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Could someone tell me What is the general idea to approach to problems in Dynamic Programming ?

ie. Given a problem How do I start my thinking :P , Im not getting a clue to deal with DP problems .  Making Recursive Equations etc .

Is DP approach is suitable for any problems ?


Example Problem

Min-Coin Change is the problem of using the minimum number of coins to make change for a particular amount of cents, n, using a given set of denominations d_{1}\ldots d_{m}.

Given the denominations 1, 5, 10, 20, 25, and wish to make change for 40 cents, the greedy algorithm would give us 25, 10, 5, but the best solution only requires 2 coins - 2 of the 20 cent coins

asked in Algorithms by Loyal (2.6k points)   | 122 views

if you really want more general idea about DP better search competitive programming problems on others sites and their different solution methods. 

These will be useful ...coin change is there in the first pdf.




By examples we get some idea of DP.
mean >?
like we cannot define it directly.

but examining floyd warshall, bellman ford, knapsack we can understand how it works.

Like a small problem able to solve a big problem.

mostly like fibonacci numbers

I just wanted to know a general method to wrte recursion equation .
See here

I didn't understand how he managed to get the cell  (i)(j)

1 Answer

0 votes
Given a problem , you have to first check whether you can apply dynamic programming in it . In order to apply the dynamic programming paradigm in a problem, the problem must have an optimal substructure and overlapping subproblems . These are the basic prerequisites for DP which you can find in any book. But what these actually mean is that the answer to the problem can only be computed once you find the answer to its subproblems or else you won't get an optimal answer to the larger problem unlike that in a greedy solution oriented problem where you can directly compute the answer to the problem.

A very important idea of approach in DP problems is that it stores all the answers to the subproblems that it once computed and it never computes the same subproblem twice thus reducing the time complexity to a great extent unlike that in divide and conquer approach where we repeatedly solve the same subproblems if they are not unique and increase the time complexity.

The coin change problem is a classical DP problem because the higher denominations depend on the lower denominations and the solution can be developed starting from the lowest denomination.

Thus DP problems always follow the golden rule that the optimal answer to the smaller subproblems leads to the optimal answer to the larger problem. If you can directly find the answer to the problem without finding the answer to its self similar subproblems then it is not a dynamic programming solution.
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