💡 Problem Formulation: In this article, we tackle the challenge of determining the minimum number of buses needed to cover all bus stops in a route. Given a list of bus stops where each stop has a certain capacity of passengers it can handle, we aim to find the least number of buses required so that all passengers at each stop can be serviced. For example, if we have an input of stops with capacities [4, 8, 1, 4] and a bus capacity of 10, the desired output would be 2 buses.
Method 1: Greedy Algorithm
This method involves a greedy approach, where we allocate buses to stops in such a way that each bus handles as many stops as it can without exceeding its capacity. The greedy algorithm sorts stops based on their capacities and assigns buses from the largest to the smallest until all stops are serviced.
Here’s an example:
def min_buses(stops, bus_capacity):
stops.sort(reverse=True)
buses_required, idx = 0, 0
while idx < len(stops):
current_load = 0
while idx < len(stops) and current_load + stops[idx] <= bus_capacity:
current_load += stops[idx]
idx += 1
buses_required += 1
return buses_required
print(min_buses([4, 8, 1, 4], 10))Output:
2
This code snippet sorts the bus stops in descending order of their passenger capacities and iteratively sums up the capacities until the bus is full. Once the current bus reaches or exceeds capacity, a new bus is allocated. The process continues until all stops are assigned a bus. This method is intuitive and implements a straightforward strategy for allocation.
Method 2: Dynamic Programming
Dynamic Programming (DP) can be applied to optimize the bus allocation process by breaking the problem into smaller sub-problems and solving them only once. The method stores solutions to sub-problems to avoid repeating work, making it efficient for certain sets of stop capacities.
Here’s an example:
# Example code coming soon...
Output:
# Corresponding output...
The DP code snippet will be explained here…
Method 3: Linear Programming
Linear programming can be used to solve optimization problems like this by constructing a set of linear inequalities to model the problem, and then by using linear programming algorithms and solvers to find the minimum number of buses.
Here’s an example:
# Example code coming soon...
Output:
# Corresponding output...
The linear programming code snippet will be explained here…
Method 4: Backtracking
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction issues. This approach incrementally builds candidates to the solutions and abandons a candidate as soon as it determines that this candidate cannot lead to a final solution.
Here’s an example:
# Example code coming soon...
Output:
# Corresponding output...
The backtracking code snippet will be explained here…
Bonus One-Liner Method 5: Heuristic Simplification
A heuristic simplification often provides a near-instantaneous but approximate solution by using a simplified calculation such as averaging the capacities and dividing by the bus capacity. While not always exact, such methods can be useful for quick estimations.
Here’s an example:
# Example code coming soon...
Output:
# Corresponding output...
The heuristic simplification code snippet will be explained here…
Summary/Discussion
- Method 1: Greedy Algorithm. Efficient for sorted stops list. May not always provide the optimal solution if buses have varied capacities or if the stops have complex capacity patterns.
- Method 2: Dynamic Programming. Can be highly efficient for large and complex datasets, but requires significant memory for storing sub-problem solutions. Implementation can become quite complex.
- Method 3: Linear Programming. Provides an optimal solution and can be tailored to different constraints. However, it requires understanding linear programming and might need third-party solvers.
- Method 4: Backtracking. Exhaustive and guarantees an optimal solution. Can lead to high computational costs for large datasets because it explores all possible combinations.
- Bonus Method 5: Heuristic Simplification. Offers a quick and simple way to get approximate results. Not suitable for precise computation where accuracy is critical.