Analysis of Algorithms - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

CS 2610

September 23, 2008

2

Algorithms

An Algorithm is a finite set of precise instructions

for performing a computation or for solving a

problem.

Properties of an algorithm:

• input: input values are from a specified set
• output: output values are from a specified set
• definiteness: each step is precisely defined
• correctness: correct output produced
• finiteness: takes a finite number of steps
• effectiveness: each step is finite & exact
• generality: applicable to various input sizes

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Pseudocode

procedure procName(argument: type)

variable := expression

informal statement

begin

statements

end

{ comment }

5

Pseudocode

if condition then statement

[ else statement2]

for variable := initial value to final value

statement

while condition statement

return expression

procName(arg1,..,argn)

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Example: Max Algorithm

procedure max( a 1 , a 2 , …, an: integers)

v := a 1 1

for i := 2 to n n-1 x

if ai > v then 1

v := ai? (max 1)

return v 1

How many times is each step executed?

Worst-Case: the input sequence is a strictly

increasing sequence

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Searching Algorithms

Searching Algorithms Problem:

Find an element x in a list a 1 ,…, an (not necessarily

ordered)

Linear Search Strategy:

Examine the sequence one element after another

until all the elements have been examined or the

current element being examined is the element x.

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Example: Linear Search

Average Case:

 x is the first element

 1 loop comparison

 x is the second element

 2 loop comparisons, 1 iteration of the loop

 x is the third element

 3 loop comparisons, 2 iterations of the loop

 x is nth element

 n loop comparisons, n-1 iterations of the loop

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Binary Search

Problem: Locate an element x in a sequence of

sorted elements in non-decreasing order.

Strategy: On each step, look at the middle

element of the remaining list to eliminate half of

it, and quickly zero in on the desired element.

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Binary Search

Suppose n=2k

The loop is executed k times, where k=log 2 (n)

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Linear Search vs. Binary Search

Linear Search Time Complexity:

 T(n) is O(n)

Binary Search Time Complexity:

 T(n) is O(log2n)

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Bubble Sort

Smallest elements “float” up to the top (front) of

the list, like bubbles in a container of liquid

Largest elements sink to the bottom (end).

See the animation at:

http://math.hws.edu/TMCM/java/xSortLab

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Example: Bubble Sort

procedure bubblesort( a 1 , a 2 , …, an : distinct integers) for i = 1 to n- for j = 1 to n-i if ( aj > aj+1 ) then swap aj and aj+

Worst-Case : The sequence is sorted in decreasing order

At step i •The loop condition of the inner loop is executed n – i + 1 times. •The body of the loop is executed n – i times

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Insertion Sort

procedure insertSort( a 1 , a 2 , …, an : distinct integers) for j=2 to n begin i=j - 1 while i > 0 and ai > ai+ swap ai and ai+ i=i- end

Worst-Case: The sequence is in decreasing order At step j, the while loop condition is executed j times the body of the loop is executed j-1 times