meljun cortes algorithm_brute force algorithm design technique

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Brute Force Algorithm Design Techniqu

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Design and An

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IC TITLE: Brute Force Algorithm Design Techni

i f ic Object ives:

e end of the topic session, the students are expect

itive:

. Define Brute Force algorithm design technique.

. Identify the importance of Brute Forcetechnique in solving different problem types.

. Explain how Brute Force algorithm design tecto various sorting and searching algorithms.

tive:

. Listen to others with respect.

. Participate in class discussions actively.

ERIALS/EQUIPMENT:

topic slidesOHP

IC PREPARATION:

Have the students research on the following:

• Definition of Brute-Force algorithm

• Applications of Brute-Force algorithm

• Difference between Selection and Bub

Brute-Force algorithm• Importance of Exhaustive search

The research work should be in a printed cofolder with a title page.Provide sample problems that can be solveForce algorithm design technique. It is imperative for the instructor to incorporateteaching strategies while discussing the sugginstructor may use the suggested learning afacilitate a thorough and creative discussion ofPrepare the slides to be presented in the class.Prepare seatwork for the students to apply thin Brute Force algorithm design technique.

IC PRESENTATION:

topic will cover the application of Brute Forceique. The following is the suggested flow of di

se topic:

. Ask the students to share their research aalgorithm design technique.

. Define Brute Force algorithm design techniquits importance.

. Ask the students to cite the different sortinalgorithms they were able to read about.

. Discuss the different sorting and searchingBrute Force algorithm design technique coulsimulate sample problems.

lysis of Algorithm

* Property o f STI

Page 1 of 12

ue

d to:

lgorithm design

hnique is applied

le sort using the

py compiled in a

by using Brute

various kinds ofsted topics. Thetivities below to

he topic.

lessons learned

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and enumerate

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lgorithms wherebe applied and


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Brute Force Algorithm DesignTechniquePage 1 of 20

BruteForce AlgorithmDesign Technique

Designand Analysisof Algorithm

* Property of STI

Page1of 20

Bru t e Fo r c e A lgo r i t hm

Des ign Techn iqu e

c Definition of Brute Force algorithm designtechnique

c Importance of Brute Force algorithm design

technique

c Application of Brute Force algorithm design

technique in the following:

q Selection sort

q Bubble sort

q Sequential search

q String matching

q Exhaustive search

c Sample problem using the application of

Exhaustive search

Brut

Thedesi

[Bru

What is Brute Force AlgorithmDesign Technique?Page 2 of 20



* Property of STI

Page2of 20

Wha t i s B r u t e Fo r c e

A l g o r i t hm De si g n

Techn ique?

c a straight forwardapproach to solvinga problem,usuallydirectly based on the problem statementanddefinitionof theconcepts involved.[LEV07]

c a simplealgorithmdesign techniquewhich worksthrough every possible answer until the correctone is found




* Property of STI

Page3of 20

Wha t i s B r u t e Fo r c eA l g o r i t hm De si g n

Techn ique?

c Sample Application Problem:

q Finding all of the factors for number 625

c Possible Solution to the Problem:

q Divide everynumber up to 624 into 625

q For each number without remainders, thenumber can be considered as a factor of 625.

Wha

Brutsolvidefin

It isposspattethatthe lthat

Appligivethink624 ibe cthealgocomtrieddividnotic

Althhaveimpo

[Wh20]

Design and An

e

e Force Algorithm Design Technique

e are the topics to be discussed under Bruten technique.

Definition of Brute Force algorithm design techn

Importance of Brute Force algorithm design tec Application of Brute Force algorithm design

following:o Selection sorto Bubble sort

o Sequential searcho String matchingo Exhaustive search

Sample problem using the application of Exhau

te Force Algorithm Design Technique, Page 1 o

t is Brute Force Algorithm Design Technique?

