The Dark Side of Greed

How to Balance Short-Term and Long-Term Goals

Photo by Henley Design Studio on Unsplash

Suppose you are playing a game with your friend. You are given a list of numbers, and you can pick from the beginning or the end of the list. Your goal is to maximize your output. After you pick, it becomes your friend’s turn to pick a number from the beginning or the end of the list. Each player can only pick from the beginning or the end of the list. The game proceeds as follows:

1. The first player chooses a number from the beginning or the end of the list.
2. The second player chooses a number from the beginning or the end of the list.
3. The output of each player is the sum of the numbers they have picked.
4. The game ends when the list is empty.

To do this, we write a function called optimize_output. The goal of the optimize_output function is to find the maximum output for the first player given a list of numbers. To do this, the function uses a greedy approach to calculate the maximum output for the first player by considering both the first and last elements of the list, and then removes the element that the first player picked from the list.

​​A greedy algorithm is a simple and intuitive approach to solving optimization problems that involves making locally optimal choices at each step in the hope of finding a global optimal solution. In other words, the greedy approach aims to maximize the immediate gain without considering the long-term consequences of the decisions made.

def optimize_output(numbers):
  # initialize the sum to zero
  total_sum = 0

  # keep track of the index of the first and last elements in the list
  start_pointer = 0
  end_pointer = len(numbers) - 1
  my_turn = True

  # keep going until the list is empty
  while start_pointer <= end_pointer:
    # if the first element is larger, select it
    if numbers[start_pointer] > numbers[end_pointer]:
      selected_value = numbers[start_pointer]
      start_pointer += 1
    # otherwise, select the last element
    else:
      selected_value = numbers[end_pointer]
      end_pointer -= 1
    
    if my_turn:
        total_sum += selected_value
    
    my_turn = not my_turn

  return total_sum

Benefits

One of the main advantages of the greedy algorithm is its simplicity and ease of implementation. It requires little effort to design and implement a greedy algorithm, and it often produces good results in practice. In addition, the greedy approach can be used to solve a wide range of optimization problems, including problems in combinatorial optimization, scheduling, and routing.

Greedy algorithms are relatively simple to understand and implement. Because they involve making locally optimal choices at each step, the logic behind them is often straightforward and easy to follow. This can make it relatively easy to design and implement a greedy algorithm for a particular problem.

Another benefit of greedy algorithms is that they can be very efficient. In many cases, a greedy algorithm will be able to find a solution to a problem in a much shorter amount of time than other algorithms. This can be particularly useful for problems where speed is a concern, such as when processing large amounts of data in real-time. They are often easy to modify and adapt to new problems. Because they involve making locally optimal choices, it is often straightforward to modify a greedy algorithm to handle different types of input data or to optimize for different objectives.

Limitations

However, the greedy approach has some limitations and may not always yield the optimal solution. One of the main drawbacks of the greedy algorithm is that it is prone to making suboptimal choices at intermediate steps, which can lead to a suboptimal overall solution. This happens because the greedy approach only considers the immediate gain and does not look ahead to see how future choices might affect the overall result.

One limitation of greedy algorithms is that they may not always find the optimal solution. For example, consider the problem of finding the shortest path through a graph from one node to another. A greedy algorithm might make the locally optimal choice at each step by always choosing the neighbor node that is closest to the destination. However, this strategy may not always lead to the shortest overall path. In some cases, it may be necessary to take a longer path at some intermediate steps in order to ultimately arrive at a shorter overall solution.

Another limitation of greedy algorithms is that they may not work for all problems. In order for a greedy algorithm to be effective, the problem must have the “greedy property,” meaning that the locally optimal choices made by the algorithm will lead to a globally optimal solution. If a problem does not have this property, a greedy algorithm may not produce a correct solution.

​​The approach for optimize_output has a time complexity of O(n), making it more efficient than the recursive solution. However, it may not always find the globally optimal solution, as it is based on making locally optimal choices rather than considering the entire problem at once.

Example

Suppose we have the following list of numbers :[5, 6, 9, 7]

If we use the greedy approach, the first player will choose the number 7, and the second player will choose the number 9. The maximum output for the first player would be 7 + 6 = 13.

However, if the first player instead chooses the number 5, and the second player chooses the number 7, the maximum output for the first player would be 5 + 9 = 14. This is a better result than the one obtained using the greedy approach.

This happens because the greedy approach only considers the immediate gain and does not look ahead to see how future choices might affect the overall result. In this case, the greedy approach makes the mistake of choosing the larger element first, while a better strategy would have been to choose the smaller element first in order to leave the larger element for the second player.

In conclusion, the optimize_output function is a simple and intuitive greedy algorithm that is easy to design and implement and often produces good results in practice. However, it is prone to making suboptimal choices at intermediate steps and may not always yield the optimal solution. Therefore, it is important to carefully consider the limitations of the greedy approach used in this function and to use other algorithms or approaches if they are more suitable for the problem at hand.

Learn more about optimization through play with Numbrail

Numbrail is an optimization puzzle that challenges players to use strategic thinking and optimization skills to collect 61 points in a structured game environment. The game is designed to spark interest in computer science and to remind players that optimization is an essential part of everyday life. Numbrail started as a technical interview question and has evolved into a complex and challenging game.