1. Sampling in Space Restricted Settings Anup Bhattacharya IIT Delhi Joint work with Davis Issac (MPI), Ragesh Jaiswal (IITD) and Amit Kumar (IITD)

2. Introduction: Sampling Select a subset of data Computations on “representative” subset would approximate computations on whole data Sampling variants: –Uniform sampling –Weighted sampling Study sampling algorithms with limited space

3. Outline

4. Sampling in Streaming Settings

5. Streaming Settings: The Model – Items/objects arrive in online fashion – #Total items not known in advance – Typically poly(log(n)) space allowed – One/multi-pass, space usage, time/item, overall time complexity, randomness, accuracy of output

6. Sampling in Streaming Settings

7. Reservoir Sampling … Throw it away Store

8. Reservoir Sampling

9. Uniform Sampling with ϵ-error

10. Lower Bound on Sampling with ϵ-error

11. Outline

12. Algorithm for Uniform Sampling ϵ-error

13. Doubling-Chopping Algorithm

14. Doubling-Chopping algorithm, ϵ=1/16

15. 0 1

16. 00 10 01 11

17. Doubling-Chopping algorithm, ϵ=1/16 000 010 100 110 001 011 101 111

18. Doubling-Chopping algorithm, ϵ=1/16 0000 0010 0100 0110 1000 1010 1100 1110 0001 0011 0101 0111 1001 1011 1101 1111

19. Doubling-Chopping algorithm, ϵ=1/16 0000 0010 0100 0110 1000 1010 1100 1110 0001 0011 0101 0111 1001 1011 1101 1111

20. Doubling-Chopping algorithm, ϵ=1/16 Chop(): Move strings from blocks to new block 0110 1000 1010 1100 1110 0111 1001 1011 1101 1111 0101 0011 0001 0100 0010 0000

21. Doubling-Chopping algorithm, ϵ=1/16 Chop(): Move strings from blocks to new block 0110 1000 1010 1100 1110 0111 1001 1011 1101 1111 0101 0011 0001 0100 0010 0000

22. Doubling-Chopping algorithm, ϵ=1/16 Chop(): Move strings from blocks to new block 0110 1000 1010 1100 1110 0111 1001 1011 1101 1111 0011 0001 0100 0010 0000 0101

23. Doubling-Chopping algorithm, ϵ=1/16 0110 1000 1010 1100 1110 0111 1001 1011 1101 1111 0011 0001 0100 0010 0000 0101

24. Doubling-Chopping algorithm, ϵ=1/16 1000 1010 1100 1110 1001 1011 1101 1111 0001 0100 0010 0000 0101 0011 0111 0110

25. Doubling-Chopping algorithm, ϵ=1/16 1000 1010 1100 1110 1001 1011 1101 1111 0001 0100 0010 0000 0101 0011 0111 0110

26. Algorithm Analysis

27. Analysis contd..

29. Sampling in Query Model

30. Space Restricted Setting: Query Model

31. Sampling in Query Model

32. Thank You Questions?