Chapter 3 Brute Force Copyright 2007 Pearson AddisonWesley

Chapter 3 Brute Force Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

ONE PARAGRAPH THAT EXPLAINS LIFE! Arthur Ashe, the legendary Wimbledon player was dying of AIDS which he got due to infected blood he received during a heart surgery in 1983. From world over, he received letters from his fans, one of which conveyed: "Why does GOD have to select you for such a bad disease"? To this Arthur Ashe replied: "The world over -- 50 million children start playing tennis, 5 million learn to play tennis, 500, 000 learn professional tennis, 50, 000 come to the circuit, 5000 reach the grand slam, 50 reach Wimbledon, 4 to semi final, 2 to the finals, when I was holding a cup I never asked GOD 'Why me? '. And today in pain I should not be asking GOD 'Why me? ' “ "Happiness keeps you Sweet, Trials keep you Strong, Sorrow keeps you Human, Failure keeps you humble and Success keeps you glowing, but only Faith & Attitude Keeps you going…” Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -1

Brute Force A straightforward approach to solving a problem, 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. 3. 4. Computing n! Multiplying two matrices Searching for a key of a given value in a list Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -2

Why Brute-force design technique? b b b Unlike other design strategies, this technique is applicable to a very wide variety of problems. Useful for solving small size instances of a problem. For some important problems ( eg. Sorting, searching, matrix multiplication, string matching) brute-force approach yields reasonable algorithms of at least some practical value with no limitation on instance size. The expense of designing a more efficient algorithm may be un-justifiable if only a few instances of a problem need to be solved and a brute-force algorithm can solve those instances with acceptable speed. It can serve as yardstick with which we can judge more efficient alternatives for solving a problem. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -3

Brute-Force Sorting Algorithm Selection Sort Scan the array to find its smallest element and swap it with the first element. 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 Example: 7 3 2 5 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -4

Analysis of Selection Sort Time efficiency: Θ(n^2) Space efficiency: Θ(1), so in place Stability: yes Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -5

Brute-Force String Matching b b b 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 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -6

Examples of Brute-Force String Matching 1. Pattern: 001011 Text: 1001010110100101111010 2. Pattern: happy Text: It is never too late to have a happy childhood. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -7

Pseudocode and Efficiency ALGORITHM Brute. Force. String. Match(T [0. . n − 1], P[0. . m − 1]) //Implements brute-force string matching //Input: An array T [0. . n − 1] of n characters representing a text and // an array P[0. . m − 1] of m characters representing a pattern //Output: The index of the first character in the text that starts a // matching substring or − 1 if the search is unsuccessful for i ← 0 to n − m do j ← 0 while j <m and P[j ]= T [i + j ] do j ←j + 1 endwhile if j = m return i endfor return − 1 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -8

Analysis of Brute. Force. String. Match( ) 1. n, the length of the Text is the measure on input’s size. 2. Comaprison P[j] = T[i+j] is the basic operation. 3. Number of times basic operation is executed depends on the value of n and the nature of input. 4. Set up the sum and solve it. m comparisons to be made for every alignment of P against T. Cbest(n) = m ( When P is found in the first m characters of text T Every time m comparisons and the pattern is found in the last n-m + 1 positions. Cworst(n) = Θ (mn) Most of such shifts happen after a few comparisons. Hence, average case efficiency is considerably better than worst case efficiency. For searching in random texts , it has been shown to be linear, ie. , Θ (m + n) = Θ(n) 5. It is a linear order complexity algorithm. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -9

Brute-Force Polynomial Evaluation Problem: Find the value of polynomial p(x) = anxn + an-1 xn-1 +… + a 1 x 1 + a 0 at a point x = x 0 Brute-force algorithm p 0. 0 for i n downto 0 do power 1 for j 1 to i do //compute xi power x p p + a[i] power return p Efficiency: 0 i n i = Θ(n^2) multiplications Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -10

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 x p p + a[i] power return p Efficiency: Θ(n) multiplications Horner’s Rule is another linear time method. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -11

Closest-Pair Problem Find the two closest points in a set of n points (in the twodimensional 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. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -12

Closest-Pair Brute-Force Algorithm (cont. ) Efficiency: Θ(n^2) multiplications (or sqrt) How to make it faster? Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Using divide-and-conquer! Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -13

Brute-Force Strengths and Weaknesses b Strengths • wide applicability • simplicity • yields reasonable algorithms for some important problems (e. g. , matrix multiplication, sorting, searching, string matching) b Weaknesses • rarely yields efficient algorithms • some brute-force algorithms are unacceptably slow • not as constructive as some other design techniques Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -14

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 • 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 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -15

