1. The Design and Analysis of Algorithms
Chapter 3: Brute Force

2. Chapter 3. Brute Force Algorithms
Basic Idea
Brute Force Search and Sort
Brute Force String Matching
Closest Pair and Convex Hull Problems
Exhaustive Search
Conclusion

3. Basic Idea
A straightforward approach to solve a problem based on the problem’s statement and definitions of the concepts involved.
Example
Computing an based on the definition of exponentiation:
an = a* a* a* …. * a
(a > 0, n a nonnegative integer)

4. Brute Force Search and Sort
Sequential Search O(n)
Selection Sort O(n2)
Bubble Sort O(n2)

5. Brute Force String Matching
Pattern: program
Text:
Write a program.
program …
Comparisons:
(n*m) in the worst possible case
(n+m)  (n) in the average case.

6. Closest Pair Problem
Find the two closest points in a set of n points in k-dimensional space.
Algorithm ClosestPairPoints (P)
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
dmin ← d
(n2)

7. Convex Hull Problem
Convex set: For any two points P and Q in the set, the entire line segment with the end points at P and Q belongs to the set
Convex hull of a set S of points is the smallest convex set containing S
The convex hull of any set S of n > 2 points is a convex polygon with the vertexes at some of the points of S.

8. Convex Hull Problem Algorithm:
for each of n(n-1)/2 pairs of distinct points
for each of the other n – 2 points
find the sign of ax + by – c
The time efficiency of this algorithm is O(n3).

9. Exhaustive Search State-space search Given an initial state,
a goal state, and
a set of operations,
find a sequence of operations that transforms the initial state to the goal state.
The solution process can be represented as a tree

10. Exhaustive Search Combinatorial problems
Traveling Salesman – permutations  ((N-1)!)
Knapsack – subsets  (2N)
Assignment problem – permutations
 (N!)

11. Conclusion - Strengths
Wide applicability, simplicity
Reasonable algorithms for some important problems such as searching, string matching, and matrix multiplication
Standard algorithms for simple computational tasks such as sum and product of n numbers, and finding maximum or minimum in a list

12. Conclusion - Weaknesses
Brute Force approach rarely yields efficient algorithms
Some brute force algorithms are unacceptably slow
Brute Force approach is neither as constructive nor creative as some other design techniques