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Chapter 3

Brute Force

Copyright 2007 Pearson Addison-Wesley. All rights reserved.

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Brute Force

A straightforward approach, usually based directly on the

problems statement and definitions of the concepts involved

Examples:

1. Computing an (a > 0, n a nonnegative integer)

2. Computing n!

3. Multiplying two matrices

4. Searching for a key of a given value in a list

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Brute-Force Sorting Algorithm

Selection Sort1. Scan the array to find its smallest element and swap it with the first

element.

2. Then, starting with the second element, scan the elements to its right to

find the smallest among them and swap it with the second elements.

3. Generally, on pass i (0 i n-2), find the smallest element inA[i..n-1]and swap it withA[i]:

A[0] . . . A[i-1] | A[i], . . . ,A[min], . . .,A[n-1]in their final positions

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Selection Sort & Bubble Sort

Selection Sort1. Start with the 1st element, scan the entire list to find its smallest element and

exchange it with the 1st element

2. Start with the 2nd element, scan the remaining list to find the the smallestamong the last (N-1) elements and exchange it with the 2nd element

3.

Example: 89 45 68 90 29 34 17

17 | 45 68 90 29 34 89

29 | 68 90 45 34 89

34 | 90 45 68 89

45 | 90 68 89

68 | 90 89

89 | 90

90

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Selection Sort

Algorithm

for i 0 to N-2 do

min i; i=0

for j i+1 to N-1 do c

if (A[j] < A[min]) min j; j=i+1

}

swap A[i] and A[min] ;

}

Analysis:

T(N) = c = c (N-1-i) = c (N-1)N / 2 in O(N )

i=0 j=i+1 i=0

N-2

N-1

N-1N-2 N-2 2

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Bubble Sort

Bubble Sort1. In each pass, we compare adjacent elements and swap them if they are out of

order until the end of the list. By doing so, the 1stpass ends up bubbling upthe largest element to the last position on the list

2. The 2nd pass bubbles up the 2nd largest, and so on until, after N-1 passes, thelist is sorted.

Example:

Pass 1 89 | 45 68 90 29 34 17 Pass 2 45 | 68 89 29 34 17

45 89 | 68 90 29 34 17 45 68 | 89 29 34 17

68 89 | 90 29 34 17 68 89 | 29 34 17

89 90 | 29 34 17 29 89 | 34 17

29 90 | 34 17 34 89 | 1734 90 | 17 17 89

17 90

45 68 89 29 34 17 90 45 68 29 34 17 89

largest 2nd largest

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Bubble Sort

Algorithm

for i 0 to N-2 do

i=0

for j 0 to N-2-i do c

j=i+1

if (A[j+1] < A[j]) swap A[j] and A[j+1] ;

}

}

Analysis:T(N) = c = c (N-1-i) = c (N-1)N / 2 in O(N )

i=0 j=0 i=0

N-2

N-2-i

N-2-iN-2 N-22

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Sec 3.2 Sequential Search and String Matching

Sequential search

Compare successive elements of a given list with a search key until1. either a match is encountered

2. or the list is exhausted without a match.

0 1 N-1 N

Algorithm:extra position

SequentialSearch(A[0..N], key){ to store the keyA[N] key;i 0;

while (i

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Brute-Force String Matching

pattern: a string ofm characters to search for

text: a (longer) string ofn characters to search in problem: find a substring in the text that matches the pattern

Brute-force algorithm

Step 1 Align pattern at beginning of textStep 2 Moving from left to right, compare each character of

pattern to the corresponding character in text until

all characters are found to match (successful search); or

a mismatch is detected

Step 3 While pattern is not found and the text is not yet

exhausted, realign pattern one position to the right and

repeat Step 2

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Examples of Brute-Force String Matching

1. Pattern: 001011Text: 10010101101001100101111010

2. Pattern: happy

Text: It is never too late to have a happy childhood.

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Brute-force String Matching

Pattern: p[0] p[1] p[m-1] m characters (m

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Brute-Force Polynomial Evaluation

Problem: Find the value of polynomial

p(x) =anxn+an-1x

n-1 + + a1x1 +a0 at a pointx =x0

Brute-force algorithm

Efficiency:

p 0.0;

for i n downto 0 do {

power 1;

for j 1 to i do //computexi

power power x;

p p +a[i] power;

}

returnp;

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Polynomial Evaluation: Improvement

We can do better by evaluating from right to left:

Better brute-force algorithm

Efficiency:

p a[0]

power 1

for i 1 ton do {power power x

p p +a[i] power

}

returnp

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Closest-Pair Problem

Find the two closest points in a set ofn points

(in the two-dimensional Cartesian plane).

