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Dynamic Programming  Minimum Coin Change Problem
Objective: Given an amount of 'A' and n coins, v1<v2<v3<...........<vn. Write a program to find out the minimum number of coins required to make the change for the amount 'A'.
Example:
Amount: 5 Coins [] = 1, 2, 3. No of ways to make the change are : { 1,1,1,1,1} , {1,1,1,2}, {2,2,1},{1,1,3} and {3,2}. So as we can see minimum number of coins required are 2 ( 3+2=5}.
Approach:
For Every coin, we have two options, whether to select it or don't select it. So we can find out the solution with both the options and then choose the optimal one, here, in this case, select the minimum among all the solutions.
First, we will break the problem into smaller problems. So what are our smaller problems?
The amount is A and coins are v1, v2, ......., vn.
Say MC(j) represents the minimum number of coins required to make a change of amount j.
Smaller Problems:
 Select 1st coin (value = v1), Now Smaller problem is the minimum number of coins required to make a change of amount( jv1), MC(jv1)
 Select 2nd coin (value = v2), Now the Smaller problem is the minimum number of coins required to make a change of amount( jv2), MC(jv2)
 Likewise up to N
 Select nth coin (value = vn), Now the Smaller problem is a minimum number of coins required to make a change of amount( jv1), MC(jvn).
We need to find the minimum number of coins required to make a change for j amount. So we will select the minimum of all the smaller problems and add 1 to it because we have selected one coin. Now smaller problems will be solved recursively.
So if we write our recursive equation
MC(j) = Min_{i=1 to n}[MC(jv_{i})] +1
NOTE: Before selecting the coin, make sure whether the value of the coin is less than equal to the amount needed.
Using Recursion: Every coin has 2 options, to be selected or not selected so
Time Complexity: O(c^{n}) which is very high.
We can reduce the Time Complexity significantly by using Dynamic programming.
Using BottomUp Dynamic Programming.
(Click here to read about Bottomup Dynamic Programming).
 We will maintain an array to store the optimal solutions for the smaller problems, say we call it as coinReq[]. length of this array will be amount+1. (starts with 0).
 So coinReq[n] will be our final answer, minimum no of coins required to make change for amount 'n'.
 Since we are using Bottomup approach, so will start filling the coinReq[] from 0 to n. we will start solving it from the smaller possible amount which is 0 here.
 We need 0 coins to make change for amount 0, so coinReq[0]=0.
 We will use another array CC[] (size = number of coins )will store the solution for amount n with using all the coins, minimum of all these will the optimal solution for the amount.
 Since we are finding the solution for all the amounts 0 to N, we have to reset CC[] every time ( for amount = 1 to N)
 See the program for better explanation.
Code:
Output:
(Dynamic Programming) Minimum Coins required to make change for 20 are: 7
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