chapter 3: brute force the design and analysis of algorithms

Chapter 3: Brute

Force

The Design and Analysis of Algorithms

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Chapter 3. 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

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Basic Idea A straightforward approach to solve a

problem based on the problem’s statement and definitions of the concepts involved.

ExampleComputing an based on the definition of exponentiation:

an = a* a* a* …. * a

(a > 0, n a nonnegative integer)

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Brute Force Search and Sort

Sequential Search O(n) Selection Sort O(n2) Bubble Sort O(n2)

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Brute Force String MatchingPattern: program

Text:

Write a program. program program … program

Comparisons:

(n*m) in the worst possible case

(n+m) (n) in the average case.

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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)

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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.

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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).

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Exhaustive SearchState-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

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Exhaustive SearchCombinatorial problems

Traveling Salesman – permutations ((N-1)!)

Knapsack – subsets (2N)

Assignment problem – permutations

(N!)

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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

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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