Brute-force algorithm

Compute the distance between every pair of distinct

points

and return the indexes of the points for which the distance

is the smallest.

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Closest-Pair Brute-Force Algorithm (cont.)

Efficiency:

How to make it faster?

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Brute-Force Strengths and Weaknesses

Strengths

wide applicability

simplicity

yields reasonable algorithms for some important problems(e.g., matrix multiplication, sorting, searching, stringmatching)

Weaknesses

rarely yields efficient algorithms

some brute-force algorithms are unacceptably slow

not as constructive as some other design techniques

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Exhaustive Search

A brute force solution to a problem involving search for an elementwith a special property, usually among combinatorial objects such aspermutations, combinations, or subsets of a set.

Method:

generate a list of all potential solutions to the problem in asystematic manner (see algorithms in Sec. 5.4)

evaluate potential solutions one by one, disqualifying infeasibleones and, for an optimization problem, keeping track of the bestone found so far

when search ends, announce the solution(s) found

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Example 1: Traveling Salesman Problem

Given n cities 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 shortestHamiltonian circuit in a

weighted connected graph

Example:

a b

c d

8

2

7

5 34

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TSP by Exhaustive Search

Tour Cost

abcda 2+3+7+5 = 17

abdca 2+4+7+8 = 21

acbda 8+3+4+5 = 20

acdba 8+7+4+2 = 21

adbca 5+4+3+8 = 20adcba 5+7+3+2 = 17

More tours?

Less tours?

Efficiency:

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Example 2: Knapsack Problem

Given n items:

weights: w1 w2 wn

values: v1 v2 vn

a knapsack of capacity W

Find most valuable subset of the items that fit into the knapsack

Example: Knapsack capacity W=16

item weight value

1 2 $20

2

5 $303 10 $50

4 5 $10

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Knapsack Problem by Exhaustive Search

Subset Total weight Total value

{1} 2 $20

{2} 5 $30{3} 10 $50

{4} 5 $10

{1,2} 7 $50

{1,3} 12 $70

{1,4} 7 $30{2,3} 15 $80

{2,4} 10 $40

{3,4} 15 $60

{1,2,3} 17 not feasible

{1,2,4} 12 $60{1,3,4} 17 not feasible

{2,3,4} 20 not feasible

{1,2,3,4} 22 not feasible

Efficiency:

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Sec 3.4 Exhaustive Search

Assignment Problem1. Assign N jobs to N people, one person per job.

2. Find an assignment such that the total cost is smallest.

Example:Let C[i, j] be the cost of assigning job j to person i.

J1 J2 J3 J4 jobs

P1 9 2 7 8C = P2 6 4 3 7

P3 5 8 1 8 cost

P4 7 6 9 4

persons

* Select one element in each row so that all selected elements are in

different columns and the total sum of the cost is the smallest.

* n-tuple notation: t =

t[i] = the column (job) # of the element selected in row i

* C(1,2) + C(2,3) + C(3,4) + C(4,1) = 2 + 3 + 8 + 7 = 20

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Few iterationsPerson 1 2 3 4

========< 1, 2, 3, 4 > 9 + 4 + 1 + 4 = 18< 1, 2, 4, 3 > 9 + 4 + 8 + 9 = 30< 1, 3, 2, 4 > 9 + 3 + 8 + 4 = 24< 1, 3, 4, 2 > 9 + 3 + 8 + 6 = 26< 1, 4, 2, 3 > 9 + 7 + 8 + 9 = 33

< 1, 4, 3, 2 > 9 + 7 + 1 + 6 = 23

< 2, 1, 3, 4 > 2 + 6 + 1 + 4 = 13< 2, 1, 4, 3 > 2 + 6 + 8 + 9 = 25

etc. j1 j2 j3 jN

Total # of permutations = N !p1 p2 p3 pN

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Final Comments on Exhaustive Search

Exhaustive-search algorithms run in a realistic amount of

time only on very small instances

In some cases, there are much better alternatives!

Euler circuits

shortest paths

minimum spanning tree

assignment problem

In many cases, exhaustive search or its variation is the

only known way to get exact solution