Example 1: Traveling Salesman Problem b b b 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: A cycle that passes 2 a b through all vertices 5 3 of a graph exactly once 4 8 c 7 d Question: How do we represent a solution (Hamiltonian circuit)? Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -16

Example 1: TSP Solution by Exhaustive Search Graph 2 a 8 c Tour 5 b 3 7 4 d 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 Efficiency : Θ((n-1)!) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -17

Example 2: Knapsack Problem Given n items: • weights: w 1 w 2 … wn • values: v 1 v 2 … 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 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -18

Knapsack Problem by Exhaustive Search Subset Total weight Total value {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} $20 $30 $50 $10 $50 $70 $30 $80 $40 $60 not feasible 2 5 10 5 7 12 7 15 10 15 17 12 17 20 22 Efficiency: Θ(2^n) Each subset can be represented by a binary string (bit vector, Ch 5). Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -19

Example 3: The Assignment Problem There are n people who need to be assigned to n jobs, –Each person exactly one job –Each job to exactly one person The cost of assigning person i to job j is C[i, j]. Find an assignment that minimizes the total cost. Example: Person 0 Person 1 Person 2 Person 3 Job 0 Job 1 Job 2 Job 3 9 2 7 8 6 4 3 7 5 8 1 8 7 6 9 4 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. How do you solve this? Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -20

Example 3: The Assignment Problem Person 0 Person 1 Person 2 Person 3 Job 0 Job 1 Job 2 Job 3 9 2 7 8 6 4 3 7 5 8 1 8 7 6 9 4 Algorithmic Plan: No obvious strategy for finding a solution works here. Opting for exhaustive search may be the solution. –Generate all legitimate assignments, –Compute their costs, and –Select the cheapest one. How many assignments are there? n! Pose the problem as one about a cost matrix: Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. cycle cover in a graph Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -21

An Assignment Problem Example Task 1 Task 2 Task 3 Task 4 Person 1 9 2 7 8 Person 2 6 4 3 7 Person 3 5 8 1 8 Person 4 7 6 9 4 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -22

Assignment Problem by Exhaustive Search Feasible solution (j 1, j 2, …, jn) indicates job numbers assigned to respective persons. 9 2 7 8 6 4 3 7 C= 5 8 1 8 7 6 9 4 Assignment (col. #s) Total Cost 1, 2, 3, 4 9+4+1+4=18 1, 2, 4, 3 9+4+8+9=30 1, 3, 2, 4 n! assignments 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 etc. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Optimal Solution 2, 1, 3, 4 Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -23

An Assignment Problem Example (cont. ) <1, 2, 3, 4> cost = 9 + 4 + 1 + 4 = 18 <2, 1, 3, 4> cost = 2 + 6 + 1 + 4 = 13 (optimal) <1, 2, 4, 3> cost = 9 + 4 + 8 + 9 = 30 <2, 1, 4, 3> cost = 2 + 6 + 8 + 9 = 25 <1, 3, 2, 4> cost = 9 + 3 + 8 + 4 = 24 <2, 3, 1, 4> cost =2 + 3 + 5 + 4 = 14 <1, 3, 4, 2> cost = 9 + 3 + 8 + 6 = 26 <2, 3, 4, 1> cost = 2 + 3 + 8 + 7 = 20 <1, 4, 2, 3> cost = 9 + 7 + 8 + 9 = 33 <2, 4, 1, 3> cost = 2 + 7 + 5 + 9 = 23 <1, 4, 3, 2> cost = 9 + 7 + 1 + 6 = 23 <2, 4, 3, 1> cost = 2 + 7 + 1 + 7 = 17 <3, 1, 2, 4> cost = 7 + 6 + 8 + 4 = 25 <4, 1, 2, 3> cost = 8 + 6 + 8 + 9 = 31 <3, 1, 4, 2> cost = 7 + 6 + 8 + 6 = 27 <4, 1, 3, 2> cost = 8 + 6 + 1 + 6 = 21 <3, 2, 1, 4> cost = 7 + 4 + 5 + 4 = 20 <4, 2, 1, 3> cost = 8 + 4 + 5 + 9 = 26 <3, 2, 4, 1> cost = 7 + 4 + 8 + 7 = 26 <4, 2, 3, 1> cost = 8 + 4 + 1 + 7 = 20 <3, 4, 1, 2> cost = 7 + 5 + 6 = 25 <4, 3, 1, 2> cost = 8 + 3 + 5 + 6 = 22 <3, 4, 2, 1> cost = 7 + 8 + 7 = 29 <4, 3, 2, 1> cost = 8 + 3 + 8 + 7 = 26 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -24

Final Comments on Exhaustive Search b Exhaustive-search algorithms run in a realistic amount of time only on very small instances b 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 b Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms, ” 2 nd ed. , Ch. 3 3 -25