1 / 16

Outline

Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property Casper Kejlberg-Rasmussen Joint work with Gerth Stølting Brodal. Outline. Problem Definitions Previous Results An Implicit Moveable Dictionary An Implicit Dictionary with the Working-Set Property.

percy
Download Presentation

Outline

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. Cache-Oblivious Implicit Predecessor Dictionarieswith the Working-Set PropertyCasper Kejlberg-RasmussenJoint work withGerthStøltingBrodal

  2. Outline • Problem Definitions • Previous Results • An Implicit Moveable Dictionary • An Implicit Dictionary with the Working-Set Property

  3. Problem Definitions • Implicit model: • All operations from the RAM • It is not allowed to create words, only to move them • All n words have to be in continuous positions • Often it is assumed that all elements are distinct • Fundamental trick: encode a bit in a pair of elements 1 n ... 0, if x=min(x,y) x y b= 1, if x=max(x,y) b

  4. Problem Definitions • Element e has a working-set number of le iff: le elements different from e have been searched for since we last searched for e • An Implicit Dictionary with the Working-Set Property: • Insert(e): insert element e into the dictionary and set le =0 • Delete(e): delete element e from the dictionary • Search(e): determine if e is in the dictionary and set le =0 • Predecessor(e): find the address of the predecessor of e • Successor(e): find the address of the successor of e le : 0 1 2 3 4 5 1 2 3 4 4 5 6

  5. Outline • Problem Definitions • Previous Results • An Implicit Moveable Dictionary • An Implicit Dictionary with the Working-Set Property

  6. Previous Results e* is the predecessor/successor of e

  7. Outline • Problem Definitions • Previous Results • An Implicit Moveable Dictionary • An Implicit Dictionary with the Working-Set Property

  8. An Implicit Moveable Dictionary • A implicit moveable dictionary laid out in memory addresses [i;j] • Interface: • Insert-left/right(e): insert element e into the dictionary which grows to the left/right • Delete-left/right(e): delete element e from the dictionary which shrinks from the left/right • Search(e): finds the address of e if e is in the dictionary • Predecessor(e): finds the address of the predecessor of e • Successor(e): finds the address of the successor of e i j i-1

  9. An Implicit Moveable Dictionary • Uses O(1) FG dictionaries as black boxes • Recall the FG interface: • Insert-right(e): insert element e into the dictionary which grows to the right • Delete-right(e): delete element e from the dictionary which shrinks from the right • Search(e): finds the address of e if e is in the dictionary • Predecessor(e): finds the address of the predecessor of e • Successor(e): finds the address of the successor of e L C R

  10. An Implicit Moveable Dictionary • L and R will shrink and grow over time • L/R might get too small • L/R might get too large compared to C • We introduce the notion of jobs • Grow-left/right – Counters when L/R gets too small • Shrink-left/right – Counters when L/R gets too large • Jobs run O(1) steps every operation: searches, updates L C R

  11. An Implicit Moveable Dictionary L C R L C R Address-mapping L L’ C R L L’ C R i j L L’ C R L L’ C R Shrink-left Grow-left L L’ C R L L’ C R k k’ i i’ L L’ C R L L’ C R L C R L C R

  12. Outline • Problem Definitions • Previous Results • An Implicit Moveable Dictionary • An Implicit Dictionary with the Working-Set Property

  13. An Implicit Dictionary with theWorking-Set Property • Exponential layout • Bi consists of O(1) moveable dictionaries • All elements e in Bi have le ≥22i-1+k or le ≥22i+k • Searched and inserted elements are moved into B0 (overflows) • These are the ideas we used in the ISAAC 2010 paper • Only gives O(log n) bounds for predecessor and successor searches: invariants do not relate e and its prede/succ-essor |Bi|=Θ(22i+k) m=O(loglog n) B0 B1 B2 Bi Bm-1 Bm ... ...

  14. An Implicit Dictionary with theWorking-Set Property • Divide the key-space into mutually disjoint intervals aligned with the points/elements • Invariant: any point/element, intersecting an interval at level i, lies in block Bi • Predecessor/Successor(e) searches can terminate when an interval at level i is intersected • Intervals to solve the predecessor and successor problems Bm Bm-1 ... B2 B1 B0 e

  15. An Implicit Dictionary with theWorking-Set Property • Representing the intervals implicitly Bm Bm-1 ... B2 B1 B0 Di Ai Ri Wi Hi Ci Gi Bi-1 Bi+1 Arriving Resting Waiting Guarding Climbing Helping le ≥22i+k le ≥22max(i,j)-1+k le ≥22i-1+k

  16. Thank you • References • Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property • GerthStøltingBrodal, Casper Kejlberg-Rasmussen • Submitted to STACS 2012 • Available at cs.au.dk/~ckr/papers/STACS2012_Submission.pdf • A Cache-Oblivious Implicit Dictionary with the Working Set Property • Gerth Stlting Brodal, Casper Kejlberg-Rasmussen, Jakob Truelsen • ISAAC 2010

More Related