1. CSE 501 Research Overview Atri Rudra atri@cse.buffalo.edu

2. 2 Research Interests Theoretical Computer Science  Coding Theory  Algorithmic Game Theory  Sublinear algorithms  Approximation and online algorithms  Computational Complexity The “algorithmic” side

3. 3 Coding Theory

4. 4 The setup C(x) x y = C(x)+error x Give up Mapping C  Error-correcting code or just code  Encoding: x  C(x)  Decoding: y  X  C(x) is a codeword

5. 5 Different Channels and Codes Internet  Checksum used in multiple layers of TCP/IP stack Cell phones Satellite broadcast  TV Deep space telecommunications  Mars Rover

6. 6 “Unusual” Channels Data Storage  CDs and DVDs  RAID  ECC memory Paper bar codes  UPS (MaxiCode) Codes are all around us

7. 7 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 1 1 1 0 011 0 0 0 01

8. 8 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

9. 9 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 ?

10. 10 A “low level” view Think of each symbol in  being a packet The setup  Sender wants to send k packets  After encoding sends n packets  Some packets get corrupted  Receiver needs to recover the original k packets

11. 11 The Optimal Tradeoff C(x) sent, y received How much of y must be correct to recover x ?  At least k packets must be correct [Guruswami, R. STOC 2006]  An explicit code along with efficient decoding algorithm  Works as long as (almost) k packets are correct

12. 12 So what is left to do? I cheated a bit in the last slide The result only holds for large packets We do not know an “optimal” code over smaller symbols (for example bits)

13. 13 Computational Complexity Collisions to lead to shallower decision trees  [Aspnes, Demirbas, O’Donnell, R.,Uurtamo 2008]

14. 14 Wireless Sensor Networks Murat Demirbas’ specialty

15. 15 Compute Aggregate Functions Each mote has one bit of information Is the temperature at least 70F? Does at least 2 motes have temperature at least 70F?

16. 16 One possible solution Ask each mote one at a time Is your temp at least 70F? Yes Is your temp at least 70F? In the worst case might have to ask ALL motes No

17. 17 Can we do any better? Formalize/Generalize this question Decision Tree model  Inputs: x 1, x 2,…,x n in {0,1}  Function: f : {0,1} n  {0,1}  Minimum # queries to the input to determine f(x 1, x 2,…,x n ) in the worst case  This worst case number of queries is called the decision tree complexity of f Very well studied complexity measure of functions

18. 18 Back to our ≥ 70F example The 2-threshold T 2,n function Previously saw D(T 2,n )  n  In fact D(f)  n  Also D(T 2,n ) ≥ n  For the t-threshold function, D(T t,n ) ≥ n Logical OR, Majority are a special case

19. 19 So are we done? The central node can broad/multi-cast Is your temp at least 70F? In the worst case might have to ask ALL motes Is your temp at least 70F? No Is your temp at least 70F?

20. 20 Replies from the motes Answer back only if answer is yes Is your temp at least 70F?

21. 21 Scenario 1: all the answers are 0 Central node hears “silence” Is your temp at least 70F?

22. 22 Scenario 2: Only one answers is 1 Central node hears a “yes” Is your temp at least 70F? Yes

23. 23 Scenario 3: ≥ 2 answers are 1 There is a collision  Central node can detect it! Is your temp at least 70F? Yes All done with ONE query!

24. 24 Feedback to complexity theory A new “decision tree” model  Inputs: x 1, x 2,…,x n in {0,1}  Function: f : {0,1} n  {0,1}  Minimum # queries to the input to determine f(x 1, x 2,…,x n ) in the worst case  Queries are more general Query any subset of bits Answer is 0, 1 or 2 + depending on #ones in the subset k + decision trees

25. 25 Our Results D 2+ (T t,n ) is O(t log (n/t)) D 2+ (T t,n ) is  (t) More general results  Understand D 2+ (f) fairly well

26. 26 Approximation Algorithms Ranking in Tournaments  [Coppersmith, Fleischer, R. SODA 2006]

27. 27 US Open 2005 Everyone plays everyone Rank the players Min #upsets Rank by number of wins  Break ties Venus WilliamsMaria Sharapova Kim Clijsters Nadia Petrova #1 #2 #3 #4

28. 28 Ranking in Tournament results [Coppersmith, Fleischer, R. SODA 2006] Ordering by number of wins is 5-approx  Ties broken arbitrarily  Problem shown to be NP-hard in 2005 Application in Rank Aggregation  Gives provable guarantee for Borda’s method (1781!) Future Directions  Try and analyze (variants) of heuristics that work well in practice

29. 29 Research Interests Theoretical Computer Science  Coding Theory  Algorithmic Game Theory  Sublinear algorithms  Approximation and online algorithms  Computational Complexity The “algorithmic” side

30. 30 For more information… My Office is Bell 123: drop by! atri@cse.buffalo.edu CSE 545 in Spring 09  Course on error correcting codes

31. 31 Algorithmic Game Theory Online auction of digital goods  [Blum, Kumar, R., Wu SODA 2003]

32. 32 Online Auctions of Digital goods Say you want to sell mp3s of a song  Can make copies with no extra cost Buyers arrive one by one  Specify how much they are willing to pay You need to decide to sell or not  At what price ? You want to make lots of money $5 OK, $4 $1 No

33. 33 However… Why not just sell at the value specified by a buyer ? Buyers are selfish  They will lie to get a better deal Why not charge a single fixed price ?  Do not know best price in advance The challenge  Build a online pricing scheme that gives buyers no incentive to cheat Our work gives pricing scheme as good as best fixed price  [Blum, Kumar, R., Wu SODA 2003]

34. 34 Problems I am interested in Problems motivated by game theory Sometimes, “old” problem with a twist  What is the best way to pair up potential couples in a dating site? Twist on the classical graph matching problem

35. 35 Sublinear Algorithms Data Streams  [Beame, Jayram, R. STOC 2007]

36. 36 Data Streams (one application) Databases are huge  Fully reside in disk memory Main memory  Fast, not much of it Disk memory  Slow, lots of it  Random access is expensive  Sequential scan is reasonably cheap Main memory Disk Memory

37. 37 Data Streams (one application) Given a restriction on number of random accesses to disk memory How much main memory is required ? For computations such as join of tables Answer: a lot  [Beame, Jayram, R. STOC 2007] Open question: computing other functions? Main memory Disk memory