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Practice

Greedy Algorithm Implementation Guide

1. Optimal Choice​

A greedy algorithm makes the choice that seems the best at each moment.

Example of Optimal Choice
def min_coins(coins, amount):
    coins.sort(reverse=True)
    count = 0
    for coin in coins:
        count += amount // coin
        amount %= coin
    return count

2. Iterative Selection Process​

Start using the largest denomination coin to get as close to the target amount as possible. Repeat this process.

Iterative Selection Process Example
for coin in coins:
    count += amount // coin
    amount %= coin

Efficiency and Application​

  1. Efficiency: Greedy algorithms quickly find a locally optimal solution at each stage, reducing computation time.

  2. Applications: They can be applied in various fields such as optimization problems, graph traversal problems, and scheduling.

  3. Cautions: While greedy algorithms choose the optimal solution at each step, they do not guarantee a global optimal solution. Thus, it's important to consider the nature of the problem when applying them.


Coin Problem Example​

  • Coin Denominations: [1, 100, 50, 500]

  • Target Amount: 800

  • Minimum Coin Count: 4 (one 500 coin, three 100 coins)


Example Execution and Result​

  • Running the example above will find the minimum number of coins needed to make 800 using the given coins. This is a typical example of a greedy algorithm, choosing the largest denomination coin at each step to quickly reach the target amount.

  • As a result, you can use one 500 coin and three 100 coins to make a total of 800 with 4 coins. This is the minimum coin count under the given conditions.


Time Complexity​

  1. Time Complexity: (O(n log n))

    • Here, n is the number of coin types. Sorting the coins in descending order takes (O(n log n)) time.

  2. Decision for Each Coin Usage: Deciding whether to use each coin takes (O(n)) time.

  3. Overall Time Complexity: Therefore, the overall time complexity is (O(n log n) + O(n)), approximately (O(n log n)).


Application Scenarios and Limitations​

  • While greedy algorithms provide quick solutions for complex problems, they do not always find the optimal solution. For example, if coin denominations are given in specific combinations, a greedy approach may fail to find the optimal solution.

  • For these reasons, it's crucial to understand the problem characteristics and requirements when applying greedy algorithms. In problems where a global optimal solution is important, caution should be exercised when using a greedy approach.

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