Maximum Number of Potholes That Can Be Fixed

Maximum Number of Potholes That Can Be Fixed

What is this problem about?

The Maximum Number of Potholes That Can Be Fixed coding problem gives you a string s representing a road, where '.' denotes a smooth section and 'X' denotes a pothole. You also have a certain budget budget to fix potholes. Fixing a pothole costs 1 unit. However, you can fix a contiguous segment of k potholes for free if k is less than or equal to a given threshold K. You want to find the maximum number of potholes you can fix within your budget.

Why is this asked in interviews?

Microsoft and Geico use this problem to test greedy algorithms, particularly when combined with string processing. The core idea is to identify which potholes are "cheapest" to fix, which often involves taking advantage of free repairs for contiguous segments. It assesses a candidate's ability to prioritize and optimize choices.

Algorithmic pattern used

Greedy Approach with Sorting/Iteration: The key to this problem is to maximize the number of fixed potholes while minimizing cost.

1. Identify Contiguous Potholes: First, iterate through the road string s to find all contiguous segments of potholes ('X's). Store the lengths of these segments.
2. Prioritize Free Fixes: Any segment of potholes with length k <= K can be fixed for free. Fix all such segments first, and add k to the total fixed potholes.
3. Sort Remaining Segments: For segments with length k > K, fixing them costs k units. Sort these segments in ascending order of length.
4. Utilize Budget: Iterate through the sorted remaining segments. For each segment, if your budget is sufficient to fix it (cost = length), fix it, deduct length from budget, and add length to total fixed potholes. Continue until the budget is exhausted.

This greedy strategy ensures that you first fix all "free" potholes, then prioritize fixing the smallest (cheapest) non-free segments first, maximizing the total fixed potholes.

Example explanation

s = "X.XX.XXX", budget = 5, K = 2.

1. Contiguous pothole segments:
   • "X": length 1
   • "XX": length 2
   • "XXX": length 3
2. Free fixes (length <= K=2):
   • "X" (length 1): Fix for free. Fixed potholes = 1.
   • "XX" (length 2): Fix for free. Fixed potholes = 1 + 2 = 3.
3. Remaining segments (length > K=2), sorted by length:
   • "XXX": length 3. Cost = 3.
4. Utilize budget:
   • Budget = 5. Current segment "XXX" costs 3. budget >= 3. Fix it.
   • Fixed potholes = 3 + 3 = 6.
   • Budget = 5 - 3 = 2.
5. No more segments. Total fixed potholes = 6.

Common mistakes candidates make

• Not identifying segments correctly: Miscounting contiguous 'X's can lead to errors.
• Incorrect prioritization: Trying to fix larger segments before smaller ones when not necessary, or not taking advantage of free fixes first.
• Off-by-one in budget usage: Ensure budget deduction is accurate.

Interview preparation tip

For the Sorting String Greedy interview pattern, problems involving resource allocation and constraints (like budget) often benefit from a greedy strategy after identifying categories of items (e.g., free vs. costly items). Sorting items by their cost or benefit is a common step in such greedy solutions.