Force algorithm design technique is a straight forng a problem, usually directly based on the probleition of the concepts involved [LEV07].

a simple algorithm design technique which worible answer until the correct one is found. Lookirns, regularities or organizational tricks will help diare more intelligent, subtle and efficient. Howeveeast desirable choice because of the enormousare involved.

ication of Brute Force algorithm can be figured outthe problem of finding all of the factors for numb

, that is very easy to answer by simply dividing evinto 625, and for each number without remaindersnsidered as a factor of 625. If that was how you

problem, you would be taking the approachithm. The algorithm may start from dividing 625s out evenly, then divide it by 3, then 4, and so foevery possible divisor, you definitely would k

ed 625 evenly. However, while you are working we that there might be a faster way to do it.

ugh brute force programming is not particularlya legitimate place in software engineering. The

rtance of using Brute Force algorithm design techn1. It always returns the correct result.2. It is useful for testing the accuracy of faster alg3. Sometimes a particular problem can be solved

brute force method that it doesn't make sendevising a more elegant solution.

4. It is applicable to a very wide variety of pr sorting, searching, string matching.

5. It yields reasonable algorithms of at least sowith no limitation on instance size.

t is Brute Force Algorithm Design Techniqu

lysis of Algorithm

* Property o f STI

Page 2 of 12

Force algorithm

ique

niquetechnique in the

stive search

20]

ard approach tom statement and

s through everyng for underlyingcover algorithmsr, it is frequentlynumber of steps

in case, you arer 625. You might

ery number up to, the number canroposed to solvef a brute forcey 2 and see if itth. After you hadow which ones

ith this you might

elegant, it doesfollowing are theique:

orithms.so quickly with ae to waste time

oblems, such as

e practical value

?, Pages 2-4 of


3/12





* Property of STI

Page4of 20

Wha t i s B r u t e Fo r c e

A l g o r i t hm De si g n

Techn ique?c Importance of using Brute Force algorithm

design technique

q Always return the correct result

q Useful for testing the accuracyof fasteralgorithms

q Sometimes a particular problem can be solvedso quicklywith a brute force method that itdoesn't make sense to waste time devising amore elegant solution.

q Applicable to a verywide varietyof problems

• sorting, searching, string matching.

q It yields reasonable algorithms of at leastsome practical value with no limitation oninstance size.

Brute Force Algorithm inSelection SortPage 5 of 20



* Property of STI

Page5of 20

B r u t e F o r c e A l g o r i t hm

i n Se l e ct i o n So r t

c Sorting

q data are arranged in according to their values

c Selection sort:

q the smallest value in the unsorted portion of alist is selected and placed at the end of thesorted portion of the list

ApplSele

Recscieimpoarrainfor num

Selethesortesorte

The

The

Design and An

e

ication of Brute Force Algorithm Designction Sort

ll that sorting is one of the most familiar applicatce where data are arranged according to their vrtant because we are bounded by data, if thged properly, we would spend hours trying

mation. Imagine the complexity of finding somber in a telephone book that is not in order.

ction sort is a type of sorting algorithm in which thensorted portion of a list is selected and placedd portion of the list. The steps are then repeatd all of the data.

igure below illustrates the process of Selection So

Figure 6.1 Flowchart for Selection Sort Proces

flowchart shows the process of selection sort whic

lysis of Algorithm

* Property o f STI

Page 3 of 12

Technique in

ions in computerlues. Sorting isdata were notto find single

one’s telephone

smallest value int the end of thed until we have

t algorithm:

s

starts by finding


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* Property of STI

Page6of 20



c Pseudo code for Selection sort algorithm




* Property of STI Page7of 20



c Example of Selection sort algorithm

q Given the original unsorted list:

q Solution:

24 79 46 9 33 57

After Pass 1After Pass 1

thewiththe c

To bsele

Pse

Sortarralist:

Exa

To fexa

33, a

Eacthe esidealgounsoexchnum

After

///

Design and An

e

mallest value from the given array list then exchathe current element in the sorted list. The processontent of the array is sorted.

etter understand the process, the pseudo codetion sort algorithm is presented below.

do code for Selection sort algorithm

List[0…last] by selecting smallest element in unand exchanging it with element at the beginnin

ple of Selection sort algorithm

urther understand Selection sort, let us considple. For instance, we have an original unsorted li

nd 57 as illustrated below.

24 79 46 9 3

Figure 6.2 Original list in the sample application o

array illustrates the repetition of the algorithm anlement in the list where the right side is the unsortis the sorted list. As stated in the definitionithm, the sorting process will start by selecting srted portion of the array which in this case, thange it with element at the beginning of theer 24.

