Chapter 4 - Tagged.pdf

Chapter 4 Brute Force Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Brute Force A straightforward approach (a particular design strategy), usually based directly on the problem’s statement and definitions of the concepts involved. In other words, “Just do it!” strateg...

Chapter 4 Brute Force Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Brute Force A straightforward approach (a particular design strategy), usually based directly on the problem’s statement and definitions of the concepts involved. In other words, “Just do it!” strategy. 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 2 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 ...  A[i-1] | A[i],... , A[min],..., A[n- 1] in their final positions Example: 7 5 4 2 3 Brute-Force Sorting Algorithm Analysis of Selection Sort Time efficiency: Θ(n^2) Space efficiency: Θ(1) extra space, Θ(n) in place Stability: yes 4  Brute-Force String Matching  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 5  Brute-Force Algorithm  The brute-force pattern Algorithm BruteForceMatch(T, P) matching algorithm compares Input text T of size n and pattern the pattern P with the text T P of size m for each possible shift of P Output starting index of a relative to T, until either substring of T equal to P or 1 a match is found, or if no such substring exists all placements of the pattern for i  0 to n  m have been tried { test shift i of the pattern }  Brute-force pattern matching runs in time O(nm) j0  Example of worst case: while j  m  T[i  j]  P[j] T  aaa … ah jj1 P  aaah if j  m may occur in images and DNA return i {match at i} sequences return -1 {no match anywhere} unlikely in English text Pattern Matching 6 Examples of Brute-Force String Matching 1. Pattern: 001011 Text: 10010101101001100101111010 2. Pattern: happy Text: It is never too late to have a happy childhood. 7 Pseudocode and Efficiency Time efficiency: Θ(mn) comparisons (in the worst case) Why? 8 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 9 Polynomial Evaluation: Improvement We can do better by evaluating from right to left: p(x) = anxn + an-1xn-1 +… + a1x1 + a0 Better brute-force algorithm p  a 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. 10 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. 11 Closest-Pair Brute-Force Algorithm (cont.) Euclidean distance Efficiency: Θ(n^2) multiplications (or sqrt) How to make it faster? Using divide-and-conquer! 12 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 13 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. Many such problems are optimization problems: they ask to find an element that maximizes or minimizes some desired characteristic such as a path length or an assignment cost. 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 14 Example 1: Traveling Salesman Problem  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 b 5 3 8 4 c 7 d How do we represent a solution (Hamiltonian circuit)? 15 TSP 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 Efficiency: Θ((n-1)!) Chapter 6 discusses how to generate permutations fast. 16 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 17 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: Θ(2^n) Each subset can be represented by a binary string (bit vector, Ch 6). 18 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 Person 0 9 2 7 8 Person 1 6 4 3 7 Person 2 5 8 1 8 Person 3 7 6 9 4 Algorithmic Plan: 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: 19 Assignment Problem by Exhaustive Search 9 2 7 8 6 4 3 7 C= 5 8 1 8 7 6 9 4 Assignment Total Cost 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 etc. (For this particular instance, the optimal assignment is: 2,1,3,4 ) 20 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 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 21