Now that we understand the fundamental properties that all MSTs must have, how do we actually find one in a complex graph? This brings us to a powerful and intuitive algorithmic approach: the greedy strategy.

  • A greedy algorithm builds a solution piece by piece, always choosing the option that offers the most immediate benefit.
  • It makes the locally optimal choice at each stage with the hope of finding the globally optimal solution.
  • For MSTs, a greedy approach means picking the cheapest available edge at every step, as long as it doesn't form a cycle.
  • The key question is whether this "shortsighted" approach can guarantee finding the true Minimum Spanning Tree for any graph $G$.
Python: Greedy Coin Change (Expected Output: [25, 10, 10, 1, 1, 1])
1# A classic greedy algorithm: making change with the fewest coins.
2# Note: This works for standard US currency, but not all coin systems!
3def greedy_coin_change(amount_cents):
4    """Returns the minimum number of coins for a given amount."""
5    coins = [25, 10, 5, 1]  # Quarters, Dimes, Nickels, Pennies
6    change = []
7    
8    for coin_value in coins:
9        while amount_cents >= coin_value:
10            change.append(coin_value)
11            amount_cents -= coin_value
12            
13    return change
14
15# Example: Make change for 48 cents
16print(f"Change for 48 cents: {greedy_coin_change(48)}")