Chapter 3
Brute Force

Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

Brute Force
A straightforward approach, usually based directly on the problem’s statement and definitions of the concepts involved Examples: 1. Computing an (a > 0, n a nonnegative integer)
2.

Computing n! Multiplying two matrices

3.

4.

Searching for a key of a given value in a list

Brute Force Sorting
What would be the most straightforward method for solving the sorting problem?

Reasonable people may disagree on the answer.

Selection Sort or Bubble Sort

Brute-Force Sorting Algorithm
Selection Sort :  Scan the array to find its smallest element and swap it with the first element.
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Then, starting with the second element, scan the elements to the right of it to find the smallest among them and swap it with the second elements. Generally, on pass i (0  i  n-2), find the smallest element in A[i..n-1] and swap it with A[i]:
A[0]  . . .  A[i-1] | A[i], . . . , A[min], . . ., A[n-1] in their final positions

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Example: 7 3 2 5

Selection Sort : Example
1 ) |89 45 68 90 29 34 17 2 ) 17 | 45 68 90 29 34 89 3 ) 17 29 | 68 90 45 34 89 4 ) 17 29 34 | 90 45 68 89 5 ) 17 29 34 45 | 90 68 89 6 ) 17 29 34 45 68 | 90 89 7 ) 17 29 34 45 68 89 | 90

Analysis of Selection Sort

Time efficiency: Space efficiency: Stability:

Θ(n^2) Θ(1), so in place yes

Bubble Sort
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Compare adjacent elements of the list and exchange them if they are out of order. By doing this repeatedly, we end up “bubbling up” the largest element to the last position The next pass bubbles up the second largest element and so on. After n-1 pass the list is sorted.

A0,….,Aj ↔ Aj+1,…, An-i-1, | An-i ≤ … ≤ An in their final positions

Example : 89 45 68 90 29 34 17

Bubble Sort : The Algorithm
Algorithm : BubbleSort(A[0..n-1]) //Sort given array by bubble sort //Input : An array A[a..n-1] with orderable elements //Output : Array A[0..n-1] sorted with ascending order for i ← 0 to n-2 do for j ← 0 to n-2-i do if A[ j + 1] < A[ j ] swap A[ j ] and A[ j+1 ]

Time efficiency worst Space efficiency is Stable Number of swaps

: : : :

Θ(n2) Θ(1) Yes n2

Bubble Sort : Example
89 ↔ 45 68 90 29 34 17 45 89 ↔ 68 90 29 34 17 45 68 89 ↔ 90 29 34 17 45 68 89 ↔ 90 ↔ 29 34 17 45 68 89 29 90 ↔ 34 17 45 68 89 29 34 90 ↔ 17 45 68 89 29 34 17 | 90 45 ↔ 68 ↔ 89 ↔29 34 17 45 68 29 89 ↔ 34 17 45 68 89 34 89 ↔ 17 45 68 89 29 17 | 89 | 90 | 90 | 90 90

First pass

Second pass

More on Bubble Sort
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Like many brute force algorithm, first version can be improved with a modest amount of effort. Specifically, if a pass of bubble sort make no changed on the list than the list has been sorted.

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Still Θ(n2) efficiency

Brute-Force String Matching
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pattern: a string of m characters to search for text: a (longer) string of n 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 text Step 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

Examples of Brute-Force String Matching
1.

Pattern: 001011 Text: 10010101101001100101111010 001011

2.

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

Pseudocode and Efficiency

Time efficiency:

Θ(mn) comparisons (in the worst case) Why?

Brute-Force Polynomial Evaluation
Problem: Find the value of polynomial p(x) = anxn + an-1xn-1 +… + a1x1 + a0 at a point x = x0 Brute-force algorithm p  0.0 for i  n downto 0 do power  1 for j  1 to i do //compute xi power  power  x p  p + a[i]  power return p Efficiency:

0in i = Θ(n^2) multiplications

Polynomial Evaluation: Improvement
We can do better by evaluating from right to left: Better brute-force algorithm p  a[0] power  1 for i  1 to n do power  power  x p  p + a[i]  power return p Efficiency: Θ(n) multiplications

Horner’s Rule is another linear time method.

