## Dynamic Programming : Few remarks

Why they still call it dynamic programing? I really do not know why people are so obsessed with continuing with the established non-sense. The name was given by Bellman, at a time when programming was an esoteric activity practised by only few crazy people. Those days it was not even worthy of a name. Bellman gave it this name because it sound impressive ( not surprising at all since he was an American!). Both of the words has little to do with how this method works. A more suitable name could have been ‘top-down optimization’ henceforth TDO.

At first look, TDO can be mistaken seen as a recursive approach to a problem. There is a subtle difference. Recursive method will provide a solution but they will be painfully slower. Recursive method works very well when the ‘divide-and-conquer’ strategy produces a smaller problem which is significantly smaller than the original one (say, half the size) e.g. merge sort etc.

TDO are suitable for the problem in which ‘divide and conquer’ can produce subproblems which are only slightly smaller than the original one. TDO at best can be approximated as a recursive strategy which remembers the solutions of previously solved smaller subproblems i.e. memoisation (yes, root word is memo).

Almost all the problems which can be solved by TDO can be represented by a Directed Acyclic Graphs (DAG). Each node of this DAG represents the subproblem and an edge between two nodes gives the ‘cost’ of reaching from one node to another. The task is now reduced to find a path which maximize or minimize this cost. Again Bellman Ford algorithm can be used to find it. For specific problems, these methods can be improved. For example, solving longest increasing subsequence problem using  TDO gives a solution of $n^2$ complexity. However a $\frac{n^2}{ log(n)}$ method was given by Masek and Paterson.

END NOTES :

[1] Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein