Type of Algorithm - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Algorithms

What is an algorithm?

• An algorithm is “a finite set of precise instructions for performing a computation or for solving a problem” - A program is one type of algorithm - All programs are algorithms - Not all algorithms are programs! - Directions to somebody’s house is an algorithm - A recipe for cooking a cake is an algorithm - The steps to compute the cosine of 90° is an algorithm

Algorithm 1: Maximum element

• Given a list, how do we find the maximum

element in the list?

• To express the algorithm, we’ll use

pseudocode

• Pseudocode is kinda like a programming language, but not really

Algorithm 1: Maximum element

• Algorithm for finding the maximum element in

a list:

procedure max ( a 1 , a 2 , …, a (^) n : integers) max := a 1 for i := 2 to n if max < a (^) i then max := a (^) i

{ max is the largest element}

Maximum element running time

• How long does this take?
• If the list has n elements, worst case scenario

is that it takes n “steps”

• Here, a step is considered a single step through the list

Properties of algorithms

• Algorithms generally share a set of properties:
  • Input: what the algorithm takes in as input
  • Output: what the algorithm produces as output
  • Definiteness: the steps are defined precisely
  • Correctness: should produce the correct output
  • Finiteness: the steps required should be finite
  • Effectiveness: each step must be able to be performed in a finite amount of time
  • Generality: the algorithm should be applicable to all problems of a similar form

Algorithm 2: Linear search

• Given a list, find a specific element in the list
  • List does NOT have to be sorted!

procedure linear_search ( x : integer; a 1 , a 2 , …, a (^) n : integers) i := 1 while ( i ≤ n and x ≠ a (^) i ) i := i + 1 if i ≤ n then location := i else location := 0

{ location is the subscript of the term that equals x, or it is 0 if x is not found}

Algorithm 2: Linear search, take 1

11

procedure linear_search ( x : integer; a 1 , a 2 , …, an : integers)

i := 1 while ( i ≤ n and x ≠ ai )

i := i + 1 if i ≤ n then location := i

else location := 0

i := 1 while ( i ≤ n and x ≠ ai )

i := i + 1 if i ≤ n then location := i

else location := 0

a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10

i 23456781

x 3

location 8

Linear search running time

• How long does this take?
• If the list has n elements, worst case scenario

is that it takes n “steps”

• Here, a step is considered a single step through the list

Algorithm 3: Binary search

• Given a list, find a specific element in the list
  • List MUST be sorted!
• Each time it iterates through, it cuts the list in half

procedure binary_search ( x : integer; a 1 , a 2 , …, an : increasing integers) i := 1 { i is left endpoint of search interval } j := n { j is right endpoint of search interval } while i < j begin m := ( i + j )/2 { m is the point in the middle } if x > am then i := m + else j := m end if x = ai then location := i else location := 0 { location is the subscript of the term that equals x, or it is 0 if x is not found}

Algorithm 3: Binary search, take 2

16

a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10

i m j

i := 1 j := n

procedure binary_search ( x : integer; a 1 , a 2 , …, a (^) n : increasing integers) while i < j begin m := ( i + j )/2 if x > am then i := m + else j := m end

if x = ai then location := i else location := 0

i := 1 j := n

while i < j begin m := ( i + j )/2 if x > am then i := m + else j := m end

if x = ai then location := I else location := 0

x 15

location 0

Algorithm 3: Binary search

• A somewhat alternative view of what a binary

search does…

Sorting algorithms

• Given a list, put it into some order
  • Numerical, lexicographic, etc.
• We will see two types
  • Bubble sort
  • Insertion sort

Algorithm 4: Bubble sort

• One of the most simple sorting algorithms
  • Also one of the least efficient
• It takes successive elements and “bubbles” them up the list

procedure bubble_sort ( a 1 , a 2 , …, a (^) n ) for i := 1 to n - for j := 1 to n - i if a (^) j > a (^) j + then interchange a (^) j and a (^) j + { a 1 , …, a (^) n are in increasing order }