Recommended Reading.
•
Goldreich, 1.3.
•
KatzLindell 8.1, 8.2.
•
Cormen, Leiserson, Rivest, Stein.
Introduction to Algorithms
(2nd ed), Ch. 13, 5, 31, 34.
•
Sipser.
Introduction to the Theory of Computation.
(various parts)
1 Algorithms
In this course, we will not use a specic model of computation when talking about algorithms.
An algorithm can be a program for a Turing machine, RAM model, or your favorite model of
computation, as they are roughly equivalent in terms of eciency.
The inputs of an algorithm are strings over an alphabet
Σ
,
Σ
being often
{0,1}
. Hence an
algorithm
A
is a function from
Σ∗
to
Σ?
.
•A
computes
a function
f:{0,1}∗→ {0,1}∗
: on input
x
in
{0,1}∗
,
A
outputs
f(x)∈ {0,1}∗
•A
decides
a language
L⊂ {0,1}∗
: on input
x
,
A
outputs 1 if
x∈L
and 0 if
x /∈L
. (equivalent
to computing a boolean function
f:{0,1}∗→ {0,1}
).
•A
runs in time
T:N→N
: for all
x
,
A(x)
halts after at most
T(kxk)
time units, where
kxk
is the
length
of the string
x
.
•A
runs in polynomial time
if it runs in time
O(nc)
for some constant
c
.
Ecient
(or
feasible
) algorithms are ones that run in polynomial time. (Although the con-
stant
c
is important for practical eciency, the polynomial time eciency criterion has the
advantage that it does not depend on the encoding of the data or the model of computation
(given reasonable choices for these).) An algorithm that does not run in polynomial time is
viewed as
infeasible
.
Note that numbers are encoded using their
binary representation
so if the input is an integer
N
, an algorithm that runs in polynomial time should run in time polynomial in
kNk ≈ log N
.
2 Polynomial time
The complexity class
P
consists of all problems solvable in polynomial time. (Technically, it is a
class of languages)
•
problems seen in CS124: sorting, shortest path, maximum ow, linear programming, ...
•
many arithmetic functions which are important in cryptography.
1
