APPLICATIONS AND
SOLUTIONS FOR COMPLEXITY
Dr. Adam P. Anthony
Lecture 27
NEXT WEEK


Have your presentations ready!
Submit electronic document/project on blackboard
ALL
PROJECTS ARE DUE NEXT
TUESDAY, AT THE START OF
CLASS REGARDLESS OF WHEN
YOU ARE PRESENTING

Presentation Order is posted on the course schedule
Complexity in the News

November 2010 issue
Talks about recent
attempt to prove P 
NP
 http://science.slashdot.o
rg/story/10/08/08/22
6227/Claimed-ProofThat-P--NP
 Talks
about using NPComplete problems to
secure elections
Dealing with NP-Completeness





Not all hope is lost with the class NP!
We typically analyze problems in the worst case
Many cases are much easier, so we can just use a slow
algorithm in this case
Even if can’t solve problems perfectly, we can still seek
something that is close to perfect
An Approximation Algorithm is one that aims to do the
best it can at solving a problem, though it may make
mistakes

Usually we can make a faster algorithm by cutting corners,
or making a lucky guess.
The Set-Cover Problem

Imagine, in a large company that we need:






And we have several vendors pushing products:






Data Storage (DS)
Data Processing (DP)
Data Imaging (DI)
Image Storage (IS)
Image Sharing (IH)
Program A can do: (DS, DP)
Program B can do: (DP, DI, IH)
Program C can do: (IS, IH)
Program D can do: (DS)
Program E can do: (IS)
In large, complex
instances this
problem is NPComplete!
Each product has the same cost, and would equally serve our needs for the
specified items. Which do we choose?
Greedy Algorithms



Remember how nondeterministic machines always
make the ‘right’ choice?
The efficiency is in the fact that they only make one
choice, and throw out all other options
Greedy algorithm: don’t know the ‘right’ choice, but
do know which one looks best at the time
 Not
always the best! (hindsight is 20/20)
 Sometimes we tie! (left to guessing)
Greedy Set Cover

Imagine, in a large company
that we need:






Data Storage (DS)
Data Processing (DP)
Data Imaging (DI)
Image Storage (IS)
Image Sharing (IH)
And we have several vendors
pushing products:





Program A can do: (DS, DP)
Program B can do: (DP, DI, IH)
Program C can do: (IS, IH)
Program D can do: (DS)
Program E can do: (IS)
GreedySetCover:
Repeat
Pick the program
which solves the
most remaining
problems
Until No Problems
Remain
This algorithm isn’t perfect,
but we can prove that it
finds answers that are VERY
close to the best answer.
Leveraging NP-Completeness for
Security


Key property of NP: solving the problem is hard,
but checking the solution is easy!
Imagine a vault with the following lock:
 Give
a particularly difficult instance of the traveling
salesman problem
 If you can solve it, the vault will be opened!


TSP is not a good problem for this task, because
you can’t ‘reverse engineer’ the problem from a
solution
A more popular usage is Integer Factorization
Cryptography

Mostly used in military applications
 Also

Idea: always assume that the enemy will intercept
our messages!
 Can

found in business, love
we still keep them from reading the message?
Cryptography: an ordered process of scrambling
(encrypting) a message such that it can only be
feasibly unscrambled (decrypted) if you possess the
proper knowledge (key)
Traditional Cryptography



Create message, encrypt using key
Deliver message to recipient
Separately deliver the key
 How
do we keep it safe?
Public Key Cryptography






Manufacture 1 key, 1 lock
Instead of the SENDER locking up the message, ask
the RECIPIENT for a lock
Put the message in a case, secure with lock
Send case in regular postal mail
Recipient already has key, so no separate courier
required!
For convenience: manufacture 1000+ locks,
distribute around country
RSA Public Key Encryption System

RSA: A popular public key cryptographic algorithm
Abbreviation of authors’ last names: Rivest, Shamir Adleman
 Relies on the (presumed) intractability of the problem of
factoring large numbers


Public key = 2 numbers, N and e: Used to encrypt
messages


Private key = 2 numbers, N and d: Used to decrypt
messages


Replaces the 1000 locks from last slide
Replaces the key for that lock
N is the same for both public, private key

Mathematical relationship between N,e,d is what makes this
work
Encrypting the Message 10111






Encrypting keys: N = 91 and e = 5
Message: 10111two = 23ten
Encryption Step 1: 23e = 235 = 6,436,343
Encryption Step 2: 6,436,343 ÷ 91 has a
remainder of 4
4ten = 100two
Therefore, encrypted version of 10111 is 100.
12-13
Decrypting the Message 100






