Confused about the for loop in the Set Covering Problem (Greedy Algorithms)

Question

This is my first time learning data structure and algorithms. I am stuck in the Grokking Algorithm Chapter 8 Greedy Algorithms, where there is a simple Set Covering Problem given to the readers.

Is there a simpler example of source code which I can refer to understand the concept behind this example?

Can somebody explain how set intersection helps to get the ultimate result of {'k4', 'k5'} in plain English? How is the array of states_needed passed in to a hash and gets converted to a set?

states_needed = set(["a", "b", "c", "d", "e", "f", "g", "h"])

stations = {}

stations["k1"] = set(["a", "b", "c"])
stations["k2"] = set(["c", "d", "e"])
stations["k3"] = set(["f", "g", "h"])
stations["k4"] = set(["a", "b", "c", "d"])
stations["k5"] = set(["e", "f", "g", "h"])

final_stations = set()

while states_needed:
  best_station = None
  states_not_covered = set()
  
  for station, states_for_station in stations.items():
    covered = states_needed & states_for_station
    if len(covered) > len(states_not_covered):
      best_station = station
      states_not_covered = covered

  states_needed -= states_not_covered
  final_stations.add(best_station)

print(final_stations)

Answer 1

1. Initialize Data:

   • states_needed: A set of states that need coverage.
   • stations: A dictionary where each key is a station name, and the corresponding value is a set of states that this station covers.

2. Greedy Algorithm Implementation:

   • final_stations: A set that will eventually contain the final selection of stations.
   • The while loop runs as long as there are states that still need coverage.
   • In each iteration of the loop, the code tries to find the best station that covers the maximum number of uncovered states.

3. Finding the Best Station:

   • best_station: Temporarily holds the best station found in the current iteration.
   • states_not_covered: A set to keep track of states covered by the best_station.
   • The for loop iterates over each station and calculates the set of uncovered states (covered) that this station would cover.
   • If the current station covers more states than the current best (comparing lengths of covered and states_not_covered), it becomes the new best_station, and states_not_covered is updated.

4. Updating States and Final Stations:

   • After finding the best station for the iteration, states_needed is updated to remove the states that are now covered.
   • The chosen best_station is added to final_stations.

5. Output:

   • Once all states are covered (i.e., states_needed becomes empty), the loop exits.
   • The final_stations set now contains the station names that collectively cover all the required states with the least number of stations (according to the greedy approach).

6. Print Result:

   • The code prints the final_stations, which is the set of stations chosen by the algorithm.