Exam 20: Minimal Spanning Tree

Correct Answer:

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A greedy algorithm is a simple, intuitive algorithmic paradigm that follows the problem-solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. In other words, a greedy algorithm makes the best immediate, or "greedy," decision at every step, choosing the most beneficial option right now without considering the broader consequences. It assumes that by choosing a local optimum at each step, the final solution will also be optimal or close to optimal.

Greedy algorithms are used when a problem exhibits the following two properties:

1. Greedy Choice Property: A global optimum can be arrived at by making a locally optimal (greedy) choice.
2. Optimal Substructure: An optimal solution to the problem contains an optimal solution to subproblems.

However, greedy algorithms do not always yield the globally optimal solution for all problems because they usually do not consider the bigger picture. They are best suited for problems where choosing locally optimal also leads to a global optimum.

**Example of a Greedy Algorithm:**

One classic example of a greedy algorithm is the coin change problem, where the goal is to make change for a particular amount of cents using the least number of coins possible. Assume you have coins of denominations 1, 5, 10, and 25 cents, and you need to give change for 63 cents. A greedy algorithm would start by selecting the largest denomination less than or equal to the remaining amount, which is 25 cents in this case. It would then subtract this value from the total amount and repeat the process with the remaining amount. The steps would be as follows:

1. Choose a 25-cent coin: 63 - 25 = 38 cents remaining.
2. Choose another 25-cent coin: 38 - 25 = 13 cents remaining.
3. Choose a 10-cent coin: 13 - 10 = 3 cents remaining.
4. Choose three 1-cent coins: 3 - 3 = 0 cents remaining.

So, the greedy algorithm would use two 25-cent coins, one 10-cent coin, and three 1-cent coins, for a total of six coins.

This greedy strategy works because the coin denominations in the United States (and many other countries) are canonical, meaning that the greedy algorithm will always produce an optimal solution. However, if the coin denominations were different, such as 1, 3, and 4 cents, the greedy algorithm might not yield an optimal solution. For example, to make 6 cents, the greedy algorithm would choose one 4-cent coin and two 1-cent coins, for a total of three coins, whereas the optimal solution is two 3-cent coins.