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2.3 Functions

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- 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.

- 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

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- Domain and Codomain
- Range
- Image
- Pre-image

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

Suppose and Then

Examples:

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

- 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

f is increasing provided whenever

f is decreasing provided whenever

f is strictly increasing provided whenever

f is strictly decreasing provided whenever

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.

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

“floor” and “ceiling” functions

- 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, …

- Examples:

The sum of the first terms of is denoted

Reindexing

- Summations and Summation Notation

- Sum of a finite geometric series

- Summation over members of a set
- Double summation

- Other useful sums

- Two sets A and B have the same cardinalityiff …
- A set A is countable provided …
- (“Aleph nought”)

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

Example:

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.

x= π↓ x =219534768

- The style of the text is Pascal-like. Example:
proceduremaxOf3(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}

Input

Output

Definiteness

Correctness

Finiteness

Effectiveness

Generality

Algorithm 1: Finding the Maximum Element in a Finite Sequence

- 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?

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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.

- 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

- 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

- Show that is where and .

- 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.

- 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) = x2

g(x) = log x

etc.

Suppose 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

g(n) = 1

g(n) = n

g(n) = n2 (and other polynomial functions n3, n4, etc.)

g(n) = log n

g(n) = n log n

g(n) = 2n

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

g(n) = n!

g(n) = nn

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 xis

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)n2is

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

- 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

- 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

- Can usually be reduced to a single integer variable, say .
- Examples:
- Payroll program
- Sort
- High-resolution Graphics

- What is depends on the algorithm

- 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.

- 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

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

index := 0

i := 1

while index = 0 and i ≤ n do

begin

if x = ai 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 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 := 0

i := 1

while index = 0 and i ≤ n do

begin

if x = ai 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 ai = x. }

- Both are for the simple linear search algorithm. (As a matter of fact both are .)

procedure SelectionSort(a1, a2, …, an: integers)

for i:=1 to n-1 do

begin

s := i

for j:=i+1 to n do

if aj < as then s := j

swap 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

begin

s := i

for j:=i+1 to n do

if aj < as then s := j

t := ai

ai := as

as := t

end

{The elements a1, a2, …, an are now in ascending order.}

- All analyses are and for Selection Sort

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

end

end

{ The sequence a1, a2, …, an now contains all the original values, but in nondecreasing order. }

- Worst-Case is
- Average-Case is
- “Best-Case” is

- 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.

- 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

- 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