Algorithms - Chapter 3 Bruteforce

8/14/2019 Algorithms - Chapter 3 Bruteforce

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Theory of Algorithms:

Brute Force

Michelle Kuttel and Sonia

Berman

(mkuttel | [email protected])


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Objectives

To introduce the brute-force mind set

To show a variety of brute-force solutions:

Brute-Force String Matching

Polynomial Evaluation

Closest-Pair

Convex-Hull

Exhaustive SearchTo discuss the strengths and weaknesses of abrute-force strategy


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Brute Force

A straightforward approach usuallydirectly based on problem statementand definitions

Motto: Just do it!Crude but often effective

Examples already encountered:

Computing an (a > 0, na nonnegative

integer) by multiplying atogether ntimes

Computation of n! using recursion

Multiplying two nby n matrices

Selection sort


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Brute-Force Sorting

What is the most straight-forward

approach to sorting?

selection sort?

(n-1)n/2 O(n2) comparisons

however, number of key swaps is n-1 O(n)

Bubble sort?

(n-1)n/2 - O(n2) comparisons number of key swaps depends on input

Worst case?

4


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Brute Force String

MatchingProblem:

Find a substring in some text that matches a pattern

Pattern: a string of mcharacters to search for

Text: a (long) string of ncharacters to search in

1. Align pattern at beginning of text

2. Moving left to right, compare each character of patternto the corresponding character in text UNTIL

All characters are found to match (successful search);

or A mismatch is detected

3. WHILE pattern is not found and the text is not yetexhausted, realign pattern one position to the right andrepeat step 2.


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Brute-Force String

Matching

Example:Pattern: AKA

Text: ABRAKADABRA

Trace: AKA AKA

AKA

AKA

Number of Comparisons:In the worst case, mcomparisons beforeshifting, for each of n-m+1tries

Efficiency: !(nm)


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Brute Force Polynomial

EvaluationProblem:

Find the value of polynomialp(x) = anx

n+ an-1xn-1 + +a1x

1 + a0

at a pointx=x0

Algorithm:

Efficiency? Improvements?


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Brute Force Polynomial

EvaluationProblem:

Find the value of polynomialp(x) = anx

n+ an-1xn-1 + +a1x

1 + a0

at a pointx=x0

Algorithm:

Efficiency? Improvements?

p"0.0

fori"n down to0 do

power"1

for j"1 toi do power"power * x

p"p + a[i] * power

returnp


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Brute Force Closest Pair


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Problem:Find the two points that are closest together in a set of n2-D points P1= (x1, y1), , Pn= (xn, yn)

Using Cartesian coordinates and Euclidean distance


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Algorithm:


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Algorithm:


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Algorithm:


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Algorithm:


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Algorithm:


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Algorithm:


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Algorithm:

Efficiency: !(n2)


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Algorithm:

Efficiency: !(n2)

dmin""

fori"1 ton-1 do forj"i+1 ton do d"sqrt((xi - xj)

2+ (yi- yj)2)

ifd < dmin dmin"d; index1"i; index2"j

returnindex1, index2


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The Convex Hull Problem

Problem:

Find the convex hull enclosing n2-D points

Convex Hull: If S is a set of points then the Convex Hull of Sis the smallest convex set containing S

Convex Set: A set of points in the plane is convex if for anytwo points P and Q, the line segment joining P and Qbelongs to the set

ConvexNon-

Convex


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Problem:




ConvexNon-

Convex


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Problem:




ConvexNon-

Convex


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Problem:




ConvexNon-

Convex


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Problem:




ConvexNon-

Convex


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Brute Force Convex Hull

Algorithm:For each pair of points p1and p2

Determine whether all other points lie to the

same side of the straight line through p1and p2Efficiency:

for n(n-1)/2point pairs, check sidedness of (n-2)others

O(n3

)

P1

P2

P3

P4

P5

P6

P7

P8

P9


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Pros and Cons of Brute

Force! Strengths:

Wide applicability

Simplicity

Yields reasonable algorithms for some important problems

and standard algorithms for simple computational tasks

A good yardstick for better algorithms

Sometimes doing better is not worth the bother

" Weaknesses:

Rarely produces efficient algorithms

Some brute force algorithms are infeasibly slow

Not as creative as some other design techniques


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Exhaustive Search

Definition:A brute force solution to the search for an element witha special property

Usually among combinatorial objects such a

permutations or subsetssuggests generating each and every element of theproblems domain

Method:1. Construct a way of listing all potential solutions to the

problem in a systematic manner All solutions are eventually listed

No solution is repeated

2. Evaluate solutions one by one (disqualifying infeasibleones) keeping track of the best one found so far

3. When search ends, announce the winner


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Travelling Salesman

Problem

Problem:Given ncities with known distances between each pair

Find the shortest tour that passes through all the citiesexactly once before returning to the starting city

Alternatively:Find shortest Hamiltonian Circuit in a weighted connectedgraph

Example: a b

c d

8

2

7

53

4


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Travelling Salesman by

Exhaustive Search Tour Cost .

a#b#c#d#a 2+3+7+5 = 17

a#b#d#c#a 2+4+7+8 = 21

a#c#b#d#a 8+3+4+5 = 20

a#c#d#b#a 8+7+4+2 = 21

a#d#b#c#a 5+4+3+8 = 20

a#d#c#b#a 5+7+3+2 = 17

Improvements:Start and end at one particular city

Remove tours that differ only in direction

Efficiency:

(n-1)!/2 = O(n!)


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Knapsack Problem

Problem: Given nitemsweights: w1, w2 wn

values: v1, v2 vn

a knapsack of capacity WFind the most valuable subset of the items that fitinto the knapsack

Example (W = 16): Item Weight Value

1 2kg R2002 5kg R300

3 10kg R500

4 5kg R100


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Knapsack by Exhaustive

Search

Efficiency:!

(2n

)

Subset TotalWgt

TotalValue

{1} 2 kg R200

{2} 5 kg R300

{3} 10 kg R500

{4} 5 kg R100

{1,2} 7 kg R500

{1,3} 12 kg R700

{1,4} 7 kg R300

{2,3} 15 kg R800

Subset TotalWgt

TotalValue

{2,4} 10 kg R400

{3,4} 15 kg R600

{1,2,3} 17 kg n/a

{1,2,4} 12 kg R600

{1,3,4} 17 kg n/a

{2,3,4} 20 kg n/a

{1,2,3,4} 22 kg n/a


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Final Comments on

Exhaustive Search

Exhaustive searchalgorithms run in a realisticamount of time only onvery small instances

In many cases there aremuch better alternatives!

In some cases exhaustive

search (or variation) is theonly known solution

and parallel solutions canspeed it up!

Algorithms - Chapter 3 Bruteforce

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