Lecture 39. CSE 331 Dec 6, 2017

Transcription

1 Lecture 39 CSE 331 Dec 6, 2017

2 Grading back on track

3 Re-grading request deadlines

4 Now relax 4

5 Randomized algorithms What is different? Algorithms can toss coins and make decisions A Representative Problem Hashing Further Reading Chapter 13 of the textbook

6 Approximation algorithms What is different? Algorithms can output a solution that is say 50% as good as the optimal A Representative Problem Vertex Cover Further Reading Chapter 12 of the textbook

7 Online algorithms What is different? Algorithms have to make decisions before they see all the input A Representative Problem Secretary Problem Further Reading

8 Data streaming algorithms What is different? One pass on the input with severely limited memory A Representative Problem Compute the top-10 source IP addresses Further Reading

9 Distributed algorithms What is different? Input is distributed over a network A Representative Problem Consensus Further Reading

10 Beyond-worst case analysis What is different? Analyze algorithms in a more instance specific way A Representative Problem Intersect two sorted sets Further Reading

11 Algorithms for Data Science What is different? Algorithms for non-discrete inputs A Representative Problem Compute Eigenvalues Further Reading

12 Johnson Lindenstrauss Lemma

13 Questions? Questions?

14 Ax = y a 0,0 a 0,2.. a 0,N-1 x 0 y 0 a N-1,0 a N-1,1.. a N-1,N-1 x N-1 y N-1 A x y Θ(N 2 ) time in worst-case

15 In practice A has structure a 0,0 a 0,2.. a 0,N-1 x 0 y 0 a N-1,0 a N-1,1.. a N-1,N-1 x N-1 y N-1 A x y Can we exploit the structure for faster algorithms?

16 Discrete Fourier Transform a 0,0 a 0,2.. a 0,N-1 b 10 a N-1,0 a N-1,1.. a N-1,N-1 b N-1 A b a x,y = exp(2πi xy/n) FFT (1965) Can compute DFT in O(N log N) time Cooley Tukey

17 Cauchy Matrix a 0,0 a 0,2.. a 0,N-1 b 10 a N-1,0 a N-1,1.. a N-1,N-1 b N-1 A b Can be computed in O(N log 2 N) time a x,y = 1 r x - s y

18 Superfast = N poly-log(n)

19 The main Question What is the largest class of matrices A for which we can have superfast algo to compute Ax?

20 Structure 1: Recurrence a x,y = exp(2πi xy/n) a 0,0 a 0,2.. a 0,N-1 b 10 a x,y+1 = a x,y exp(2πi x/n) a N-1,0 a N-1,1.. a N-1,N-1 b N-1 Multiplier matrix only depends on x Multiplier only depends a 0,0 a.. 0,2 on ax 0,N-1 a N-1,0 a N-1,1 a x,y.. a N-1,N-1 A b

21 Structure 2: Low Displacement Rank a x,y = 1 r x - s y r x a x,y - a x,y s y = 1 LA AR has low rank r A A s 0 r N-1 s N-1

22 Known Results Recurrences Low Displacement Rank O(N log 2 N) O(N log 2 N) Driscoll Healy Rockmore FFT OlshevskyShokrollahi

23 Our Main Result* One Result that recovers all existing results* Recurrences Low Displacement Rank O(N log 2 N) O(N log 2 N) Driscoll Healy Rockmore FFT OlshevskyShokrollahi

24 Questions? Questions?

25 Coding Theory 25

26 Communicating with my 3 year old C(x) x Code C Kiran English C(x) is a codeword y = C(x)+error x Give up 26

27 The setup C(x) x Mapping C Error-correcting code or just code Encoding: x C(x) Decoding: y x C(x) is a codeword x y = C(x)+error Give up 27

28 Different Channels and Codes Internet Checksum used in multiple layers of TCP/IP stack Cell phones Satellite broadcast TV Deep space telecommunications Mars Rover 28

29 Unusual Channels Data Storage CDs and DVDs RAID ECC memory Paper bar codes UPS (MaxiCode) Codes are all around us 29

30 Redundancy vs. Error-correction Repetition code: Repeat every bit say 100 times Good error correcting properties Too much redundancy Parity code: Add a parity bit Minimum amount of redundancy Bad error correcting properties Two errors go completely undetected Neither of these codes are satisfactory 30

31 Two main challenges in coding theory Problem with parity example Messages mapped to codewords which do not differ in many places Need to pick a lot of codewords that differ a lot from each other Efficient decoding Naive algorithm: check received word with all codewords 31

32 The fundamental tradeoff Correct as many errors as possible with as little redundancy as possible Can one achieve the optimal tradeoff with efficient encoding and decoding? 32

33 Interested in more? CSE 545, Spring

34 Whatever your impression of the 331 IT WAS 34

35 Hopefully it was fun! 35

36 Thanks! Except of course, HW 10, presentations and the final exam 36

37 The simplest non-trivial join query Intersection of R and S R A S Assume R and S are sorted Let us concentrate on comparison based algorithms Assume R = S = N

38 Not all inputs are created equal R S R S Ω(N) comparisons 1 comparison!

39 We need a faster/adaptive algorithm

40 The MERGE algorithm works R S 1 N

41 An assumption Output of the join is empty

42 MERGE is (near) instance optimal Benchmark: Minimum number of comparisons (C) to certify output 1 Demaine Lopez-Oritz Munro C logn comparions (and time) R S Value not ruled out yet Need a comparison to rule the value out Each value involved with 2 comparisons Once the pointer moves the value is N never seen again Each move takes log N comparisons