Chapter 1 Introduction Definition of Algorithm An algorithm is a finite sequence of precise...

Chapter 1 IntroductionDefinition of Algorithm An algorithm is a finite sequence of precise instructions for performing a computation or for solving a problem.

Example 1 Describe an algorithm for finding the maximum (largest) value in a finite sequence of integers.Solution

1. Set the temporary maximum equal to the first integer in the sequence.

2. Compare the next integer in the sequence to the temporary maximum, and set the larger one to be temporary maximum.

3. Repeat the previous step if there are more integers in the sequence.

4. Stop when there are no integers left in the sequence. The temporary maximum at this point is the maximum in the sequence.

maxreturn

:max then max if

to2for

;:max

) of maximum (themax :Output//

integers ://Input

Sequence Finite ain Element Maximum theFinding 1. Algorithm

1

21

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ii

n

n

aa

n i:

a

,...,a,aa

,...,a,aan

Instead of using a particular computer language, we use a form of pseudocode.

Algorithmic Problem Solving

Understand the problem

Design an algorithm and proper data structures

Analyze the algorithm

Code the algorithm

• Ascertaining the Capabilities of a Computational Device

• Choosing between Exact and Approximate Problem Solving

• Deciding on Appropriate Data Structures

• Algorithm Design

• Algorithm Analysis

• Coding

Important Problem Types

• Sorting

• Searching

• String processing (e.g. string matching)

• Graph problems (e.g. graph coloring problem)

• Combinatorial problems (e.g. maximizes a cost)

• Geometric problems (e.g. convex hull problem)

• Numerical problems (e.g. solving equations )

Fundamental Data Structures

1. Array1. Arraya[1]a[1] a[2]a[2] a[3]a[3] a[n]a[n]

2. Link List2. Link List Head of list1a 2a 3a na

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4

4

7

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3. Binary Tree

1. Stack1. Stack

1a2a

na• Operations a stack supportsOperations a stack supports: last-in-first-out (pop, push) : last-in-first-out (pop, push)

• Implementation: Implementation: using an array or a list (need to check using an array or a list (need to check underflows and overflows)underflows and overflows)

Fundamental Abstract Data Structures

2. Queue

• Operations a queue supports: first-in-first-out (enqueue, dequeue)

• Implementation: using an array or a list (need to check underflows and overflows)

ma 1ma 2ma na

front rear

3. Priority Queue (each elements of a priority queue has a key)

• Operations a priority queue supports: inserting an element, returning an element with maximum/minimum key.

• Implementation: heap

Assignment

(1) Read the sections of the text book about array, linked list, stack, queue, heap and priority queue.

(2)

(i) Implement stack S and queue Q using both of array and linked list.

(ii) Write a main function to do the following operations for S: pop, push 10 times, pop, pop, repeat push and pop 7 times. Do the following operations for Q: dequeue, enqueue 10 times, dequeue, depueue, repeat enqueue and dequeue 7 times. When you use array, declare the size of the array to be 10. Printout all the elements in S and in Q.

1. Graph

• Operations a graph support: finding neighbors

• Implementation: using list or matrix

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d e

c b

f

)},(),,(),,(),,(),,(),,(),,(),,{(

),,,,,(

),(

feedfbceecadcbcaE

fedcbaV

EVG

0 0 0 0 0 0 f

1 0 0 1 0 0 e

0 1 0 0 0 1 d

0 1 0 0 0 0 c

1 0 0 1 0 0 b

0 0 0 1 0 0 a

f e d c b a abcdef

cc fea ef c

Adjacent listsAdjacent matrix

Some Advanced Data Structure

2. Binary Search Tree

Definition 1 Binary Search Tree is a Binary Tree satisfying the following condition:

(1) Each vertex contains an item called as key which belongs to a total ordering set and two links to its left child and right child, respectively.

(2) In each node, its key is larger than the keys of all vertices in its left subtree and smaller than the keys of all the vertices in its right subtree.

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Operations a binary search tree support: search, insert, and delete an element with a given key.

How to select a data structure for a given problem?How to select a data structure for a given problem?

Definition 2

A Dynamic Dictionary is a data structure of item

with keys that support the following basic operations:

(1) Insert a new item

(2) Remove an item with a given key

(3) Search an item with a given key

What data structure is the best?

Example:Example: Select a data structure for supporting Dynamic Select a data structure for supporting Dynamic DictionaryDictionary

Implementation and Empirical Analysis

Challenge in empirical analysis:

• Develop a correct and complete implementation.

• Determine the nature of the input data and other factors influencing on the experiment. Typically, There are three choices: actual data, random data, or perverse data.

• Compare implementations independent to programmers, machines, compilers, or other related systems.

• Consider performance characteristics of algorithms, especially for those whose running time is big.

Can we analyze algorithms that haven’t run yet?