Closest-Pair Problem
Find the two closest points in a set of n 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.

Closest-Pair Brute-Force Algorithm (cont.)

Efficiency:

Θ(n^2) multiplications (or sqrt)) Using divide-and-conquer!

How to make it faster?

Brute-Force Strengths and Weaknesses
 Strengths  wide applicability  simplicity  yields reasonable algorithms for some important problems (e.g., matrix multiplication, sorting, searching, string matching)  Weaknesses  rarely yields efficient algorithms  some brute-force algorithms are unacceptably slow  not as constructive as some other design techniques

Exhaustive Search
A brute force solution to a problem involving search for an element with a special property, usually among combinatorial objects such as permutations, combinations, or subsets of a set. Method:  generate a list of all potential solutions to the problem in a systematic manner (see algorithms in Sec. 5.4)  evaluate potential solutions one by one, disqualifying infeasible ones and, for an optimization problem, keeping track of the best one found so far  when search ends, announce the solution(s) found

Example 1: Traveling Salesman Problem
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Given n cities with known distances between each pair, find the shortest tour that passes through all the cities exactly once before returning to the starting city Alternatively: Find shortest Hamiltonian circuit in a weighted connected graph Example:
2

a
5 3

b
4

8

c

7

d

How do we represent a solution (Hamiltonian circuit)?

TSP by Exhaustive Search
Tour a→b→c→d→a a→b→d→c→a a→c→b→d→a a→c→d→b→a a→d→b→c→a a→d→c→b→a Efficiency: Cost 2+3+7+5 = 17 2+4+7+8 = 21 8+3+4+5 = 20 8+7+4+2 = 21 5+4+3+8 = 20 5+7+3+2 = 17 Θ((n-1)!)

Chapter 5 discusses how to generate permutations fast.

The Nim Method

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 $30 3 10 $50 4 5 $10

Knapsack Problem by Exhaustive Search
Subset Total weight
{1} {2} {3} {4} {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,3,4} 2 5 10 5 7 12 7 15 10 15 17 12 17 20 22

Total value
$20 $30 $50 $10 $50 $70 $30 $80 $40 $60 not feasible $60 not feasible not feasible not feasible

Efficiency: Θ(2^n)

Each subset can be represented by a binary string (bit vector, Ch 5).

Example 3: The Assignment Problem
There are n people who need to be assigned to n jobs, one person per job. The cost of assigning person i to job j is C[i,j]. Find an assignment that minimizes the total cost.
Job 0 Job 1 Job 2 Job 3 9 2 7 8 6 4 3 7 5 8 1 8 7 6 9 4

Person 0 Person 1 Person 2 Person 3

Algorithmic Plan: Generate all legitimate assignments, compute their costs, and select the cheapest one. How many assignments are there? n! cycle cover Pose the problem as one about a cost matrix: in a graph

Assignment Problem by Exhaustive Search
9 2 7 8 6 4 3 7

C= 5 8 1 8
7 6 9 4 Total Cost 9+4+1+4=18 9+4+8+9=30 9+3+8+4=24 9+3+8+6=26 9+7+8+9=33 9+7+1+6=23 etc. (For this particular instance, the optimal assignment can be found by exploiting the specific features of the number given. It is: 2,1,3,4 ) Assignment (col.#s) 1, 2, 3, 4 1, 2, 4, 3 1, 3, 2, 4 1, 3, 4, 2 1, 4, 2, 3 1, 4, 3, 2

Final Comments on Exhaustive Search
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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 The Hungarian method runs in O(n^3) time. In many cases, exhaustive search or its variation is the only known way to get exact solution

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