Greedy Algorithms Questions Long
The coin change problem is a classic optimization problem in computer science and mathematics. It involves finding the minimum number of coins needed to make a certain amount of change, given a set of coin denominations and their corresponding values.
In this problem, we are given a set of coin denominations, such as {1, 5, 10, 25}, and their corresponding values in cents. The goal is to find the minimum number of coins needed to make a certain amount of change, such as 99 cents.
A greedy algorithm is a simple and intuitive approach to solve the coin change problem. The idea behind the greedy algorithm is to always choose the largest denomination coin possible at each step, until the desired amount of change is reached.
To solve the coin change problem using a greedy algorithm, we follow these steps:
1. Sort the coin denominations in descending order. This is important as it allows us to always choose the largest denomination coin first.
2. Initialize a variable, let's call it "count", to keep track of the number of coins used.
3. Start with the largest denomination coin and repeatedly subtract it from the desired amount of change until it is no longer possible to subtract it.
4. If the current denomination coin cannot be subtracted anymore, move on to the next smaller denomination coin and repeat step 3.
5. Increment the count variable by 1 each time a coin is used.
6. Repeat steps 3-5 until the desired amount of change is reached.
7. Return the count variable, which represents the minimum number of coins needed to make the desired amount of change.
For example, let's consider the coin denominations {1, 5, 10, 25} and the desired amount of change is 99 cents.
1. Sort the coin denominations in descending order: {25, 10, 5, 1}.
2. Initialize count = 0.
3. Start with the largest denomination coin, 25 cents. Subtract it from 99 cents: 99 - 25 = 74. Increment count by 1.
4. Since 74 cents is still greater than 25 cents, subtract another 25 cents: 74 - 25 = 49. Increment count by 1.
5. Repeat step 4 until it is no longer possible to subtract 25 cents.
6. Move on to the next smaller denomination coin, 10 cents. Subtract it from 49 cents: 49 - 10 = 39. Increment count by 1.
7. Repeat step 6 until it is no longer possible to subtract 10 cents.
8. Move on to the next smaller denomination coin, 5 cents. Subtract it from 39 cents: 39 - 5 = 34. Increment count by 1.
9. Repeat step 8 until it is no longer possible to subtract 5 cents.
10. Move on to the next smaller denomination coin, 1 cent. Subtract it from 34 cents: 34 - 1 = 33. Increment count by 1.
11. Repeat step 10 until it is no longer possible to subtract 1 cent.
12. The desired amount of change, 99 cents, has been reached. The count variable is now 7, which represents the minimum number of coins needed to make 99 cents using the given coin denominations.
In conclusion, the coin change problem with denominations and values can be solved using a greedy algorithm by always choosing the largest denomination coin possible at each step until the desired amount of change is reached. The greedy algorithm provides a simple and efficient solution to this optimization problem.