2.3 Functions A function is an assignment of each element of one set to a specific element of some other set. Synonymous terms: function, assignment, map Examples: –Each pixel on this screen is assigned exactly one integer: its color as a mixture of various levels of red, green, and blue. –Each person is assigned exactly one birth mother. –Each non-negative real number is assigned exactly one square root.

Notations and Drawings

a b c d 2 1 AB a b c d 2 1 A B a b c d A B a b c d A B

Domain and Codomain Range Image Pre-image

“Arithmetic” on Functions If f 1 and f 2 are functions whose codomain is the real numbers, then we can define f 1 + f 2 and f 1 f 2

Image of a Set

One-to-One (Injective) Functions: Onto (Surjective) Functions:

One-to-One Correspondences and Inverse Functions

f is increasing provided whenever f is decreasing provided whenever f is strictly increasing provided whenever f is strictly decreasing provided whenever

Examples: Proving functions are 1-1, onto and bijections

Composition of Functions

The “Graph” of a Function

Some Important Functions “floor” and “ceiling” functions

2.4 Sequences and Summations Sequences and sequence notation

Geometric progressions Arithmetic progressions

Finding the pattern… –Examples: 3, 10, 31, 94, … … 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, …

Summations and Summation Notation

Sum of a finite geometric series

Summation over members of a set Double summation

Other useful sums

Cardinality

Countability of the Rationals

3.1 Algorithms “A precise set of written instructions for performing a computation or for solving a problem”

Example: x= π↓ x =

Pseudo-code The style of the text is Pascal-like. Example: procedure maxOf3(a, b, c: integers) max := a if b > max then max := b if c > max then max := c {max now contains the largest of a, b, and c}

Properties of an Algorithm Input Output Definiteness Correctness Finiteness Effectiveness Generality

Algorithm 1: Finding the Maximum Element in a Finite Sequence

Greedy Algorithms A greedy algorithm is a class of algorithm used when a problem can be solved by making a sequence of decisions, and each such decision moves us closer to an overall solution to the problem The greedy algorithm, at any given stage, always makes the decision that moves us closest to that overall solution Do greedy algorithms always produce the “best” solution?

Greedy Algorithm for Travelling from one City to Another start finish 10km 6km 3km 7km 4km 13km 12km

Example: Describe an algorithm that puts the first three terms of a sequence of integers of arbitrary length in increasing order

Example: Describe an algorithm for determining whether a string of n characters is a palindrome.

Example: Devise an algorithm that finds the first term of a sequence of integers that equals some previous term in the sequence.

3.2 The Growth of Functions

Example

Functions of the Same Order

Use a Simpler Function as a “Yardstick” Whenever possible, we want to use as our g(x) function a relatively simple function whose behavior we are quite familiar with. Examples: g(x) = 1 g(x) = x g(x) = x 2 g(x) = log x etc.

Theorem

Some Important “Ideal Functions” g(n) = 1 g(n) = n g(n) = n 2 (and other polynomial functions n 3, n 4, etc.) g(n) = log n g(n) = n log n g(n) = 2 n g(n) = a n (any constant a > 1) g(n) = n! g(n) = n n

Combinations If f(x) = f 1 (x)+f 2 (x) where f 1 (x) = O(g 1 (x)) and f 2 (x) = O(g 2 (x)), then f(x) is O(max(g 1 (x),g 2 (x)). Example: f(x) = 2 x + log x is If f(x) = f 1 (x)f 2 (x) where f 1 (x) = O(g 1 (x)) and f 2 (x) = O(g 2 (x)), then f(x) is O(g 1 (x)g 2 (x)). Example: f(n) = (log n + 17)n 2 is

More Examples

3.3 Complexity of Algorithms Complexity is loosely defined as the degree of sensitivity of an algorithm to the size of the problem to be solved –Time complexity –Space complexity

The “Size of the Problem”

Critical Operations In any algorithm there is usually at least one critical operation, i.e. an operation which is performed at least as often as any other operation appearing in the algorithm Examples: –Comparison –Swap –Arithmetic operation such as +, *, etc.

Operation Count Analysis

Example procedure search(x, a 1, a 2, …, a n : integers) index := 0 i := 1 while index = 0 and i ≤ n do begin if x = a i then index := i i := i+1 end { ‘index’ contains 0 if x is not in the list; otherwise index is the first value of i between 1 and n for which a i = x. } Using comparison for equality as the critical operation, what is the worst-case critical operation count f(n)?

procedure search(x, a 1, a 2, …, a n : integers) index := 0 i := 1 while index = 0 and i ≤ n do begin if x = a i then index := i i := i+1 end { ‘index’ contains 0 if x is not in the list; otherwise index is the first value of i between 1 and n for which a i = x. }

Worst-Case and Average-Case Analysis for Linear Search Both are for the simple linear search algorithm. (As a matter of fact both are.)

Another Example procedure SelectionSort(a 1, a 2, …, a n : integers) for i:=1 to n-1 do begin s := i for j:=i+1 to n do if a j < a s then s := j swap a i and a s end {The elements a 1, a 2, …, a n are now in ascending order.} Using comparisons for order (, ≤, ≥) as the critical operation, what is the worst-case critical operation count f(n)?

procedure SelectionSort(a 1, a 2, …, a n : integers) for i:=1 to n-1 do begin s := i for j:=i+1 to n do if a j < a s then s := j t := a i a i := a s a s := t end {The elements a 1, a 2, …, a n are now in ascending order.}

Worst-Case, Average-Case, and Best- Case Analyses for Selection Sort All analyses are and for Selection Sort

Example 2 procedure BetterInsertionSort(a 1, a 2, …, a n : integers) for j := 2 to n do begin m  := a j { Insert a j into the sorted sequence a 1, a 2, …, a j-1 } i := j-1 while i > 0 and a i > m do begin a i+1 := a i i := i-1 a i+1 := m end { The sequence a 1, a 2, …, a n now contains all the original values, but in nondecreasing order. }

Worst-Case and Average-Case Analysis for Insertion Sort Worst-Case is Average-Case is “Best-Case” is

Commonly Used Terminology for the Complexity of Algorithms ComplexityTerminology Constant Complexity Logarithmic Complexity Linear Complexity Polynomial Complexity Exponential Complexity Factorial Complexity

Classes of Problems Intractable problem: A problem is intractable if there is a mathematical proof that no polynomial algorithm exists for solving it Unsolvable problem: A problem is unsolvable if there is a mathematical proof that no algorithm at all exists for solving it.

Classes P and NP Class P: A problem is in class P if there is a known algorithm that solves the problem in polynomial time Class NP: A problem is in class NP if there is no known polynomial algorithm for solving it, but it is known that a non- deterministic Turing Machine can check the correctness of a potential solution in polynomial time

Class NP-Complete A problem is in class NP-Complete if it belongs to a certain set of NP problems for which, if any one of them is found to be solvable with a polynomial algorithm, then all of them can be solved in polynomial time Examples –Traveling salesman problem –3-coloring problem