Greedy Algorithms Questions Long
The minimum number of coins needed to make change for a given amount with limited denominations can be calculated using a greedy algorithm known as the "Coin Change" problem.
The greedy approach for this problem involves repeatedly selecting the largest denomination coin that is less than or equal to the remaining amount, until the remaining amount becomes zero.
Here is the step-by-step process to calculate the minimum number of coins using a greedy algorithm:
1. Sort the available coin denominations in descending order, so that the largest denomination is at the beginning of the list.
2. Initialize a variable, let's say "count", to keep track of the total number of coins used.
3. Start with the largest denomination coin and check if it is less than or equal to the remaining amount.
4. If the current coin denomination is less than or equal to the remaining amount, subtract the coin value from the remaining amount and increment the count by 1.
5. Repeat step 4 until the remaining amount becomes zero.
6. If the remaining amount becomes zero, return the count as the minimum number of coins needed.
7. If the remaining amount is not zero and there are no more denominations left to check, it means that it is not possible to make exact change for the given amount with the available denominations.
8. In such cases, return a special value (e.g., -1) to indicate that it is not possible to make change.
The greedy algorithm for the coin change problem provides an optimal solution when the available coin denominations have the "greedy choice property." This property means that at each step, choosing the largest denomination coin will not lead to a suboptimal solution.
However, it is important to note that the greedy algorithm may not always provide an optimal solution for all coin systems. There are cases where dynamic programming algorithms are required to find the minimum number of coins needed.
In conclusion, the minimum number of coins needed to make change for a given amount with limited denominations can be calculated using a greedy algorithm by repeatedly selecting the largest denomination coin until the remaining amount becomes zero.