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• If is a function from set to set , we state

that fact with the symbolism .• For each element of set , we represent its

assigned element of setas

a

b

c

d

2

1

A B

a

b

c

d

2

1

A B

a

b

c

d

2

1

3

A B

a

b

c

d

2

1

3

A B

Terminology: Given • Domain and Codomain

• Range

• Image

• Pre-image

“Arithmetic” on Functions

• If f1 and f2 are functions whose codomain is the real numbers, then we can define f1+ f2 and f1f2

Image of a SetSuppose and Then

Examples:

One-to-One (Injective) Functions:

is said to be one-to-one (injective) provided that for each whenever then

Onto (Surjective) Functions:

is said to be onto (surjective) provided that

One-to-One Correspondences and Inverse Functions

• A map that is 1-1 and onto, i.e. both injective and surjective, is called a

• Inverse of (when it exists) is the function

If is onto then

If is 1-1 then

If is both 1-1 and onto then

Monotonic Functions on 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

1. Show defined by is a bijection.

2. Let be the set of even integers and be the set of odd integers. Define by Determine if is a bijection.

Composition of Functions

The “Graph” of a Function

• The graph of function is the set of all ordered pairs for which .

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, …

... ,321,16

1,81,4

1,21

• Summations and Summation Notation

The sum of the first terms of is denoted

Reindexing

• Sum of a finite geometric series

𝑆=∑𝑖=0

𝑛

𝑎𝑟 𝑖

• Summation over members of a set

• Double summation

𝑓 : 𝐴→𝑅 ∑𝑖∈ 𝐴

𝑎𝑖

∑𝑖=1

3

∑j=1

4

𝑖− j

• Other useful sums

Cardinality

• Two sets A and B have the same cardinality iff …

• A set A is countable provided …

• (“Aleph nought”)

Countability of the Rationals

3.1 Algorithms

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

Algorithm 3.12. (x = π↓ x to π↑ x)

• Step 1: Label the descending runs through , with the first on the left and the last on the right.

• Step 2: Starting with , label the leftmost element of as . As a group, move all of the elements of leftward until you reach the largest element to the left of that is still smaller than (i.e. move all of directly to the right of the element where > such that j < i and there is no element with k < i such that < < ). Call the new permutation x(2).

• Step 3: Repeat step 2 in x(2) with .

• Step 4: Continue the process with right successive descending runs (i.e. repeat step 2 in x(i) with for successive i’s) until you create x(r). The permutation x(r) is the top element of the congruence class, x(r) = π↑x.

Example:

x= π↓ x =219534768

Pseudo-code

• The style of the text is Pascal-like. Example:

procedure maxOf3(a, b, c: integers)max := aif b > max then max := bif 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

7km4km

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• Suppose we have two functions and

mapping into the set of real numbers, each of whose domain is some subset of the set of real numbers

• We say that “” (“is big-oh of ”) provided there exists a positive constant and a positive real number for which, whenever , then

The meaning of • How do we measure the “rate of growth” of

a function? Not just the rate of growth at a single point, but over its entire domain?

•

• When we say “we are saying that in some general way the rate of growth of is ultimately no worse than that of a constant multiple of

Example

• Show that is where and .

Functions of the Same Order

• Is it possible to have two functions and for which each is “big-oh” of the other?

• Yes, and in that case we say the two functions are 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) = 1g(x) = xg(x) = x2

g(x) = log xetc.

TheoremSuppose is a polynomial of degree , i.e. f(x) = anxn+an-1xn-1+…+a1x+a0 for some series of constants a0 , a1 , …, an . Then f(x) is

Some Important “Ideal Functions”

g(n) = 1g(n) = ng(n) = n2 (and other polynomial functions n3, n4, etc.)g(n) = log ng(n) = n log ng(n) = 2n

g(n) = an (any constant a > 1)g(n) = n!g(n) = nn

Combinations

If f(x) = f1(x)+f2(x) where f1(x) = O(g1(x)) and f2(x) = O(g2(x)), then f(x) is O(max(g1(x),g2(x)).

Example: f(x) = 2x + log x is

If f(x) = f1(x)f2(x) where f1(x) = O(g1(x)) and f2(x) = O(g2(x)), then f(x) is O(g1(x)g2(x)).

Example: f(n) = (log n + 17)n2 is

More Examples

• is

• Find a big-oh upper bound for …(x3+1)log x + (x2+7)(log x)2

Big- and Big-

• We say that is (“big omega of ”) precisely when is

• To say that is (“big theta of ”) is to say that is and is

• In other words, big-theta is a symmetric relationship. It means and are functions of the same order

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”

• Can usually be reduced to a single integer variable, say .

• Examples:– Payroll program– Sort– High-resolution Graphics

• What is depends on the algorithm

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• Suppose we have an algorithm for which the “size

of the problem” is measured by some integer variable .

• Suppose also that a particular critical operation has been chosen.

• Let be defined as the number of times the critical operation is performed for a problem of size .

• An analysis of the time complexity of the algorithm can be performed by studying the nature of the function

Exampleprocedure search(x, a1, a2, …, an: integers)

index := 0i := 1while index = 0 and i ≤ n do

beginif x = ai then index := ii := 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 ai = x. }

Using comparison for equality as the critical operation, what is the worst-case critical operation count f(n)?

procedure search(x, a1, a2, …, an: integers)index := 0i := 1while index = 0 and i ≤ n do

beginif x = ai then index := ii := 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 Exampleprocedure SelectionSort(a1, a2, …, an: integers)

for i:=1 to n-1 dobegin

s := ifor j:=i+1 to n do

if aj < as then s := jswap ai and as

end{The elements a1, a2, …, an 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(a1, a2, …, an: integers)for i:=1 to n-1 do

begins := ifor j:=i+1 to n do

if aj < as then s := jt := ai

ai := as

as := tend

{The elements a1, a2, …, an 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(a1, a2, …, an: integers)for j := 2 to n do begin m:= aj

{ Insert aj into the sorted sequence a1, a2, …, aj-1 } i := j-1 while i > 0 and ai> m do begin ai+1 := ai

i := i-1 ai+1 := m endend{ The sequence a1, a2, …, an 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

Complexity Terminology

Constant Complexity

Logarithmic Complexity

Linear Complexity

Complexity

Polynomial Complexity

, where 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