Chapter 2 Fundamentals of the Analysis of Algorithm

Example 1 Describe an algorithm for finding an element x in a list of distinct elements .,...,, 21 naaa

found}not is if 0 isor , equals that term

ofsubscript theis {

;0 else

then if

;1

) and ( while

;1

integers)distinct :,...,integer,:( Procedure

Algorithm.Search linear The 1 Algothm

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location

locatiton:

ilocation:ni

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i:

a,aaxrchlinear sea

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n


ofsubscript theis {

;0 else

then if

end;

; else

1 then if

;2

begin

)( while

;

;1

integers) increasing:,...,integer,:( Procedure

Algorithm.Search Binary The 2 Algorithm

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location

location:

ilocation:ax

mj:

mi:ax

j)/(im:

ji

nj:

i:

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m

n

Linear Search and Binary Search written in C++

1. Linear search int lsearch(int a[], int v, int l, int r) { for (int i=l; i<=n; i++) if (v==a[i]) return i; return -1; }

2. Binary search int bsearch(int a[], int v, int l, int n) { while (r>=l) {int m=(l+r)/2; if (v==a[m]) return m; if (v<a[m]) r=m-1; else l=m+1; } return -1; }

Complexity of Algorithms

Assume that both algorithms A and B solve the problem P. Which one is better?

• Time complexity: the time required to solve a problem of a specified size.

• Space complexity: the computer memory required to solve a problem of a specified size.

The time complexity is expressed in terms of the number of operations used by the algorithm.

• Worst case analysis: the largest number of operations needed to solve the given problem using this algorithm.

• Average case analysis: the average number of operations used to solve the problem over all inputs.


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Algorithm.Search Linear

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locatiton:

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a,aaxrchlinear sea

i

n

12(n+1)

n

2

3n+5

Example 2 Analyze the time complexities of linear search algorithm and binary search algorithm


ofsubscript theis {

;0 else

then if

end;

; else

1 then if

;2

begin

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Algorithm.Search Binary

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location

location:

ilocation:ax

mj:

mi:ax

j)/(im:

ji

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i

m

n

times.n)log(kk most at repeatsIt

.2nLet

2

k

1

1

2

2

11

Number of operations = 4 log n+42

Orders of Growth

Running

Time

necessary operations

Operation

Per second

instant 1 second 11.5 days Never end

days

instant Instant 1 second Never end

days

Running time for a problem with size 610n

2n n2nlg n

6101210

593502

593402

Using silicon computer, no matter how fast CPU will be you can never solve the problem whose running time is exponential !!!

Asymptotic Notations: O-notation

Definition 2.1 A function t(n) is said to be O(g(n)) if there exist some constant . allfor )()(such that 0 and 0 0000 nnngcntnc

)(nt

)(0 ngc

. then , )constant a is 0(c )(

)(lim If O(g(n))t(n)c

ng

ntn

0n n

Example 5

List the following function in O-notation in increasing order:

.2,!,,lg,,,lg 32 nnnnnnnn

Example 3 Prove 2n+1=O(n)

)(51210 Prove 4 Example 22 nOnn

n

nnnnn

n

n

20001.0

,1000lg100lg

,10001005n

functions? following theofoh -big theis What 6 Example

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25


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Algorithm.Search Linear

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Example 7 Analyze the time complexities of linear search algorithm and binary search algorithm asymptotically.

Asymptotic Analysis of algorithms (using O-notation)

Totally(addition): O(n)

times. repeatsIt n


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then if

end;

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;2

begin

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Algorithm.Search Binary

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location:

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m

n

times.n)log(kk most at repeatsIt

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2

k

Totally(comparison): O(log n)

Example 8 Analyze the time complexities of following algorithm asymptotically.

Matrix addition algorithm

Procedure MatricAddition(A[0..n-1,0..n-1],B[0..n-1,0..n-1])

for i=0 to n-1 do

for j=0 to n-1 do

C[i,j] = A[i,j] + B[i,j];

return C;

Repeat n times

Repeat n times

Totally(addition): )( 2nO

Recursive Algorithms

Example 9 Computing the factorial function F(n)=n!.

F(n) can be defined recursively as follows:

nnFnF

F

)1()(

1)0(

Factorial Algorithm

Procedure factorial(n)

if n = 0 return 1

else return factorial(n-1) * n;

Time complexity(multiplication): T(0) = 0

T(n) = T(n-1) +1 when n>0

Algorithm factorial calls itself in its body!

recurrence

Basic Recurrences

n)(n

ncncncnnT

nkkcnnT

icnnT

cnnTcncnncnT

cncnnTcnncnT

cnnTnT

kk

ii

logO

log'log

)log( )2/(2

......

)2/(2

......

3)2/(2))2/()2/(2(2

)2/(2))2/()2/(2(2

)2/(2)(

1

33232

222

Example 10

Solving the following recurrence

T(0) = 1

T(n) = T(n-1) + 1 n>0

T(n) = T(n-1) + 1

= T(n-2) + 1 + 1 = T(n-2) + 2

= T(n-3) + 1 + 2 = T(n-3) + 3

…

= T(n-i) + i

…

= T(n-n) + n

= n

c'T

nnTT nn

1

2/

1 2

recurrence following theSolve

11 Example

Example 12 Solve the recurrence

1

1 )1()(

1

T

nnnTnT

2/)1(

...321

)1()2(...32

......

)1()2()3(

)1()2(

)1()(

1

nn

n

nnnT

nnnnT

nnnT

nnTnT).log where,2 is,That

.2 ofpower a isn that (Assume

1)1(

1n 1)2/()(

recurrence theSolve 13 Example

nkn

T

nTnT

k

n

nkknT

inT

nT

nT

nTnT

k

i

log1

)log( )2/(

......

)2/(

......

111)2/(

11)2/(

1)2/()(

3

2

Chapter 1 Introduction Definition of Algorithm An algorithm is a finite sequence of precise...

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