Introduction to Algorithms
Week 1: Notes

pronunciation

Note how these are pronounced in English:

x² – "x squared"

x³ – "x cubed"

x⁴ – "x to the fourth"

x¹⁰⁰ – "x to the one hundredth"

xⁿ – "x to the n" (or, less commonly, "x to the nth")

a / b – "a over b"

z – "zee" (American English) or "zed" (British)

π – "pie"

exponentiation

You should be familiar with these identities:

• a^{u + v} = a^u · a^v

• a^{u – v} = a^u / a^v

• a^uv = (a^u)^v

logarithms

You should be familiar with these identities:

• log_b(uv) = log_b u + log_b v

• log_b(u / v) = log_b u - log_b v

• log_b(u^p) = p log_b u

• log_a b · log_b c = log_a c

powers of 10

10^-9 = 1 / 1,000,000,000, prefix = nano (e.g. nanosecond)

10^-6 = 1 / 1,000,000, prefix = micro (e.g. microsecond)

10^-3 = 1 / 1,000, prefix = milli (e.g. millisecond)

10⁰ = 1

10³ = 1,000 = one thousand, prefix = kilo (e.g. kilogram = 1000 g)

10⁶ = 1,000,000 = one million, prefix = mega, (e.g. megawatt)

10⁹ = 1,000,000,000 = one billion, prefix = giga (e.g. gigawatt)

10¹² = 1,000,000,000,000 = one trillion, prefix = tera (e.g. terabyte)

powers of 2

You should know by heart:

• 2⁰ = 1

• 2¹ = 2

• 2² = 4

• 2³ = 8

• 2⁴ = 16

• 2⁵ = 32

• 2⁶ = 64

• 2⁷ = 128

• 2⁸ = 256

• 2⁹ = 512

• 2¹⁰ = 1,024 ≈ 1,000 = 10³ = 1 K = kilo

• 2²⁰ = 1,048,576 ≈ 1,000,000 = 10⁶ = 1 M = mega

• 2³⁰ = 1,073,741,824 ≈ 1,000,000,000 = 1 G = giga

The fact that 2¹⁰ ≈ 10³ allows you to convert between powers of 2 and 10 easily, at least approximately. For example, 2²⁶ = 2⁶ · 2²⁰ = 64 M ≈ 64,000,000.

Computers store information as a series of bytes. Every byte stores a number from 0 … 2⁸ – 1 , i.e. 0 ... 255.

Is 1K equal to 1,000 (a decimal power) or 1,024 (a binary power)? For example, is a kilobyte 1,000 bytes, or 1,024 bytes? In practice, both definitions are used sometimes. In many situation the difference does not matter much, since 1,024 is only 2.4% greater than 1,000. Similarly, in some contexts 1M means 1,000,000, and in others it is 1,048,576.

integers

integers = ℤ

… , -2, -1, 0, +1, +2, +3, ...

natural numbers = non-negative integers = ℕ

= 0, 1, 2, 3, 4, ...

The mathematical study of the integers is called number theory.

divisibility

Let a, d, q ∈ ℤ. If a = dq then a is divisible by d. a is a multiple of d. d is a factor of a.

Example: What are all factors of 6?

1, 2, 3, 6, -1, -2, -3, -6

(Division Theorem) For any a, b ∈ ℤ with b > 0, there exist unique q, r ∈ℤ such that

a = q · b + r and 0 ≤ r < b

We write

q = quotient of a and b, denoted by a div b
r = remainder, denoted by a mod b

For example, if a = 17 and b = 5, then

q = 3
r = 2
17 = 3 · 5 + 2

So 17 div 5 = 3, and 17 mod 5 = 2.

Or if a = -17 and b = 5, then

q = -4
r = 3
-17 = -4 · 5 + 3

So -17 div 5 = -4, and -17 mod 5 = 3.

For any a ∈ ℤ,

a mod 2 = 0 <==> a is even
a mod 2 = 1 <==> a is odd

For any a ∈ ℤ, a div 1 = a, and a mod 1 = 0.