Decrypting keys: N = 91, d = 29
Encrypted Message: 100two = 4ten
Decryption Step 1: 4d = 429 =
288,230,376,151,711,744
Decryption Step 2: 288,230,376,151,711,744 ÷
91 has a remainder of 23
Original Message: 23ten = 10111two
Therefore, decrypted version of 100 is 10111.
12-14
Establishing an RSA public key
encryption system

1.
Where did N = 91, e = 5, and d = 29 come from?
Pick two prime numbers p, and q

2.
3.
7 and 13
Multiply them together to get N: 7*13 = 91
e and d can be any numbers that satisfy this
equation:
e*d = k(p-1)(q-1)
4.
Short story: Pick p and q and the rest of the
numbers fall into place, and the system works!
Security of RSA





Remember, d is kept secret, and we can’t decrypt
without it!
But d is computed using p and q!
And n = p*q (Uh Oh!)
A crook can compute d but only if he/she can factor
n into its prime components!
Factoring large integers is NP-Complete
 Easy
for small numbers (7,13)
 Hard for large numbers—RSA uses p,q values with
150+ digits!
A (rejected) Homework Problem

Find the factors of 107,531
 How
can we solve this?
 Maybe we can write a program?
 http://homepages.bw.edu/~apanthon/courses/CSC18
0/slides/PrimeFactors.sb
Parallel Computing

If one computer is too slow, try 2!
 Or

4, or 8, or 16 or 32 or 64 or 128 or 256…
Parallel computing involves the process of solving a
single problem by:
 Breaking
it into smaller pieces
 Solving each smaller piece in parallel
 Putting the small solutions back together

Many computers today have 2-8 separate
processors, giving it the apparent strength of
multiple computers!
Types of Parallel Computing
Distributed Memory




Each processor has its own
private RAM storage
Total RAM in a distributed
computer: Sum of all private
RAM sources
Must request access to
another’s RAM
Advantages:



different processors can’t
interfere with another’s work
Easy to expand
Shared Memory


All processors share a single
source of RAM
To access memory, just send out a
Load/Store command to the Bus


Advantages:



Hard to program
Easier to program, locate data
Millions of example computers
available
Disadvantages:

Disadvantages:

But what if someone else was
using that data???

Memory Bottleneck can serialize
the parallelism
Have to take care in how data is
accessed, used
Cost of Parallel Computing



Question: A problem is split into 3 parts. Each part can be
solved in 5 steps. How long will it take to solve the problem
if we do it in parallel?
Trick question!!
Answer:

Cost of parallelism =
Cost of Splitting Data +
Cost of computing the single largest part +
Cost of reassembling partial solutions


For Shared memory: cost of splitting/combining is very
small, but cost of computing can be much higher
For Distributed memory: cost of splitting/combining is larger,
and cost of computing is (hopefully) faster
Problems with Parallelism

Splitting data is not always possible!


And when it is, it may be tricky
Concurrency problems:
Parallel writes
 Race Conditions
 Parallel Reads are safe



Combining results may be as much work as completing
the non-parallel algorithm
For these reasons—and more—NP-complete problems
remain unsolvable in the infinite case

BUT we may be able to solve larger problems than would
otherwise be possible
Obvious Parallelism


Obvious parallelism occurs when data is in a List
and the task involves executing a simple action on
each item in the list where the action is independent
of actions on all other items in the list
Many vendors now have limited support for obvious
parallelism:
 apply_parallel(
2
1 3
2 4
ADD-1, List)
3 5
4
5
6 6
7 7
8 8
9
Example problems

Which of these are obviously parallel, which are not? Which
could still be made parallel?
Convert a list of Fahrenheit temperatures to Celsius
For each item in a list, compute the product of that number times all
the numbers that appear before it in the list
1.
2.

Take two lists, multiply each corresponding element in each list, then
add the resulting products together.
3.

4.
5.
[2 5 8 9 10] computes [2 10 80 270 2700]
[2 4 6] and [4 5 9] computes 8 + 20 + 54 = 82
Take a list and confirm whether it is sorted
Search dictionary.com and record the definition of each word in a list
Quantum Computing


Bits can’t get any smaller
But electrons can be in multiple
quantum states simultaneously
(“superpositioning”)
qubit: can be in 2 states at once
 2 qubits: 4 states at once
 n qubits: 2n states at once!


 In effect, we can build massively parallel computers!

SciAm Special: How Do Quantum Computers Work?

http://www.youtube.com/watch?v=hSr7hyOHO1Q&feature=related
Images: ams.org