24 79 46 9 3

Figure 6.3 Content of the list before Pass 1 in the sam

the first pass, the list will look like the following fig

9 79 46 24 3

Figure 6.4 Content of the list after Pass 1 in the samp

LGORITHM SelectionSort(List[0..n - 1])/The algorithm sorts a given array by selection sort/Input: An array List[0..n - 1] of orderable elements/Output: Array List[0..n - 1] sorted in ascending ord

1 for iß 0 to n - 2 do

1 minß i2 for jß i + 1 to n - 1 do

1 if List[j ]


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* Property of STI

Page8of 20



c Analysis of algorithmof Selection sort usingtheBrute Force approach

q the algorithm’s basic operation is the keycomparisonList [ j] < List [min ]

q the number of times it is executed dependsonlyon the array’s size

q selection sort is a O(n 2 ) algorithm on all inputs

q the number of keyswaps is onlyO(n) or, moreprecisely,n -1

Thefollo

Afte

Afte

Afte

Afte

As ypassusenum

Anal

The

inpuoperthe swe cTheeachposit

[Bru

ApplSort

Bubof elunsodata,posit

The

Design and An

e

algorithm will be repeated until the list is sorted aing figures:

Pass 2

9 24 46 79 3

Pass 3

9 24 33 79 4

Pass 4

9 24 33 46 7

Pass 5

9 24 33 46 5

igure 6.5 Content of the list after Pass 2 to Pass 5 in the sam

ou study the figure, you will see that the arrayes, one less than the number of elements in the a

loop to control the sorting, our loop has one lesser of elements in the array.

sis of algorithm of Selection sort using the Brute F

analysis of Selection sort is straightforward whi

’s size is given by the number of elements n. Theation is the key comparison List [ j] < List [min.. Theorting algorithm is executed depends on the array’an simply say that a selection sort is a O(n

2 ) algori

number of key swaps is only O(n) or, more precirepetition of the i loop). This property distinguis

ively from many other sorting algorithms.

te Force Algorithm in Selection Sort, Pages 5-8

ication of Brute Force Algorithm Design Tech

le sort is a type of sorting algorithm that comparesements of the list. Swapping is necessary if the prted and steps are then repeated until we have, this shows the ‘bubbling up’ of the largest eleion on the list.

igure below illustrates the process of Bubble Sort

lysis of Algorithm

* Property o f STI

Page 5 of 12

illustrated in the

57

57

57

79

le Selection sort

sorted after fiverray. Thus, if weiteration than the

orce approach

h considers the

algorithm’s basicnumber of times

s size. Therefore,thm on all inputs.ely, n-1 (one fores selection sort

of 20]

nique in Bubble

the adjacent pairaired elements issorted all of thement to the last

lgorithm:


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Brute Force Algorithm in BubbleSortPage 9 of 20



* Property of STI

Page9of 20


i n B u b b l e So r t

c Bubble Sort

q compares the adjacent elements of the listand swap if elements are out of order




* Property of STI

Page10of 20

c Pseudo code for Bubble sort algorithm



Pse

Exa

To fexalist:

Eacthe

AL//T//In//O

Design and An

e

Figure 6.6 Flowchart for Bubble Sort Proc

do code for Bubble sort algorithm

ple of Bubble sort algorithm

urther understand Selection sort, let us considple as illustrated below. For instance we have an4, 79, 46, 9, 57 and 33 as illustrated below.

24 79 46 9 5

Figure 6.7 Original list in the sample application of Bub

array illustrates the repetition of the algorithm anlement in the list where the right side is the sorte

GORITHM BubbleSort (L ist [0..n - 1] ) he algorithm sorts array List [0..n - 1] by bubble sortput: Array List [0..n - 1] of orderable elementsutput: Array List [0..n - 1] sorted in ascending order

1 for iß 0 to n - 2 do1 for jß 0 to n - 2 - i do

1 if List [ j + 1]


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* Property of STI

Page11of 20

c Example of Bubble sort algorithm


q Solution:

After Pass 1






* Property of STI

Page12of 20

c Analysis of algorithmof Bubble sort usingtheBrute Force approach

q the outer loop runs n times and the onlycomplexityin this analysis in the inner loop

q the number of keycomparisons for the bubblesort version given above is the same for allarrays of size n






* Property of STI

Page13of 20

c Analysis of algorithmof Bubble sort usingtheBrute Force approach

q The number of keyswaps depends on theinput.

q For the worst case of decreasing arrays, it isthe same as the number of keycomparisons.

