1 / 22

Data Structures: Range Queries - Space Efficiency

Data Structures: Range Queries - Space Efficiency. Pooya Davoodi Aarhus University. PhD Defense July 4, 2011. Thesis Overview. Range Minimum Queries in Arrays (ESA 2010, Invited to Algorithmica ) Path Minima Queries in Trees (WADS 2011)

lanai
Télécharger la présentation

Data Structures: Range Queries - Space Efficiency

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Thesis Overview • Range Minimum Queries in Arrays (ESA 2010, Invited to Algorithmica) • Path Minima Queries in Trees (WADS 2011) • Range Diameter Queries in 2D Point Sets(Submitted to ISAAC 2011) • Succinct -ary Trees(TAMC 2011) 4 6 7 4 3 10 2 5 4 20 a b e c d f

  3. Range Minimum Queries • Database systems • Lowest average-salary: Year Age Minimum: 65,000at [3,1]

  4. Definition • Input: an array • Query: where is minimum in ? ()

  5. Naïve Solution • Brute force search • Query time: time • Worst case : time ()

  6. Data Structures • Preprocess and store some information • Naïve: store the answers of all queries • query time • Size of the table: bits Tabulation

  7. Space-Efficient Data Structures

  8. 1D vs. 2D Lowest Common Ancestor 2 • 1D: Cartesian Trees • bits per element (Tarjan et al., STOC’84) • bits per element (Sadakane, ISAAC’07) • 2D: Nothing like Cartesian Trees 7 5 20 8 6 10 16 bits per element (Our Result, ESA’10)

  9. Indexing Data Structures • Popular in Succinct Data Structures Read-only Index Input Array bits per element (Our Results, ESA’10)

  10. bits with query time 2 7 5 Cartesian Trees 20 8 6 16 10 Cartesian Tree: Tabulation Atallah and Yuan (SODA’10)

  11. bits Per Element • bits query time • bitsquery time • Proof: queries distinguish inputs in time • query with time C

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

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

  14. Naïve Structures • Brute Force Search • Worst case query time: • Update time: • Tabulation • Query time: • Update time: i 4 6 30 e j 7 c 3 b 4 2 10 4 5 h g f a k d 30 4

  15. Dynamic Weights • Reduction from Range Minimum Queries in 1D arrays Comparison Based Optimal: Brodal et al. (SWAT’96) Optimal: Alstrup et al. (FOCS’98) RAM Optimal by conjecture: Patrascu and Thorup (STOC’06) Optimal: Alstrup et al. (FOCS’98)

  16. Dynamic Leaves i 4 6 30 e j 7 c 3 b 4 2 10 4 5 h g f Optimal: Pettie (FOCS’02) a k d 4

  17. Updates with link and cut i 4 6 30 cut(c,e) e j 7 c 3 b 4 2 link (d,i,12) 10 4 12 5 h g f a k d 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) 4 6 7 3 4 10 2 5 4 20

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

  20. Known Results Set Intersection Problem Conjecture: Set Intersection problem is difficult (Patrascu and Roditty, FOCS’10)

  21. Set Intersection QueriesReduction ? Arithmetic on real numberswith unbounded precisions Diameter = 3 Diameter < 5

  22. Publications • Range Minimum Queries(ESA 2010, Invited to Algorithmica) • Path Minima Queries(WADS 2011) • Range Diameter Queries(Submitted to ISAAC 2011) • Succinct -ary Trees(TAMC 2011) 4 6 7 3 4 10 2 5 4 20 b a e c d f

More Related