Data Structures and Algorithm Analysis in C - SVS

Structures, Algorithm Analysis: CHAPTER 10: ALGORITHM DESIGN TECH...
from this.
Skip lists are similar to hash tables, in that they require an estimate of the number of elements
that will be in the list (so that the number of levels can be determined). If an estimate is not
available, we can assume a large number or use a technique similar to rehashing. Experiments have
shown that skip lists are as efficient as many balanced search tree implementations and are
certainly much simpler to implement in many languages.
10.4.3. Primality Testing
In this section we examine the problem of determining whether or not a large number is prime. As
was mentioned at the end of Chapter 2, some cryptography schemes depend on the difficulty of
factoring a large, 200-digit number into two 100-digit primes. In order to implement this scheme,
we need a method of generating these two primes. The problem is of major theoretical interest,
because nobody now knows how to test whether a d-digit number n is prime in time polynomial in d.
For instance, the obvious method of testing for the divisibility by odd numbers from 3 to
requires roughly divisions, which is about 2 d/2 . On the other hand, this problem is not
thought to be NP-complete; thus, it is one of the few problems on the fringe--its complexity is
unknown at the time of this writing.
Figure 10.61 Before and after an insertion
In this chapter, we will give a polynomial-time algorithm that can test for primality. If the
algorithm declares that the number is not prime, we can be certain that the number is not prime.
If the algorithm declares that the number is prime, then, with high probability but not 100
percent certainty, the number is prime. The error probability does not depend on the particular
number that is being tested but instead depends on random choices made by the algorithm. Thus,
this algorithm occasionally makes a mistake, but we will see that the error ratio can be made
arbitrarily negligible.
The key to the algorithm is a well-known theorem due to Fermat.
THEOREM 10.10.
Fermat's Lesser Theorem: If p is prime, and 0 < a < p, then a p-1 1(mod p).
PROOF:
A proof of this theorem can be found in any textbook on number theory.
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页码,53/83
2006-1-27