1. Data Structures: Range Queries - Space Efficiency Pooya Davoodi Aarhus University PhD Defense July 4, 2011

2. Thesis Overview 2 5 7 4 4 2 3 4 6 10 20 2512681065713 12437631176295 142318798410840 46984511385825 c d e b f a

3. Range Minimum Queries  Database systems – Lowest average-salary: 3 80,00085,000115,000120,000118,000 81,00073,00090,000100,00094,000 65,00075,00086,00092,00095,000 72,00091,00089,000103,000102,000 35,00045,00042,00050,00041,000 60 50 40 30 20 19901995200020052010 Age Year Minimum: 65,000 at [3,1]

4. Definition 4

5. Naïve Solution 5

6. Data Structures 6 Top-LeftBottom-RightMinimum (1,1) (1,1): 12 (1,1)(1,2)(1,2): 8 (1,1)(2,1)(2,1): 5 (1,1)(2,2)(2,1): 5 (2,1) (2,1): 5 (2,1)(2,2)(2,2): 5 (1,2) (1,2): 8 (1,2)(2,2)(1,2): 8 (2,2) (2,2): 10 Tabulation 128 510 12 1 2

7. ReferenceSpace (bits)Query Time Tabulation Tarjan et al. (STOC’84) Chazelle & Rosenberg (SoCG’89) Lewenstein et al. (CPM’07) Demaine et al. (ICALP’09) - Sadakane (ISAAC’07) Our Result (ESA’10) - Our Result (ESA’10) Space-Efficient Data Structures 7

8. 1D vs. 2D 8 72021085166 2 7 20 8 10 6 16 Lowest Common Ancestor 5

9. Indexing Data Structures  Popular in Succinct Data Structures 9 Index Read-only Size of InputSize of IndexQuery Time (Our Results, ESA’10)

10. 10 27302902815186 1320935417111612 7743962854623 613868109879821 72021085166 2 7 20 5 8 10 6 16 Cartesian Trees Cartesian Tree: Atallah and Yuan (SODA’10) Tabulation

11. 11 11011111101110101111 10111011111111010111 11101101111110111011 11110110111101111011 01111110111011111110 10111101110111110111 C

12. Outline  Range Minimum Queries (ESA 2010, Invited to Algorithmica)  Path Minima Queries (WADS 2011)  Range Diameter Queries (Submitted to ISAAC 2011) 12 5 7 4 4 2 3 4 6 10 20 2512681065713 12437631176295 142318798410840 46984511385825

13. Path Minima/Maxima Queries  The most expensive connection between two given nodes? – between b and k = (c,e) – between e and k = (j,k)  Update(c,e) = 4 13 Tree-Topology Networks 30 a b d c e f g h k j i 5 7 4 4 2 3 4 6 10 4 Trees with Dynamic Weights

14. 4 Naïve Structures 14 a b d c e f g h k j i 5 7 4 4 2 3 4 6 10 30

15.  Reduction from Range Minimum Queries in 1D arrays ReferenceQuery TimeUpdate Time Tabulation Brute Force Search Sleator and Tarjan (STOC’81) Our Result (WADS’11) Our Result (WADS’11) Dynamic Weights 15 Optimal: Brodal et al. (SWAT’96) Optimal by conjecture: Patrascu and Thorup (STOC’06) Optimal: Alstrup et al. (FOCS’98) Comparison Based RAM Optimal: Alstrup et al. (FOCS’98)

16. Dynamic Leaves 16 Reference Query Time Update TimeComment Alstrup and Holm (ICALP’00) and Kaplan and Shafrir (ESA’08) RAM Our Results (WADS’11) Comparison based Optimal: Pettie (FOCS’02) a b d c e f g h k j i 5 7 4 4 2 3 4 6 10 30 4

17. Updates with link and cut 17 a b d c e f g h k j i 5 7 4 4 2 3 4 6 10 30 link (d,i,12) cut(c,e) 12 ReferenceQuery Time Update Time Comment Sleator and Tarjan (STOC’81) Comparison Based Our Results (WADS’11) Cell Probe Proof: by reduction from connectivity problems in graphs

18. Outline  Range Minimum Queries (ESA 2010, Invited to Algorithmica)  Path Minima Queries (WADS 2011)  Range Diameter Queries (Submitted to ISAAC 2011) 18 5 7 4 4 2 3 4 6 10 20 2512681065713 12437631176295 142318798410840 46984511385825

19. Range Diameter Queries  Farthest pair of points 19 A Difficult Problem

20. Known Results 20 Cohen and Porat (2010) Set Intersection Problem Conjecture: Set Intersection problem is difficult ( Patrascu and Roditty, FOCS’10 ) ReferenceQuery TimeSpace Tabulation Smid et al. (CCCG’08) Our Results (Submitted to ISAAC’11) Reduction from Set Intersection

21. Set Intersection Queries Reduction 21 Diameter = 3Diameter < 5 Arithmetic on real numbers with unbounded precisions Points in Convex Position Our Results (Submitted to ISAAC’11) ReferenceQuery TimeSpace

22. Publications 22 5 7 4 4 2 3 4 6 10 20 2512681065713 12437631176295 142318798410840 46984511385825 c d e b f a