atri@cse.buffalo.edu

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

Coding Theory
3

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
4

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

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

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
1 1 1 0 0 1
1 0 0 0 0 1
 Neither of these codes are satisfactory
7

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
8

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 ?
9

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
10

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
11

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)
12

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

Wireless Sensor Networks

Murat Demirbas’ specialty
14

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

One possible solution
 Ask each mote one at a time
Is your temp at least 70F?
In the worst case might have to ask
ALL motes
Is your temp at least 70F?
16

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
17

Back to our ≥ 70F example


The 2threshold 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
18

So are we done?
 The central node can broad/multi-cast
Is your temp at least 70F?
ALL motes
No
Is your temp at least 70F?
19

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

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

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

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

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
24

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
25

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

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

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
28

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

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

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

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
32

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 ]
33

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
34

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

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
36

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
37