Greedy Algorithms Questions Long
The minimum number of coins needed to make change for a given amount with limited denominations and values can be calculated using a greedy algorithm known as the "Coin Change" problem.
The greedy approach for this problem involves selecting the largest denomination coin possible at each step until the desired amount is reached. The steps to calculate the minimum number of coins needed are as follows:
1. Sort the available coin denominations in descending order.
2. Initialize a variable "count" to keep track of the number of coins used.
3. Start with the largest denomination coin and check if it can be used to make change for the given amount.
4. If the coin value is less than or equal to the remaining amount, subtract the coin value from the amount and increment the count by 1.
5. Repeat step 4 until the remaining amount becomes zero.
6. If the remaining amount is not zero, move to the next smaller denomination coin and repeat steps 4 and 5.
7. Continue this process until the remaining amount becomes zero or all denominations have been checked.
The final value of "count" will represent the minimum number of coins needed to make change for the given amount.
It is important to note that the greedy algorithm may not always provide the optimal solution for all cases. There can be scenarios where the greedy approach fails to give the minimum number of coins. For example, if the available denominations are 1, 5, and 10, and the desired amount is 12, the greedy algorithm would select one coin of denomination 10 and two coins of denomination 1, resulting in a total of three coins. However, the optimal solution would be to use two coins of denomination 5, resulting in a total of two coins.
To overcome this limitation, dynamic programming techniques can be used to find the optimal solution for the minimum number of coins needed.