Greedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. In other words, at each step, a greedy algorithm selects the best available option without considering the overall consequences or future steps.
A recursive greedy algorithm is a variant of the greedy algorithm that uses recursion to solve a problem by making greedy choices at each step. It breaks down the problem into smaller subproblems and solves them recursively.
Let's explain a recursive greedy algorithm with an example:
Problem: Consider a scenario where you have a set of coins with different denominations and you want to find the minimum number of coins needed to make a certain amount of change.
Recursive Greedy Algorithm for Coin Change:
1. Sort the coins in descending order based on their denominations.
2. Start with the largest denomination coin.
3. If the current coin's denomination is less than or equal to the remaining amount, subtract the coin's value from the remaining amount and increment the count of coins used.
4. Recursively repeat steps 3 and 4 with the updated remaining amount until the remaining amount becomes zero.
5. Return the total count of coins used.
Example:
Consider the following set of coins: {1, 5, 10, 25}
Let's say we want to make a change for the amount 47.
1. Sort the coins in descending order: {25, 10, 5, 1}
2. Start with the largest coin, 25.
3. Since 25 is less than the remaining amount (47), subtract 25 from 47, resulting in 22. Increment the count of coins used by 1.
4. Recursively repeat steps 3 and 4 with the updated remaining amount (22).
- The largest coin, 25, is not applicable as it is greater than the remaining amount.
- Move to the next coin, 10. Subtract 10 from 22, resulting in 12. Increment the count of coins used by 1.
- Recursively repeat steps 3 and 4 with the updated remaining amount (12).
- Continue this process until the remaining amount becomes zero.
5. Return the total count of coins used, which is 3 (25 + 10 + 10 + 1 + 1).
In this example, the recursive greedy algorithm finds the minimum number of coins needed to make the change for the amount 47 using the available coin denominations.