two integers is another example of the divide
and conquer strategy.A divide and conquer
problem solving strategy
works by recursively
breaking down a problem into two or more
sub-problems of the same (or related) type,
until these become simple enough to be
solved directly. The solutions to the sub-
problems are then combined to give a solution
to the original problem.

Job Assignment Problem
Exhaustive Search

Knapsack Problem
Exhaustive Search

Problem solving by exhaustive search
•
Suppose we have a Boolean expression
•
F(a, b, c) =
ab`c` + abc` + ac`
•
One way of finding all the tuples (a, b, c) that
yield true is to try out all the possibilities for
the Boolean variables a, b, c.
•
This gives 2^3 possibilities.

Problem solving by exhaustive search
•
Suppose we have n variables then we get 2^n
possibilities.
•
This is an example of exhaustive search.
•
Here the number of searches grows
exponentially with n.
•
Exhaustive search is also known as brute-force
search.

Trial and Error Method

Problem solving by trial and error
•
Suppose we have to factor the polynomial x^2
+ 4x + 3.
•
The original binomials must have looked like
this:
•
(x + m)(x + n),
•
where m and n are integers.We need to figure
out the values of m and n. The constant term
of the original polynomial is 3, so we need
m*n = 3.

Problem solving by trial and error
•
What integers multiply together to give 3? The
only choices are 1 and 3, or perhaps -1 and -3.
•
The coefficient of the x term in the original
polynomial is 4, so we also need
•
m + n = 4.
•
Since 1 and 3 multiply to give 3 and add
together to give 4, we have m = 1 and n = 3.
Therefore, we can factor our original
polynomial as
•
(x + 1)(x + 3)

Problem solving by trial and error
•
If we let m = 3 and n = 1 we'll have the same
factorization, except with the factors written in
a different order. Either way is correct,
•
The problem was solved by the problem solving
strategy of trial and error.
Trial and error can be
used for solving puzzles such as Sudoku.

Map Coloring Problem

Problem solving by backtracking
•
Suppose we have to arrange 8 queens in a chess
board such that no two queens attack each
other.

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- Fall '16
- Jegan
- Eight queens puzzle