B r u t e F o r c e A l g o r i t hmi n B u b b l e So r t

sidecomnumwillelem

F

Sincsamlargeillust

Theelemuntil

Anal

Clea

analruns,thantime

Theis th

Thedecr

Design and An

e

is the unsorted list. Looking at the original listaring the adjacent element in the list, which iers 24 and 79. Since these are already sorted, throceed to the next element to be compared to tent.

24 79 46 9 5

gure 6.8 Content of the list after the first comparison of adjac sample Bubble sort

e 79 is greater than 46, exchange the position ofprocess should be applied to the preceding el

st element value is placed at the last elemerated below:

24 79 46 9 5

24 46 79 9 5

24 46 9 79 5

24 46 9 57 7

24 46 9 57 3

Figure 6.9 Content of the list after the first ass in the samp

illustration shows that at the end of the first pass tent value 79 is at the sorted position. The stepswe have sorted all of the data.

sis of algorithm of Bubble sort using the Brute For

rly, the outer loop runs n times and the only c

sis in the inner loop. If we think about a single ti, we can get a simple bound by noting that it cann times. Since the outer loop will make the inner, the comparison can't happen more than O(n WL

number of key comparisons for the bubble sort versame for all arrays of size n.

number of key swaps depends on the input. For teasing arrays, it is the same as the number of key

lysis of Algorithm

* Property o f STI

Page 7 of 12

we will start byin this case aree sorting processhe next adjacent

33

nt elements in the

the element andlements until thet of the list as

33

33

33

33

79

le Bubble sort

e position of there then repeated

ce

omplexity in this

e the inner loopnever loop moreloop complete n HV

sion given above

he worst case ofomparisons.


8/12


If aand

somsortsimp

[Bru

Brute Force Algorithm inSequential SearchPage 14 of 20





in Sequen t i a l Sea rch

c Sequential search

q compares successive elements of a given listwith a given search keyuntil either a match isencountered or the list is exhausted withoutfinding a match

c Pseudo code for Sequential search algorithm

Brute Force Algorithm inSequential SearchPage 15 of 20





in Sequen t i a l Sea rch

c Example of Sequential search algorithm


q Solution:

ApplSeq

Seqsuccmatc

findi

Pse

Exa

To fthe f46, 7

Theor thsimu

A//////IO//

Design and An

e

ass through the list makes no exchanges, the listwe can stop the algorithm. Though the new versi

inputs, it is still in O(n2

) in the worst and averais not very good for big set of input. However, ble to code.

te Force Algorithm in Bubble Sort, Pages 9-13

ication of Brute Force Algorithm Designential search

ential search is a type of search algorithmessive elements of a given list with a given searchh is encountered (successful search) or the list is e

g a match (unsuccessful search).

do code for Sequential search algorithm

ple of Sequential search algorithm

rther understand the Sequential search algorithollowing example. For instance we have an original9, 9, 57 and 33 as illustrated below.

24 46 79 9 5

Figure 6.10 Original list in the sample application of Se

most important parameter in searching algorithm ie target, this time we set the value 9 as the target.lates the processing of Sequential search algorith

9 is not equal to 24

LGORITHM SequentialSearch2(List [0..n], K)

he algorithm implements sequential search withsearch key as a sentinel

nput: An array List of n elements and a search keyutput: The position of the first element in List [0.alue is equal to K or -1 if no such element is found

1 List [n]ß K 2 iß 03 while List [i ] = K do

1 iß i + 12 if i < n return i3 else return -1

lysis of Algorithm

* Property o f STI

Page 8 of 12

has been sortedn runs faster on

e cases. Bubblebble sort is very

f 20]

Technique in

that compareskey until either axhausted without

, let us considerunsorted list: 24,

33

quential Search

s the search keyThe figure below.

