Foundations of Data Science

Multiplication is modulo an irreducible polynomial. Thus
(x 7 + x 5 + x)(x 6 + x 5 + x 4 ) = x 13 + x 12 + x 11 + x 11 + x 10 + x 9 + x 7 + x 6 + x 5
= x 13 + x 12 + x 10 + x 9 + x 7 + x 6 + x 5
= x 6 + x 4 + x 3 + x 2 mod x 8 + x 4 + x 3 + x + 1
Division of x 13 + x 12 + x 10 + x 9 + x 7 + x 6 + x 5 by x 6 + x 4 + x 3 + x 2 is illustrated below.
x 13 +x 12 +x 10 +x 9 +x 7 +x 6 +x 5
−x 5 (x 8 + x 4 + x 3 + x 2 + 1) = x 13 +x 9 +x 8 +x 6 +x 5
x 12 +x 10 +x 8 +x 7
−x 4 (x 8 + x 4 + x 3 + x 2 + 1) = x 12 +x 8 +x 7 +x 5 +x 4
x 10 +x 5 x 4
−x 2 (x 8 + x 4 + x 3 + x 2 + 1) = x 10 x 6 +x 5 x 3 x 2
12.9.3 Hash Functions
Universal Hash Families
ADD PARAGRAPH ON MOTIVATION integrate material with Chapter
Let M = {1, 2, . . . , m} and N = {1, 2, . . . , n} where m ≥ n. A family of hash functions
H = {h|h : M → N} is said to be 2-universal if for all x and y, x ≠ y, and for h chosen
uniformly at random from H,
x 6 +x 4 +x 3 +x 2
P rob [h (x) = h (y)] ≤ 1 n
Note that if H is the set of all possible mappings from M to N, then H is 2-universal. In
fact P rob [h (x) = h (y)] = 1 . The difficulty in letting H consist of all possible functions
n
is that a random h from H has no short representation. What we want is a small set H
where each h ∈ H has a short representation and is easy to compute.
Note that for a 2-universal H, for any two elements x and y, h(x) and h(y) behave as
independent random variables. For a random f and any set X the set {f (x) |x ∈ X} is
a set of independent random variables.
12.9.4 Application of Mean Value Theorem
The mean value theorem states that if f(x) is continuous and differentiable on the
interval [a, b], then there exists c, a ≤ c ≤ b such that f ′ (c) = f(b)−f(a) . That is, at some
b−a
point between a and b the derivative of f equals the slope of the line from f(a) to f(b).
See Figure 12.9.4.
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