Prime Numbers

348 Chapter 7 ELLIPTIC CURVE ARITHMETIC
is empty? And if nonempty, what is the precise intersection A ∩ B? Anaive
method is simply to check A1 against every Bi, then check A2 against every
Bi, and so on. This inefficient procedure gives, of course, an O(N 2 ) complexity.
Much better is the following procedure:
(1) Sort each list A, B, say into nondecreasing order;
(2) Track through the sorted lists, logging any comparisons.
As is well known, the sorting step (1) requires O(N ln N) operations
(comparisons), while the tracking step (2) can be done in only O(N)
operations. Though the concepts are fairly transparent, we think it valuable
to lay out an explicit and general list-intersection algorithm. In the following
exposition the input sets A, B are multisets, that is, repetitions are allowed,
yet the final output A ∩ B is a set devoid of repetitions. We shall
assume a function sort() that returns a sorted version of a list, having
the same elements, but arranged in nondecreasing order; for example,
sort({3, 1, 2, 1}) ={1, 1, 2, 3}.
Algorithm 7.5.1 (Finding the intersection of two lists). Given two finite
lists of numbers A = {a0,...,am−1} and B = {b0,...,bn−1}, this algorithm
returns the intersection set A ∩ B, written in strictly increasing order. Note
that duplicates are properly removed; for example, if A = {3, 2, 4, 2},B =
{1, 0, 8, 3, 3, 2}, thenA ∩ B is returned as {2, 3}.
1. [Initialize]
A = sort(A); // Sort into nondecreasing order.
B = sort(B);
i = j =0;
S = {}; // Intersection set initialized empty.
2. [Tracking stage]
while((i