) Input Data (Relatively Big)

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

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

9. Encoding Approach Input Data (Relatively Big) Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still)

10. Encoding Approach Succinct Data Structure: Minimum Space Possible Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) Input Data (Relatively Big) Input Data (Relatively Big) Preprocess w.r.t. Some Query

11. Encoding Approach 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) Input Data (Relatively Big) Preprocess w.r.t. Some Query

12. This Talk: Maximum-Sum Segments

13. Range Maximum-Sum Segment Queries

19. Now find the minimum in this range

22. Candidate Pairs

23. What Do They Store?

24. How to answer a query: the easy case

27. How to answer a query: the not so easy case

32. Reducing the Space

35. Nested Is Good 12345678 ()((())(()(()))())((()(()))())

40. Recall The Query Algorithm

44. Problem: cannot store the left siblings explicitly

45. Idea: try to find something that is nested

49. What Do We Store?