A Brief History of
Computer Science
William Klostermeyer
Typical Conversation
A: What do you do?
B: I’m a computer scientist.
A: How come when I’m on screen X
in MS Y.Z I can’t print file Q?
“Computer science is no more about
computers than astronomy is about
telescopes.”
E. Dijkstra (1972 Turing Award winner)
What is computer science?
Sometimes called computing or
computing science.
Not so much about
computers, but
computing.
Areas of Computer Science
Hardware/Architecture
Building chips, machines, devices
Software
Building programs, systems, databases
Theory
Determines what is possible: underlies
all other areas of computing.
History of computer science
Pre-dates modern computers by more
than 2000 years!
Digital computers make it more practical
to compute.
“Computer” was a job title for people
around time of WWII
Computers

Electronic computers created because
of need for them:



Ballistics computations
Codebreaking
Census calculations
Algorithm

Precise set of instructions to solve a
problem.

How to play blackjack:
draw two cards; compute total
while (total < 17)
draw a card
add cards to total
end
Mathematics of Algorithms


Hilbert (1928): “Can every problem be
solved by a mechanical procedure?”
Turing (1936): NO! There exist
problems no computer can solve.
Alan Turing: developed
model of computation
Programs


Computer programs implement
algorithms and are written in a
computer language (Java, C, etc.)
Interested in efficient (FAST)
algorithms/programs
Hard Problems

Hilbert’s 10th problem (1900): Does
there exist an algorithm to find integer
solutions to Diophantine equations?

x + 2y2 = 0 (use quadratic formula)

x6 + y6 = z6
Hilbert’s 10th problem


Matiyasevich proved in 1970 that no
algorithm exists to solve arbitrary
Diophantine equations.
Some problems yield themselves to
algorithms, some don’t. Some yield
themselves to EFFICIENT algorithms!
“Computer science is the systematic
study of algorithmic processes”
From an ACM study (1989)
Ancient Algorithms




Babylonians knew how to approximate
square roots (500 B.C.)
Newton’s method (1600’s) generalizes
this to find zeroes of polynomial
“Numerical” algorithms
Gaussian elimination
Euclid’s Algorithm


300 B.C.
GCD(x, y)
greatest common
divisor of x & y
Still in use today!
GCD Algorithm

Euclid(12, 9) =
Euclid(9, 3) =
Euclid(3, 0) = 3
Euclid(a, b):
if b=0 return b
else return Euclid (b, a mod b)
Prime Numbers

x is prime if nothing divides x evenly
except 1 and x
Some primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31


Big primes useful in cryptography
Sieve of Eratosthenes
250 B.C.
 Algorithm to determine if x is prime:
x=17. List all numbers from 2 to 16.
Start with 2. If 2 does not divide 17,
cross 2 and all multiples of 2 off list.
Go to next uncrossed number on list and
repeat.
If all numbers crossed off, then x is prime.

Primality Testing
Eratosthenes method not
fast for LARGE numbers
(hundreds of digits)
 Fast probabilistic methods
developed in ’70’s, ’80’s
(have a small chance of error)

Breakthrough!
2002: after centuries of searching, a
fast algorithm found (polynomial-time
algorithm, no chance of error)
IIT professor and two undergrads:

Uses of Prime Numbers


Large primes needed in cryptography
(to securely send information over
internet)
RSA encryption algorithm based on
assumption that factoring large integers
into primes is difficult.
RSA Algorithm
1.
2.
3.
Find P and Q, two large (e.g., 1024bit) prime numbers.
Choose E such that E is greater than
1, E is less than PQ, and E and (P1)(Q-1) are relatively prime. E must
be odd.
Compute D such that (DE - 1) is
evenly divisible by (P-1)(Q-1).
RSA cont.
The encryption function is
C = (T^E) mod PQ, where C is the
ciphertext (a positive integer), T is the
plaintext (a positive integer).
2.
Your public key is the pair (PQ, E). Your
private key is the number D. D used in
decrypting C back into T.
No known easy methods of calculating D, P, or
Q given only (PQ, E) (your public key).
1.
Factoring



RSA security based on assumption that
FACTORING large integers is hard.
Note that we can determine if an
integer is prime or not quickly, but
factoring seems to require more work
In other words: can prove an integer is
composite w/o showing factors
Graph Algorithms

Euler 1736: Bridges of Konigsberg
Konigsberg = Graph
Graphs
“Concepts from graph theory may hold
the key for everything…”
Business Week, January 22, 2002
Shortest Routes


Dijkstra’s Algorithm (1959): used by
www.mapquest.com
Faster algorithms found in 1990’s
Traveling Salesman


Salesman wants to visit N cities and
return home
Minimize total distance traveled
TSP Algorithms?



No fast algorithm known to compute
best route for salesman
Computing optimal route for 1000 cities
would take centuries on fast computer
Must settle for near-optimal solutions:
Can get very close to optimal if cities
are in Euclidean plane (Arora 1999)
Map Coloring

Can map be colored with 4 colors so
neighboring regions have different
colors (1850’s)
4-color theorem



Yes! (Appel & Haken 1970’s)
Can be done quickly (1990’s)
But some maps require only 3 colors!
Biological Problems

Find best match for criminal’s DNA
sequence in a database:
AATCCGATAGGAT
ATTCCAGATCGAT
TACCGATAGACAT
GTACAGGCAATCA
AAACCC
GATACAAATCCGA
Sequence Assembly

Assemble small overlapping sequences
into single long sequence
AAAATCGC, CGCATAAA, GCATCTCATT
CGCATAAAATCGCATCTATT
Can be hard if you start with lots of fragments
Easy & Hard

Which is harder?




Taking a test
Grading a test
Proving a theorem
Checking a proof
Hilbert and von Neumann pondered this.
P vs. NP
P: problems that can be solved quickly
(shortest route)
 NP: problems whose solutions can be
checked quickly (TSP, 3-coloring)
Is this a 3-coloring?

P = NP?


Believed P is not equal to NP
$1,000,000 reward for proof:
www.claymath.org
Are certain problems intrinsically hard?
NP-complete problems


Class of hard problems (unless P=NP)
Occur in real world:



Scheduling problems, routing problems,
Biological problems, etc.
Optimal solutions take too long to compute
(we believe, if P not equal to NP)
Must settle for sub-optimal solutions.
Complexity Theory



Interested in categorizing how hard
(complex) problems are.
Key concept is a “proof” that an object
satisfies some conditions.
How might we “prove” that 11 is prime?
See that none of 2, 3, 5, 7 divide 11.
Checking Proofs


How do we grade a test?
How do we check a proof?
xn + yn = zn,
no solutions for n > 2
By reading it! By reading every word of it!
PCP Theorem



1990’s was found that proofs can be
checked (with high probability) by
reading only a few characters.
(Proof must be in a standard form)
“Holographic” proofs: each symbol
contains essence of proof
Amazing connection


PCP Theorem also tells us that for
certain hard problems, finding suboptimal solutions quickly is hard!
Hard to quickly find a route that is close
to the longest possible route between
two cities!
Conclusion

Computer science is about:
Algorithms to solve problems!