Question

I am reading up dynamic programming chapter of Introduction to Algorithms by Cormen et al. I am trying to understand how to characterize the space of subproblems . They gave two examples of dynamic programming . Both these two problems have an input of size n

1. Rod cutting problem (Cut the rod of size n optimally)
2. Matrix parenthesization problem .(Parenthesize the matrix product A₁ . A₂ .A₃ …A_n optimally to get the least number of scalar multiplications)

For the first problem , they choose a subproblem of the form where they make a cut of length k , assuming that the left subproblem resulting from the cut can not be cut any further and the right subproblem can be cut further thereby giving us a single subproblem of size (n-k) .

But for the second problem that choose subproblems of the type A_i…A_j where 1<=i<=j<=n . Why did they choose to keep both ends open for this problem ? Why not close one end and just consider on subproblems of size (n-k)? Why need both i and j here instead of a single k split?

Answer 1

It is an art. There are many types of dynamic programming problems, and it is not easy to define one way to work out what dimensions of space we want to solve sub-problems for.

It depends on how the sub-problems interact, and very much on the size of each dimension of space.

Dynamic programming is a general term describing the caching or memoization of sub-problems to solve larger problems more efficiently. But there are so many different problems that can be solved by dynamic programming in so many different ways, that I cannot explain it all, unless you have a specific dynamic programming problem that you need to solve.

All that I can suggest is to try when solving a problem is:

• if you know how to solve one problem, you can use similar techniques for similar problems.
• try different approaches, and estimate the order of complexity (in time and memory) in terms of input size for each dimension, then given the size of each dimension, see if it executes fast enough, and within memory limits.

Some algorithms that can be described as dynamic programming, include:

• shortest path algorithms (Dijkstra, Floyd-Warshall, …)
• string algorithms (longest common subsequence, Levenshtein distance, …)
• and much more…