K .n - 1] whose


9/12


Thecomtheelemwhicnext

Seqbecagive

Anotsearstopis en Seqappr(infeaver

Appl14-1

ApplMat Strinthe gpartprocor aIn ssubsanytimpo

Design and An

e

24 46 79 9 5

24 46 79 9 5

24 46 79 9 5

24 46 79 9 5

Figure 6.11 Representation of Sequential search alg

figure illustrates the simulation of searching algoritaring the first element, 24, to the target element,alue is not equal to 9, the comparison will proent which is 46; still the elements are not equal. Sh happen to be the next element. Finally, the targelement which is 9.

ential search is a good application of Bruteuse it simply compares succeeding elements of a search key or target until either a match is encou

her straightforward improvement can be incorporah, if a given list is known to be sorted; searchinged as soon as an element greater than or equal

countered.

ential search provides an excellent illustration ooach with its characteristics, strength (simplicityior efficiency). The efficiency result is linear in

age cases.

ication of Brute Force Algorithm in Sequentiaof 20]

ication of Brute Force Algorithm Design Teching

g matching is the process of finding the exact copiven text. We can simply say that string matching aof our daily life like using a spell checker comssing application software while you are working

signments in school.arching a line of text, we will process it by se

tring or pattern in the text to locate the patterning else. The larger the text that is being searc

rtant is the efficiency of the searching algorithm.

9 is not e ual to 46

9 is eq

9 is not equal to 79

lysis of Algorithm

* Property o f STI

Page 9 of 12

33

33

33

33

orithm

hm. It starts fromwhich is 9. Sinceeed to the nextme goes with 79

et is found at the

Force algorithmlist based on a

tered or not.

ted in sequentialuch a list can beo the search key

f the brute force) and weakness

both worst and

l Search, Pages

nique in String

y of pattern(s) inpplication can be

and in a Wordwith your reports

rching for somer replace it withed for, the more

ual to 9


10/12


Brute Force Algorithm in StringMatchingPage 16 of 20



* Property of STI

Page16of 20


i n S t r i n g Ma t c h i n g

c String matching

q process of finding the exact copyof pattern(s)in the given text

c Pseudo code for Pattern matchingalgorithm




* Property of STI

Page17of 20


i n S t r i n g Ma t c h i n g

c Example of Pattern matchingalgorithm

q Given a string of n characters, find a substringof the text that matches the pattern, that is tofind i—the index of the leftmost character ofthe first matching substring in the text—suchthat:

Solution:




* Property of STI

Page18of 20

B r u t e F o r c e A l g o r i t hmi n S t r i n g Ma t c h i n g

c Analysis of algorithmof Pattern matchingusingthe Brute Force approach

q the algorithm shifts the pattern almost alwaysafter a single character comparison

q the algorithm mayhave to make all mcomparisons before shifting the pattern, andthis can happen for each of the n -m +1 tries

Pse

Exa

Give(m =patteleftm

t

The

Notesinglthepatte

[Bru

A

// // // // // //

Design and An

e

do code for Pattern matching algorithm

ple of String matching algorithm

n a string of n characters called the text and a strinn) called the pattern, find a substring of the text

rn. To put it more precisely, we want to find i ost character of the first matching substring in the t

t i = p0, . . . , t i + j = p j , . . . , t i +m-1 = pm-1:. . . t i . . . t i + j . . . t i +m-1 . . . t n-1 text T

p0 . . . p j . . . pm-1 pattern P

ollowing illustrates the processing of Pattern Matc

Figure 6.12 Pattern matching example

that in this example, the algorithm shifts the pattee character comparison. However, the worst cas

lgorithm may have to make all m comparisons brn, and this can be for each of the n-m + 1 tries.

te Force Algorithm in String Matching, Pages 1

LGORITHM BruteForceStringMatch(T [0..n - 1], P[ The algorithm implements brute-force string matchi nput: An array T [0..n - 1] of n characters represenan array P[0..m - 1] of m characters representing autput: The position of the first character in the tex

irst matching substring if the search is successful-1 otherwise.

1 for iß 0 to n - m do

1 jß 0

2 while j


11/12


Brute Force Algorithm inExhaustive SearchPage 19 of 20

Brute Force Algorithm Design Technique

Design and Analysis of Algorithm

* Property of STI

Page 19of20


in Exhaus t i ve sea rch

c Exhaustive search:

q is a brute-force approach to combinatorialproblem

c Example of Exhaustive search

q Traveling Salesman Problem

• find the shortest tour through a given set of ncities that visits each cityexactlyonce beforereturning to the citywhere it started

• in what order should he visit themto minimizethe distance travelled?

