We covered a lot!

Algorithms

What is an Algorithm?

Basic Algorithms

• Find the largest/smallest number in a list
• Sum the numbers in a list
• Find the even numbers in a list
• Making change (greedy algorithm)

Searching and Sorting

• Non-recursive
• Linear search and binary search
• Bubble sort / Insertion sort

Growth of functions

• Big O notation
• Upper bound on the function
• O(1) < O(log n) < O(n^k) < O(bⁿ)
• Other notations

Complexity of Algorithms

• Time Complexity
• Worst case complexity
• Counting the number of operations.
• What is the complexity of linear search? Binary search? Sorting algorithms?

Data Representations

Numbers in other Bases

• Binary, Octal, and Hexadecimal
• How do we convert these to decimal?
• How do we convert numbers from decimal?
• Signed and unsigned integers

Numbers in other Bases

• Binary, Octal, and Hexadecimal
• How do we convert these to decimal?
• How do we convert numbers from decimal?
• Signed integers

Adding and Subtracting

• How do we do these in base 10?
• Translate this process into any base.

Modulo Arithmetic

• When we divide two integers, we get a quotient and a remainder
• Modulo congruence (8 mod 12 = 20 mod 12)
• There are multiple theorems associated with congruence

Primes

Prime Numbers

• Algorithm for determining if a number is prime
• Sieve of Erastothenes

Greatest Common Divisor

• Can break down into prime factors
• Use the Euclidean Algorithm
• Let a = bq + r, where a, b, q, and r are integers. Then gcd(a, b) = gcd(b, r).

Induction

Inductive Proofs

• Basis step and the Inductive step (P(k) → P(k + 1) )
• Strong Induction
  [P(b) ^ P(b + 1) ^ P(b + 2) ... P(k)] -> P(k + 1)

Recursive Definitions

• Define a function in terms of itself
• Base case, inductive case
• Consider the Fibonacci Sequence
• Strings

Recursive Algorithms

• Thinking recursively
• Define linear/binary search recursively
• Recursive Euclidean Algorithm
• Sorting (Merge Sort and Quick Sort)

Counting

Basic Counting

• Product Rule
• Sum Rule
• Inclusion-Exclusion Principle
• Pigeonhole Principle

Permutations and Combinations

• Permutations are an ordered subset of a set
• Combinations are an unordered subset of a set
• Consider when order matters and when it does not.

With Replacement

• Permutations are straight product rule.
• Combinations with replacement are trickier
  C(n + k - 1, k) = (n + k - 1)! / k! (n - 1)!
• Permutations with indistinguishable objects