Greedy Algorithms Questions Long
The minimum number of coins needed to make change for a given amount can be calculated using a greedy algorithm known as the "Coin Change" problem. The greedy approach aims to find the optimal solution at each step by making locally optimal choices.
To calculate the minimum number of coins needed, we can follow these steps:
1. Sort the available coin denominations in descending order. This is important as the greedy algorithm always selects the largest denomination first.
2. Initialize a variable, let's call it "count," to keep track of the total number of coins used.
3. Start with the largest denomination and repeatedly subtract it from the given amount until it is no longer possible to subtract the full denomination. Each time a subtraction is made, increment the count variable.
4. Move to the next smaller denomination and repeat step 3 until the given amount becomes zero.
5. Return the count variable, which represents the minimum number of coins needed to make change for the given amount.
Here is an example to illustrate the greedy algorithm:
Let's say we have the following coin denominations: [25, 10, 5, 1] and we want to make change for the amount 47.
1. Sort the coin denominations in descending order: [25, 10, 5, 1].
2. Initialize count = 0.
3. Start with the largest denomination, 25. Subtract it from 47: 47 - 25 = 22. Increment count to 1.
4. Move to the next denomination, 10. Subtract it from 22: 22 - 10 = 12. Increment count to 2.
5. Move to the next denomination, 5. Subtract it from 12: 12 - 5 = 7. Increment count to 3.
6. Move to the next denomination, 1. Subtract it from 7: 7 - 1 = 6. Increment count to 4.
7. Move to the next denomination, 1. Subtract it from 6: 6 - 1 = 5. Increment count to 5.
8. Move to the next denomination, 1. Subtract it from 5: 5 - 1 = 4. Increment count to 6.
9. Move to the next denomination, 1. Subtract it from 4: 4 - 1 = 3. Increment count to 7.
10. Move to the next denomination, 1. Subtract it from 3: 3 - 1 = 2. Increment count to 8.
11. Move to the next denomination, 1. Subtract it from 2: 2 - 1 = 1. Increment count to 9.
12. Move to the next denomination, 1. Subtract it from 1: 1 - 1 = 0. Increment count to 10.
13. The given amount is now zero. Return the count, which is 10. Therefore, the minimum number of coins needed to make change for 47 is 10.
In this example, the greedy algorithm selected the largest denomination at each step, which resulted in the minimum number of coins needed. However, it's important to note that the greedy algorithm may not always provide the optimal solution for every coin system.