Greedy Algorithms Questions Long
The minimum number of coins needed to make change for a given amount with limited denominations, values, and constraints can be calculated using a greedy algorithm known as the "Coin Change" problem.
The greedy algorithm for the Coin Change problem works by always 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, so that the largest denomination is at the beginning of the list.
2. Initialize a variable "count" to keep track of the number of coins used.
3. Start iterating through the sorted denominations from the largest to the smallest.
4. At each iteration, check if the current denomination is less than or equal to the remaining amount to be changed.
5. If the current denomination is less than or equal to the remaining amount, divide the remaining amount by the current denomination to get the maximum number of coins that can be used.
6. Add the maximum number of coins to the "count" variable.
7. Subtract the product of the maximum number of coins and the current denomination from the remaining amount.
8. Repeat steps 4-7 until the remaining amount becomes zero.
9. Finally, return the value of 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:
Consider the following denominations: [1, 5, 10, 25] and the amount to be changed is 47.
1. Sort the denominations in descending order: [25, 10, 5, 1].
2. Initialize "count" to 0.
3. Start iterating through the denominations.
4. The first denomination is 25, which is less than the remaining amount (47). Divide 47 by 25 to get 1.88, but since we are dealing with integers, take the floor value of 1.88, which is 1.
5. Add 1 to the "count" variable.
6. Subtract 1 * 25 from the remaining amount, which becomes 22.
7. Repeat steps 4-6 for the next denominations.
8. The second denomination is 10, which is less than the remaining amount (22). Divide 22 by 10 to get 2.2, but take the floor value of 2.2, which is 2.
9. Add 2 to the "count" variable.
10. Subtract 2 * 10 from the remaining amount, which becomes 2.
11. The third denomination is 5, which is less than the remaining amount (2). Divide 2 by 5 to get 0.4, but take the floor value of 0.4, which is 0.
12. Add 0 to the "count" variable.
13. Subtract 0 * 5 from the remaining amount, which remains 2.
14. The last denomination is 1, which is less than the remaining amount (2). Divide 2 by 1 to get 2.
15. Add 2 to the "count" variable.
16. Subtract 2 * 1 from the remaining amount, which becomes 0.
17. The remaining amount is now zero, so the algorithm terminates.
18. The value of the "count" variable is 1 + 2 + 0 + 2 = 5, which represents the minimum number of coins needed to make change for the given amount of 47.
Therefore, the minimum number of coins needed to make change for a given amount with limited denominations, values, and constraints can be calculated using the greedy algorithm as described above.