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Dynamic Programming

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Dynamic Programming Concept

A *writes down "1+1+1+1+1+1+1+1 =" on a sheet of paper*

A : "What's that equal to?"
B : *counting* "Eight!"

A *writes down another "1+" on the left*
A : "What about that?"
B : *quickly* "Nine!"
A : "How'd you know it was nine so fast?"
A : "You just added one more"
A : "So you didn't need to recount because you remembered there were eight! Dynamic Programming is just a fancy way to say 'remembering stuff to save time later'"

This conversation has the essence of dynamic programming.

The idea is very simple, If you have solved a problem with the given input, then save the result for future reference, so as to avoid solving the same problem again.. shortly ‘Remember your Past’. If the given problem can be broken up in to smaller sub-problems and these smaller subproblems are in turn divided in to still-smaller ones, and in this process, if you observe some overlapping subproblems, then its a big hint for DP. Also, the optimal solutions to the subproblems contribute to the optimal solution of the given problem

Following are steps to coming up with a dynamic programming solution :

1. Think of a recursive approach to solving the problem.
    Essentially expressing the problem P(X) in terms of P(Y) or an expression involving P(Yi)
            where Yi is less than X.
    The "less than" here could mean multiple things. if X is an integer, then it could mean less than arithmetically.
    If X is a string, it could mean a substring of X.
    If X is an array, it could mean a subarray of X, and so forth.

2. Write a recursive code for the approach you just thought of.
            Lets say your function definition looks like this :
                    solve(A1, A2, A3 ... )

3. Save the results you get for every function run so that if solve(A1, A2, A3, ... ) is called again, you do not recompute the whole thing.

4. Analyze the space and time requirements, and improve it if possible.

    And voila, we have a DP solution ready.

Lets explore this using an example where we see how DP improves the time complexity of solving the same problem.

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Dynamic Programming Problems

Greedy or dp
Problem Score Companies Time Status
Tushar's Birthday Bombs 200
Jump Game Array 225 41:16
Min Jumps Array 300 71:44
Tree dp
Problem Score Companies Time Status
Max edge queries! 200 56:34
Max Sum Path in Binary Tree 400 55:07
Suffix / prefix dp
Derived dp
Problem Score Companies Time Status
Chain of Pairs 200 42:09
Max Sum Without Adjacent Elements 225 58:10
Merge elements 300 57:43
Problem Score Companies Time Status
Flip Array 200
Tushar's Birthday Party 200 70:50
0-1 Knapsack 200 47:57
Equal Average Partition 350 71:49
Problem Score Companies Time Status
Best Time to Buy and Sell Stocks II 225 40:18
Dp optimized backtrack
Problem Score Companies Time Status
Word Break II 350
Multiply dp
Problem Score Companies Time Status
Unique Binary Search Trees II 400 36:06
Count Permutations of BST 400
Breaking words
Problem Score Companies Time Status
Palindrome Partitioning II 400 62:02
Word Break 400

Additional Practice

Problem Score Companies Time Status
Potions 200 56:48
Dice Throw 400 49:19
Double Increasing Series 200 45:07
Dice Rolls 300 27:36
Palindromic Substrings 200
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