Cache-Oblivious Dynamic Dictionaries with Update/Query Tradeoff

1. Cache-Oblivious Dynamic Dictionaries with Update/Query Tradeoff GerthStøltingBrodal Erik D. Demaine Jeremy T. Fineman John Iacono Stefan Langerman J. Ian Munro Resultpresentedat SODA 2010

2. Dynamic Dictionary Search(k) Insert(e) Delete(k)

3. I/O Model[Aggarwal, Vitter 88] B Slow Memory Fast Memory CPU block M/B B • Cost:the number of block transfers (I/Os)

4. Cache-Oblivious Algorithms[Frigo, Leiserson, Prokop, Ramachandran 99] • Algorithms not parameterized by M or B • Analyze in ideal-cache model — I/O model, except optimal replacement policy is assumed B Slow Memory Fast Memory CPU block M/B B

5. Cache-Oblivious Dynamic Dictionaries * amortized † assumes M = Ω(B2)

6. Building an xDict ( = 1/2) … 221-box 222-box 22lglgN-box lglgNx-boxesof squaring capacities Insert: insert into smallest box • When a box reaches capacity, Flush it and Batch-Insertinto the next box • O((1/√B) logBx) cost is dominated by largest box O((1/√B) logBN) Search: search in each x-box • O(logBx) cost is dominated by largest box O(logBN)

7. x-Box = dictionary with capacity x2 Batch-Insert(D,A): insert Θ(x) presorted objects — cost O((1/√B)logBx) per element Search(D,κ): — cost is O(logBx) Flush(D): produce a size-x2 sorted array A containing all the elements in the x-box D — cost is O(1/B) per element

8. Recursive x-Box size-x input buffer Upper level: at most x1/2/4 subboxes … √x-box size-x3/2 middle buffer … Lower level: at most x/4 subboxes √x-box size-x2 output buffer … … input middle output subboxes stored contiguously in arbitrary order Unused (currently empty) subboxes are preallocated

9. x-Box Space Usage size-x input buffer Theorem: An x-Box uses at most cx2space (within constant factor of capacity/output buffer) Upper level: at most x1/2/4 subboxes … √x-box size-x3/2 middle buffer … Lower level: at most x/4 subboxes √x-box size-x2 output buffer

10. Fractional Cascading within x-Box size-x input buffer Propagate samples upwards + Lookahead pointers Upper level: at most x1/2/4 subboxes √x-box size-x3/2 middle buffer Lower level: at most x/4 subboxes √x-box size-x2 output buffer

11. Searching in an x-Box size-x input buffer Describe searches by the recurrence S(x) = 2S(√x) + O(1) with base case S(<√B) = 0 Solves to O(logBN) Upper level: at most x1/2/4 subboxes √x-box size-x3/2 middle buffer Lower level: at most x/4 subboxes √x-box size-x2 output buffer

12. Flush size-x input buffer • Moves all real elements to the output buffer in sorted order. Upper level: at most x1/2/4 subboxes √x-box size-x3/2 middle buffer Lower level: at most x/4 subboxes √x-box size-x2 output buffer • Lookahead pointers are rebuilt to facilitate searches. Most subboxes remain empty.

13. Batch-Insert + input buffer • Merge sorted input into input buffer. middle buffer output buffer

14. Batch-Insert • Merge sorted input into input buffer. • If input buffer is “full enough,” Batch-Insert into upper-level subboxes (in chunks of Θ(√x)) middle buffer output buffer

15. Batch-Insert input buffer • Merge sorted input into input buffer. • If input buffer is “full enough,” Batch-Insert into upper-level subboxes (in chunks of Θ(√x)) • Whenever a subbox is near capacity, Flush it, then split it into two subboxes middle buffer output buffer

16. Batch-Insert input buffer • Merge sorted input into input buffer. • If input buffer is “full enough,” Batch-Insert into upper-level subboxes (in chunks of Θ(√x)) • Whenever a subbox is near capacity, Flush it, then split it into two subboxes • If no empty subboxes remain, Flush all of them and merge output buffers into middle buffer. middle buffer output buffer

17. Generalizing to O((1/εB 1-ε)logBN) size-x input buffer Parameterize by 0 < α ≤ 1, where α=ε/(1-ε) Upper level: at most x1/2/4 subboxes √x-box size-x1+α/2 middle buffer Lower level: at most x1/2+α/2/4 subboxes √x-box size-x1+α output buffer 2(1+α)1 2(1+α)2-box 2(1+α)i-box • 1/ε overhead comes from geometric sum in xDict

18. Results Summary * amortized † assumes M = Ω(B2)