Brute Force Algorithm inExhaustive SearchPage 20 of 20

Brute Force Algorithm Design Technique




in Exhaus t i ve Sea rch

c Determining the tour and the total cost asillustrated in the table below:

App

Exh

ExhFor

imposolut

Sam

Thetheexaccanvertidistacom

nodeupdista

TheD) teachbetwhowairplfor tthe tmusthe stotal

A

A

A

A

A

A

Design and An

e

icat ion of Brute Force Algor i thm Design

ust ive search

ustive search is a brute-force approach to combidistinct problem where the efficient solution is no

rtant to test each possible solution in order toion.

ple problem using Exhaustive search

ollowing example is about the Traveling Salesmanhortest tour through a given set of n cities tha

tly once before returning to the city where it star be conveniently modeled by a weighted graph,

es representing the cities and the edge weighnces. A traveling salesman problem is a deinatorial problem. A salesman spends his time vi

s) cyclically. In one tour he visits each city just ohere he started. In what order should he visit thence travelled?

Figure 6.13 Travel Salesman problem exa

figure illustrates the number of cities represented be salesman need to visit. Some of those cities arother thru airplanes or by road or railway. Ea

een the cities has one or more weights (cost) attamuch it takes to travel the route, for examplene ticket or train ticket. The question is how to fie salesman to go to the cities. The salesman w

ravel costs, as well as the distance he travels as logo to each city exactly once, and at the end, he

tarted in. To do this, let us consider determiningcost as illustrated in the table below:

TOUR

B C D A 3 + 9

B D C A 3 + 4

C B D A 6 + 9

C D B A 6 + 2

D B C A 8 + 4

D C B A 8 + 2

Figure 6.1 Solution for Travel salesman pr

lysis of Algorithm

* Property o f STI

Page 11 of 12

Technique in

natorial problem.t yet defined, it is

identify the best

Problem. To findt visits each cityed. The problemwith the graph’ss specifying theceptively simplesiting n cities (or

nce, and finishesto minimize the

mple

y nodes (A, B, C,e connected withh of those links

ched. The cost is, the cost of and the best routents to keep both

as possible. Heust be in the citythe tour and the

ENGTH

+ 2 + 8 = 22

+ 2 + 6 = 15

+ 4 + 8 = 27

+ 4 + 3 = 15

+ 9 + 6 = 27

+ 9 + 3 = 22

blem


12/12


Nowshocoulexa

cons

[Bru

GEN

REF

Design and An

e

, the salesman has identified different routes for ths three pairs of tours that differ only by their dire

cut the number of vertex permutations by halfple, choose any two intermediate vertices; sa

ider any permutation in which B precedes C.

te Force Algorithm in Exhaustive Search, Page

ERALIZATION:

Brute Force algorithm design technique is aapproach to solving a problem, usually direcproblem statement and definition of the concepThe principal strengths of the brute-forcetechnique are wide applicability and simpliweakness is the subpar efficiency of most brute

Selection sort, sequential search and string m

are examples of Brute-force approach. String matching is the process of finding thpattern(s) in the given text. Exhaustive search is a brute-force approach

problem. For a distinct problem where the efnot yet defined, it is important to test each poorder to identify the best solution.Traveling salesman problem is an example ofalgorithm.

ERENCES:http://www.algorithmist.com/index.php/Exhausti

http://www.cs.utsa.edu/~bylander/cs3343/chapt http://www.c3.lanl.gov/mega-math/gloss/compu http://www.webopedia.com/TERM/B/brute_forc http://simple.wikipedia.org/wiki/Travelling_Sale Emily Somach,(2008, April 21), String Matching

Course Material, Bryn Mawr College Retrievedfrom http://mainline.brynmawr.edu/Courses/cs3ImportantAlgorithms/StringMatching(ESomach)

Anany Levitin,(2007), The design and analysised.), Pearson Education Inc.

lysis of Algorithm

* Property o f STI

Page 12 of 12

e tour. The tabletion. Hence, we. We could, for

B and C then

19-20 of 20]

straight forwardly based on thes involved. algorithm designity; its principal-force algorithm. atching algorithm

e exact copy of

to combinatorial

ficient solution is

ssible solution in

exhaustive search

ve_Search

er3handout.pdf te/brute.html e.htmlman_Problem Algorithm. ADDMay 18, 200930/spring2008/ .pdff algorithm (2nd

meljun cortes algorithm_brute force algorithm design technique

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