1 / 20

Orthogonal Range Reporting

Optimal 3-d Orthogonal Range Reporting Kasper Dalgaard Larsen Joint work with Peyman Afshani and Lars Arge. Orthogonal Range Reporting. Input: n points in d -dimensional space. Task: Build structure that reports all points inside an axis-aligned query box efficiently. Our Focus:

roland
Télécharger la présentation

Orthogonal Range Reporting

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. Optimal 3-d Orthogonal Range ReportingKasper Dalgaard LarsenJoint work with PeymanAfshani and Lars Arge

  2. Orthogonal Range Reporting • Input: npoints in d-dimensional space. • Task: Build structure that reports all points inside an axis-aligned query box efficiently. • Our Focus: • Pointer Machine Model • Near-linear space, Polylog(n)+t queries.

  3. Previous 2-d Results • Upper Bounds: • O(lgn+t) query O(n) space for 3-sided [McCreight 85]. • O(lgn+t) query O(n lgn/lglgn) space [Chazelle 86]. • Lower Bounds: • Ω(n lgn/lglgn) space for O(logO(1)n+t) query [Chazelle 90]. Solved

  4. Infinite Range Variants • Query Variants: • Q(d,k): d-dimensional space, k ranges are finite.

  5. Previous Results: Adding Sides • Range Trees [Bentley 80]: Q(d,k) to Q(d,k+1) Q(d,k) to Q(d+1,k+1) • Binary- and fanout (lgn)e trees. • Recently (FOCS’09) we showed.

  6. Previous Results 2-d opt. space • Upper Bounds in 2 and 3-d: • Lower Bounds Q(3,3): • Ω(n (lgn/lglgn)2) space for O(logO(1)n+t) query [Chazelle 90]. 3-d opt. queries

  7. Our Results • We achieve: • 3-d closed, but problem still open for d≥4.

  8. How To Get Better Reductions • We want: Q(d,k) to Q(d,k+1) Q(d,k) to Q(d+1,k+1) • Define new and harder variant of range reporting. • Give reductions with lgn/lglgn penalty. • Solve base case efficiently.

  9. Goals • Fanout (lgn)e • Perform O(1) queries at level. • Store points in one structure at level. • Observations: • Same Q(d,k) query on subset of children. • Only O((lgn)2e) possible subsets to query. Q(d+1,k+1) Q(d+1,k) Q(d,k) Q(d+1,k) Q(d,k) (lgn)e

  10. Concurrent Range Reporting • Color points based on child. • Perform Q(d,k) query on all the points. • Report only with colors { , }. • Recurse. • Observations: • (lgn)e colors. • O((lgn)2e) possible color subsets to query. Q(d+1,k+1) Q(d+1,k) Q(d,k) Q(d+1,k) Q(d,k)

  11. Concurrent Range Reporting • Concurrent Q(d,k) (CQ(d,k)): • n points in d-dimensional space. • Points have colors from a set C. • Set P of subsets of C (possible color queries). • Queries: • A Q(d,k) box q and a set p from P. • Output all points in q with color in p. p={ } C={ } P={ , , , }

  12. Concurrent Range Reporting • We reduce Q(3,3) to CQ(2,2): • |C|=(lgn)e , |P|= (lgn)2e • O(lgn/lglgn) levels. • Points stored in one CQ(2,2) at level. • Two CQ(2,2) queries at level. • CQ(d,k) solution with: • Sd,k(n,|C|,|P|) space. • Qd,k(n,|C|,|P|)+O(t) queries. Q(3,3) Q(3,2) Q(2,2) Q(3,2) Q(2,2) • Q(d+1,k+1) solution with: • Sd,k(n,|C|,|P|) lgn/lglgn • Qd,k(n,|C|,|P|) lgn/lglgn+O(t)

  13. Concurrent Range Reporting • CQ(d+1,k+1) to CQ(d,k): • New colors C’ are pairs (old color,child). • P’ is product of P and ranges of child colors. • |C’|=(lgn)e |C| • |P’|=(lgn)2e|P| Q(d+j,k+j) to CQ(d,k) with |C|=(lgn)ej and |P|=(lgn)2ej Q(d+1,k+1) C={ } P={ , } C’={ } Q(d+1,k) Q(d,k) Q(d+1,k) Q(d,k) P’={ , , , , , } , , , p={ } to p’={ } ... , , ...

  14. Overview of Optimal 3-d Solution • Combination of many previous techniques: • [Afshani 08],[Afshani et. al 09],[Chazelle 86],[Chazelle and Guibas 86],[Karpinski and Nekrich 09],[Makris and Tsakalidis 98]. • CQ(3,0): O(n) space, O(lgn+t) queries. • [Afshani 08],[Afshani et. al 09],[Chazelle 86],[Makris and Tsakalidis 98]. • CQ(3,2): O(nlgn/lglgn) space, O(lgn+t) queries. • [Afshani et. al 09],[Chazelle and Guibas 86],[Karpinski and Nekrich 09]. • CQ(3,3): O(n(lgn/lglgn)2) space, O(lgn+t) queries. • [Afshani et. al 09].

  15. Sketch of CQ(3,0) • lgn-shallow cutting [Afshani 08] for every color. • Q(3,0) structure in every box and one for every color. • O(n) space, O(|C|lglgn+t) queries. • 2-d rect. point location finds boxes [Makris and Tsakalidis 98]. • Point Enclosure [Chazelle 86] finds all boxes at same time. • O(n) space, O(lgn+|C|) queries.

  16. Sketch of CQ(3,2) • A grid of size yxy where y=√(n/|C||P|lg2n) [KN 09]. • n/y points in slabs. • Query: • grid query. • 4 recursive. • Need to answer grid queries. • Recursive: • grid query. • 1 recursive. • 2 CQ(3,0) queries.

  17. Sketch of CQ(3,2): Grid Queries • Answering grid queries. • Snap points to grid. • Linked lists for each color. • Q(3,2) structure [Chazelle and Guibas 86] for every pєP on bottom points. • Link to lists. • S(n)=2n/yS(y)+O(n), S(lgn)=lgn S(n)=nlgn/lglgn • Q(n)=Q(y)+lgn Q(n)=lgn

  18. Summary • We thus have: Solved

  19. Open Problems • Pointer Machine Model: • 4-d queries with O(lgn+t) query time? • Query bounds of the form O(lgd/2+O(1)n+t)? • External Memory: • We have CQ(3,0) (modification). • CQ(3,2) not space optimal. • Q(3,3) with O(lgBn+t/B) queries and O(n(lgn/lglgBn)2) space?

  20. Thank You

More Related