Greedy Algorithms (General Structure and Applications)
The general structure of a greedy algorithm can be summarized in the following steps:
- Identify the problem as an optimization problem where we need to find the best solution among a set of possible solutions.
- Determine the set of feasible solutions for the problem.
- Identify the optimal substructure of the problem, meaning that the optimal solution to the problem can be constructed from the optimal solutions of its subproblems.
- Develop a greedy strategy to construct a feasible solution step by step, making the locally optimal choice at each step.
Prove the correctness of the algorithm by showing that the locally optimal choices at each step lead to a globally optimal solution.
Some common applications of greedy algorithms include:
- Coin change problem: Given a set of coins with different denominations, find the minimum number of coins required to make a given amount of change.
Fractional knapsack problem: Given a set of items with weights and values, fill a knapsack with a maximum weight capacity with the most valuable items, allowing fractional amounts of items to be included.
Huffman coding: Given a set of characters and their frequencies in a message, construct a binary code with minimum average length for the characters.
Shortest path algorithms: Given a weighted graph, find the shortest path between two nodes.
Minimum spanning tree: Given a weighted graph, find a tree that spans all nodes with the minimum total weight.
Greedy algorithms can be very efficient and provide fast solutions for many problems. However, it is important to keep in mind that they may not always provide the optimal solution and to analyze the problem carefully to ensure the correctness of the algorithm.
- Greedy Algorithms work step-by-step, and always choose the steps which provide immediate profit/benefit. It chooses the “locally optimal solution”, without thinking about future consequences. Greedy algorithms may not always lead to the optimal global solution, because it does not consider the entire data. The choice made by the greedy approach does not consider future data and choices. In some cases making a decision that looks right at that moment gives the best solution (Greedy), but in other cases, it doesn’t. The greedy technique is used for optimization problems (where we have to find the maximum or minimum of something). The Greedy technique is best suited for looking at the immediate situation.