Dynamic Programming Questions Long
Dynamic Programming can be used to solve the coin change problem by breaking it down into smaller subproblems and using the solutions to those subproblems to build up the solution to the original problem.
The coin change problem involves finding the minimum number of coins needed to make a certain amount of change. Given a set of coin denominations and a target amount, the goal is to determine the minimum number of coins required to make the target amount.
To solve this problem using Dynamic Programming, we can follow these steps:
1. Define the subproblem: Let's define the subproblem as finding the minimum number of coins needed to make a certain amount of change for a given target amount.
2. Identify the base case: The base case for this problem is when the target amount is 0. In this case, no coins are needed, so the minimum number of coins required is 0.
3. Determine the recurrence relation: To find the minimum number of coins needed for a target amount, we can consider each coin denomination individually and calculate the minimum number of coins required for the remaining amount after subtracting the value of the current coin denomination. We then take the minimum of these values and add 1 to account for the current coin.
Let's say we have a set of coin denominations [c1, c2, c3, ..., cn] and a target amount T. The recurrence relation can be defined as follows:
minCoins(T) = min(minCoins(T - c1), minCoins(T - c2), ..., minCoins(T - cn)) + 1
4. Build the solution using bottom-up approach: We can use a table or an array to store the minimum number of coins required for each target amount from 0 to the given target amount. We start by initializing the table with a large value (e.g., infinity) for all target amounts except 0, which is initialized as 0.
Then, we iterate through each target amount from 1 to the given target amount and calculate the minimum number of coins required using the recurrence relation mentioned above. We update the table with the calculated values.
Finally, the value at the last index of the table represents the minimum number of coins required to make the given target amount.
5. Return the solution: The minimum number of coins required to make the target amount can be obtained from the table or array created in the previous step.
By using Dynamic Programming and following these steps, we can efficiently solve the coin change problem and find the minimum number of coins required to make a given target amount. The time complexity of this approach is O(T * n), where T is the target amount and n is the number of coin denominations.