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Perform brute force

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Perform brute force of the weak 64-bit WEP generator

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Chapter 3 Brute Force Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
Brute Force ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Brute-Force Sorting Algorithm ,[object Object],[object Object],[object Object]
Analysis of Selection Sort ,[object Object],[object Object],[object Object],Θ ( n^2 ) Θ ( 1 ), so in place yes
Brute-Force String Matching ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Examples of Brute-Force String Matching ,[object Object],[object Object]
Pseudocode and Efficiency ,[object Object],Θ (mn) comparisons (in the worst case) Why?
Brute-Force Polynomial Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],p  0.0 for i  n downto 0 do power  1 for j  1 to i do //compute x i power  power  x p  p + a [ i ]  power return p  0  i  n i = Θ (n^2) multiplications
Polynomial Evaluation: Improvement ,[object Object],[object Object],[object Object],[object Object],p  a [0] power  1 for i  1 to n do power  power  x p  p + a [ i ]  power return p Θ ( n) multiplications
Closest-Pair Problem ,[object Object],[object Object],[object Object],[object Object]
Closest-Pair Brute-Force Algorithm (cont.) Efficiency: How to make it faster? Θ (n^2) multiplications (or sqrt) Using divide-and-conquer!
Brute-Force Strengths and Weaknesses ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Exhaustive Search ,[object Object],[object Object],[object Object],[object Object],[object Object]
Example 1: Traveling Salesman Problem ,[object Object],[object Object],[object Object],How do we represent a solution (Hamiltonian circuit)? a b c d 8 2 7 5 3 4
TSP by Exhaustive Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Θ ((n-1)!) Chapter 5 discusses how to generate permutations fast.
Example 2: Knapsack Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Knapsack Problem by Exhaustive Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Efficiency: Θ (2^n) Each subset can be represented by a binary string (bit vector, Ch 5).
Example 3: The Assignment Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],n! cycle cover in a graph
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Assignment Problem by Exhaustive Search C = 2,1,3,4
Final Comments on Exhaustive Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],The Hungarian method runs in O(n^3) time.

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Perform brute force

  • 1. Chapter 3 Brute Force Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
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  • 11. Closest-Pair Brute-Force Algorithm (cont.) Efficiency: How to make it faster? Θ (n^2) multiplications (or sqrt) Using divide-and-conquer!
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Notas do Editor

  1. Draw example.
  2. The basic operation of the algorithm is computing the Euclidean distance between two points. The square root is a complex operation who’s result is often irrational, therefore the results can be found only approximately. Computing such operations are not trivial. One can avoid computing square roots by comparing distance squares instead.