Theoretical and Experimental DNA Computation (Natural ...

2.8 Summary 43
Table 2.6. Running times of different algorithms for different values of n
n O(n) O(n 2 ) O(2 n ) O(n n )
1 0.000001 seconds 0.000001 seconds 0.000002 seconds 0.000001 seconds
5 0.000002 seconds 0.000025 seconds 0.000032 seconds 0.003125 seconds
10 0.00001 seconds 0.0001 seconds 0.001024 seconds 2.778 hours
50 0.00005 seconds 0.0025 seconds 35.7 years 2.87 × 10 70 years
2.7 P and NP
We can therefore call an algorithm “fast” if the number of steps to solve a
problem of size n is (no more than) some polynomial involving n. We define
the complexity class P to mean the set of all problems (not algorithms) that
have polynomial-time solutions. Therefore, the problem of sorting numbers is
in P , since some solution (e.g., bubble sort) runs in O(n 2 )time,andn 2 is a
polynomial.
By the late 1960s it became apparent that there were several seemingly
simple problems for which no fast algorithms could be found, despite the best
efforts of the algorithms community. In an attempt to classify this set of problems,
Cook observed that in order for a problem to be solved in polynomial
time one should be able (at the very least) to verify a given correct solution in
polynomial time [47]. This observation holds because if we have a polynomialtime
algorithm for a problem and someone gives us a proposed solution, we can
always re-run the algorithm to obtain the correct solution and then compare
the two, in polynomial time.
This led to the creation of the complexity class NP containing decision
problems for which one can verify the solution in polynomial time. Cook also
showed that within NP lies a set of problems that are the hardest of them
all. If a polynomial-time algorithm exists for any one of these problems then
all NP problems can be solved in polynomial time. This fact is known as
Cook’s Theorem [47], and is one of the most profound results in theoretical
computer science. The class of those “hardest” problems in NP is known as
NP-complete problems, of which satisfiability is the archetype.
Soon after Cook’s paper, Karp [89] proved that several interesting problems
could also be shown to be NP-complete. The list of NP-complete problems
has grown considerably since, and the standard text may be found at
[63]. The list includes many problems of great theoretical and practical significance,
such as network design, scheduling, and data storage.
2.8 Summary
In this chapter we provided an introduction to the theory of computer science.
We described the fundamental “building blocks” of computers, logic gates,