Question

Please discuss the similarities and differences between decrease and conquer, divide and conquer, and transform and conquer approaches of problem solving with an example (do not use the same example used by your peers) pointing out the advantages and disadvantages. When would it be appropriate to choose one approach to problem solving over the other?

Answer 1

(A)Decrease and Conquer strategy is used if we could reduce a problem into its smaller state by a factor and solving for it. The solution to this smaller instance, would form the basis to larger instances of the same problem.

Suppose a problem of size n
Solve for n - c ( c = constant factor)
Repeat the above and eventually the problem is reduce to something trivial that we can solve immediately. i.e: n = 1 or n = 0

(B)Divide and Conquer strategy is used if we could break a larger problem into several smaller sub-problems. These sub problems could then be combined if necessary to obtain the solution to the larger problem.

Divide and conquer VS Decrease and conquer:

(1)Decrease and Conquer involves reducing the problem into 1 sub problem, while divide and conquer reduces the problem into several sub-problems.
(2) Divide and Conquer would yield something along the lines of T(n/x), where x is amount of sub-problems. Whereas Decrease and Conquer would yield something like T(n - c) where c is a constant factor.

(C)Transform and conquer involves changing the problem into something that is much easier to solve, or to an instance in which you know the solution to.

In the celebrity problem, if you are a graph expert, you would have interpreted the problem as a graph problem, and model your solution base on vertices and edges.
You could also model the problem using arrays, stacks etc. Some representations are inherently easier to digest. (Knowing the right data structure to use is hence crucial)
You could also simplify a problem first. In the case of a search problem, we could make our lives easier by sorting the array.