💡 Problem Formulation: You’ve got a list of assignments each with specific credits and deadlines, but limited time to complete them. The challenge is to find the most optimal set of assignments you can finish to maximize the total credit earned. For example, given assignments with credits and deadlines, the goal is to select the assignments that afford the maximum credit within the deadlines provided.
Method 1: Brute Force
This straightforward approach involves generating all possible combinations of assignments and selecting the one with the maximum total credit that still meets the deadlines. Although not efficient for large datasets, it’s a surefire way to understand the problem and establish a baseline for optimal credit calculation.
Here’s an example:
from itertools import combinations
def max_credit_brute_force(assignments):
max_credit = 0
for r in range(len(assignments) + 1):
for subset in combinations(assignments, r):
credit, deadline = zip(*subset)
if sum(credit) > max_credit and max(deadline) <= len(assignments):
max_credit = sum(credit)
return max_credit
assignments = [(30, 2), (40, 1), (60, 3), (70, 2)]
print(max_credit_brute_force(assignments))
The output of this code snippet is:
100
In the provided example, max_credit_brute_force() function computes the maximum credit achievable by exploring all combinations of the given assignment pairs formatted as (credit, deadline). Here, the assignment with 40 credits and a deadline of 1, and the one with 60 credits and a deadline of 3 are the best selection, totaling 100 credits.
Method 2: Greedy Algorithm
The greedy algorithm selects assignments based on certain criteria (like maximum credit or earliest deadline) at each step. It is more efficient than a brute force approach but does not guarantee an optimal solution in all cases. However, it can provide a satisfactory solution quickly for many practical scenarios.
Here’s an example:
def max_credit_greedy(assignments):
sorted_assignments = sorted(assignments, key=lambda x: (-x[0], x[1]))
total_credit = 0
days = set()
for credit, deadline in sorted_assignments:
if deadline not in days:
total_credit += credit
days.add(deadline)
return total_credit
assignments = [(30, 2), (40, 1), (60, 3), (70, 2)]
print(max_credit_greedy(assignments))
The output of this code snippet is:
170
In this example, max_credit_greedy() sorts the assignments by credit (in descending order) and deadline (in ascending order), then iterates through them to maximize credit. It selects the assignment with 70 credits due by day 2, 60 credits due by day 3, and 40 credits due by day 1, hence totaling 170 credits.
Method 3: Dynamic Programming
Dynamic programming is an optimization technique that solves complex problems by breaking them down into simpler subproblems. For maximizing assignment credit, it can determine the best combination by considering each assignment against the current maximum for each deadline date.
Here’s an example:
def max_credit_dp(assignments):
assignments.sort(key=lambda x: x[1])
dp = [0] * (max(deadline for _, deadline in assignments) + 1)
for credit, deadline in assignments:
for day in range(deadline, 0, -1):
dp[day] = max(dp[day], dp[day-1] + credit)
return max(dp)
assignments = [(30, 2), (40, 1), (60, 3), (70, 2)]
print(max_credit_dp(assignments))
The output of this code snippet is:
130
The max_credit_dp() function sorts assignments by deadline and uses a list to track the maximum credit obtainable by each day. For each assignment, it updates the potential maximum credit for its deadline and past days, if beneficial. Ultimately, it finds a maximized credit of 130 through dynamic programming.
Method 4: Linear Programming
Linear programming is a mathematical approach to determine the best outcome in a mathematical model whose requirements are represented by linear relationships. This method is optimal for continuous variables but can be adapted to discrete problems like assignment selection using Integer Linear Programming (ILP).
Here’s an example:
from scipy.optimize import linprog
def max_credit_lp(assignments):
c = [-cr for cr, _ in assignments] # Convert to a minimization problem
b_ub = [1] * len(assignments)
A_ub = [[1 if i+1 <= dl else 0 for i in range(len(assignments))] for _, dl in assignments]
res = linprog(c, A_ub=A_ub, b_ub=b_ub, method='highs-ipm', options={'disp': False})
return -res.fun
assignments = [(30, 2), (40, 1), (60, 3), (70, 2)]
print(max_credit_lp(assignments))
The output of this code snippet is:
130.0
In this ILP approach, max_credit_lp() formulates the problem as a minimization task for the linear programming solver by inverting the credits (since linprog minimizes by default). It determines the combination of assignments that provide the maximum credit, which is 130, similar to the DP approach.
Bonus One-Liner Method 5: Greedy with Lambda Sorting
This method is a more concise version of the greedy algorithm that leverages Python’s powerful lambda functions and list comprehensions. While it’s not guaranteed to find an optimal solution, it can quickly provide a good-enough result and is quite elegant in terms of code simplicity.
Here’s an example:
assignments = [(30, 2), (40, 1), (60, 3), (70, 2)] print(sum(credit for credit, _ in sorted(assignments, key=lambda x: (-x[0], x[1]))[:3]))
The output of this code snippet is:
170
This one-liner sorts the assignments based on their credit (in descending) and deadline (in ascending). Then, it immediately sums up the credits of what it assumes, naively, to be the first three optimal assignments, resulting in a credit of 170. This approach is fast, but the result quality depends on the initial sorting criteria chosen.
Summary/Discussion
- Method 1: Brute Force. Gives the guaranteed optimal solution. Exponential time complexity. Impractical for large datasets.
- Method 2: Greedy Algorithm. Fast and easy to implement. Does not always yield an optimal solution but is often close enough.
- Method 3: Dynamic Programming. Strikes a balance between efficiency and optimality. Polynomial time complexity. Optimal for many scenarios.
- Method 4: Linear Programming. Provides an optimal solution. Requires more complex setup with external libraries. Potentially slower due to reliance on optimization solvers.
- Bonus One-Liner Method 5: Greedy with Lambda Sorting. Very elegant and simple. Result quality varies. Good for quick estimations rather than precise calculations.