Brute Force A straightforward approach usually based on

Brute Force A straightforward approach usually based on problem statement and definitions Examples: 1. Computing an (a, n > 0, n integer) 2. GCD with consecutive integer checking 3. Computing n! 4. Multiply two n by n matrices 5. Selection sort 6. Sequential search Design and Analysis of Algorithms - Chapter 3 1

Bubble Sort Algorithm Bubble. Sort(A[0. . n-1]) // Input: an array A[0. . n-1] of orderable elements // Output: Array A[0. . n-1] sorted ascendingly 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] Analysis for comparisons and swaps Improvements Design and Analysis of Algorithms - Chapter 3 2

String matching b b b pattern: a string of m characters to search for text: a (long) string of n characters to search in Brute force algorithm: 1. Align pattern at beginning of text 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 3. while pattern is not found and the text is not yet exhausted, realign pattern one position to the right and repeat step 2. Design and Analysis of Algorithms - Chapter 3 3

Brute force string matching – Algorithm Brute. Force. String. Match(T[0. . n-1], P[0. . m-1]) // Input: array T[0. . n-1] for text and array P[0. . m-1] for pattern // Output: the position of the first character in the text starts the first matching substring if the search is successful, and -1 otherwise for i 0 to n-m do j 0 while j<m and P[j]=T[i+j] do j j+1 if j=m return I return -1 Design and Analysis of Algorithms - Chapter 3 4

Brute force string matching – Examples: 1. Pattern: 001011 Text: 1001010110100101111010 1. Pattern: happy Text: It is never too late to have a happy childhood. Number of comparisons: Efficiency: Design and Analysis of Algorithms - Chapter 3 5

Brute force polynomial evaluation b 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 b Algorithm p : = 0. 0 for i : = n down to 0 do power : = 1 for j : = 1 to i do power : = power * x p : = p + a[i] * power return p b Efficiency: Design and Analysis of Algorithms - Chapter 3 6

Polynomial evaluation: improvement b Improvement: evaluation from right to left: b Algorithm: p : = a[0] power : = 1 for i : = 1 to n do power : = power * x p : = p + a[i] * power return p b Efficiency: Design and Analysis of Algorithms - Chapter 3 7

Closest Pair Problem b b b Problem: find closest points among n ones in k-dimensional space Algorithm: // Input: A list P of n>1 points P 1=(x 1, y 1), …, Pn=(xn, yn) //Output: Indices id 1 and id 2 of the closest pair of points dmin ∞ for i 1 to n-1 do for j i+1 to n do d sqrt[(xi-xj)2+(yi-yj)2] if d<dmin d; id 1 i; id 2 j; return id 1, id 2 Efficiency Design and Analysis of Algorithms - Chapter 3 8

The Convex Hull Problem b Problem: find smallest convex polygon enclosing n points on the plane b Algorithm: For each pair of points p 1 and p 2 determine whether all other points lie to the same side of the straight line through p 1 and p 2 b Efficiency: Design and Analysis of Algorithms - Chapter 3 9

Brute force strengths and weaknesses b Strengths: • • • wide applicability simplicity yields reasonable algorithms for some important problems – searching, string matching, matrix multiplication • yields standard algorithms for simple computational tasks – sum/product of n numbers, finding max/min in a list b Weaknesses: • rarely yields efficient algorithms • some brute force algorithms unacceptably slow • not as constructive/creative as some other design techniques Design and Analysis of Algorithms - Chapter 3 10

Exhaustive search b Exhaustive search is a brute force approach when searching for an element with a special property, usually among combinatorial objects such a permutations, combinations, or subsets of a set. b Method: • construct a way of listing all potential solutions to the problem in a systematic manner – all solutions are eventually listed – no solution is repeated • Evaluate solutions one by one, disqualifying infeasible ones and keeping track of the best one found so far • When search ends, announce the winner Design and Analysis of Algorithms - Chapter 3 11

Example 1: Traveling salesman problem 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. b Alternatively: Find shortest Hamiltonian circuit in a weighted connected graph. b Example: 2 a 5 3 8 c b 7 4 d Design and Analysis of Algorithms - Chapter 3 12

Traveling salesman by exhaustive search b 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 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 b Efficiency: Design and Analysis of Algorithms - Chapter 3 13

Example 2: Knapsack Problem: Given n items with weights w 1, w 2, …, wn and values v 1, v 2, …, vn and a knapsack of capacity W Find the most valuable subset of the items that fit into the knapsack Example: item weight 1 2 2 5 3 10 4 5 value $20 $30 $50 $10 Knapsack capacity W=16 Design and Analysis of Algorithms - Chapter 3 14

Knapsack by exhaustive search Subset Total weight {1} 2 {2} 5 {3} 10 {4} 5 {1, 2} 7 {1, 3} 12 {1, 4} 7 {2, 3} 15 {2, 4} 10 {3, 4} 15 {1, 2, 3} 17 {1, 2, 4} 12 {1, 3, 4} 17 {2, 3, 4} 20 {1, 2, 3, 4} 22 Total value $20 $30 $50 $10 $50 $70 $30 $80 $40 $60 not feasible Efficiency: Design and Analysis of Algorithms - Chapter 3 15

Assignment by exhaustive search Problem: Given n people and n jobs, whereas C[i, j] is the cost if person i takes job j Assign one person per job with the minimum cost Example: Job 1 Job 2 Job 3 Job 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 Design and Analysis of Algorithms - Chapter 3 16

Assignment by exhaustive search Assignment <1, 2, 3, 4> <1, 2, 4, 3> <1, 3, 2, 4> <1, 3, 4, 2> <1, 4, 2, 3> <1, 4, 3, 2> ………… 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 ……………. Design and Analysis of Algorithms - Chapter 3 Efficiency: 17

Final comments: b Exhaustive search algorithms run in a realistic amount of time only on very small instances b In many cases there are much better alternatives! • • Euler circuits shortest paths minimum spanning tree assignment problem b In some cases exhaustive search (or variation) is the only known solution Design and Analysis of Algorithms - Chapter 3 18