What I have learnt is that dynamic programming (DP) is of two kinds: top-down and bottom-up.
In top-down, you use recursion along with memoization. In bottom-up, you just fill an array (a table).
Also, both these methods use same time complexity. Personally, I find top-down approach more easier and natural to follow. Is it true that a given question of DP can be solved using either of the approaches? Or will I ever face a problem which can only be solved by one of the two methods?
Well I believe theoretically you should be able to solve a DP problem with either approach. However, there are instances when bottom up approach can become too expensive. Consider a knapsack problem with the knapsack_size = 200,000 and the num_items = 2000. To fill in a two dimensional DP table with just ints is not going to be possible. You'll exhaust the main memory of an ordinary computer. Moreover, you do not require to fill in all the entries in a table to achieve the desired final computation. A recursive top-down approach is far superior in a case like this. Hope it helps.
bottom-up DP requires you to see how the recursion was precisely building the complete solution i.e what kind of subproblems were being created how they were filled by the base cases hence it is somewhat difficult to write bottom-up dynamic programming while in top-down have to write a backtrack solution(which is still hard to think of) and then see the states of backtrack solution.states means those arguments to the recursive function that were varying during subsequent recursive calls.
Now coming to the time complexities:
Bottom-up DP is faster than top-down since it doesn't involve any function calls. It completely depends on the table entries while in top-down DP it requires function calls and thereby causing an implicit stack formation.
PS: to see difference between time complexities of top-down and bottom-up you need to solve ASSIGNMENTS problem on SPOJ.
