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Maximize Minimum Scaled Value

UberSoftware Engineer · Online Assessment
HardSoftware EngineerOnline AssessmentCoding & Algorithms

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This problem tests your ability to use binary search on the answer space to solve a constrained resource allocation problem effectively.

Problem Statement

You are managing a fleet of vehicles, each represented by an initial capacity arr[i]. You have k additional capacity units to distribute among these vehicles. Each unit added to a vehicle i increases its capacity by exactly 1. You want to maximize the minimum capacity among all vehicles after distributing all k units.

Write a function maximizeMinCapacity(arr, k) that returns the highest possible minimum capacity achievable.

Input

  • arr: A list of non-negative integers representing current capacities.
  • k: A non-negative integer representing the total units available to distribute.

Constraints

  • 1 <= len(arr) <= 10^5
  • 0 <= arr[i] <= 10^9
  • 0 <= k <= 10^18

Examples

Example 1: Input: arr = [2, 3, 5], k = 4 Output: 4 Explanation: We can add 2 units to the first vehicle and 1 unit to the second, resulting in [4, 4, 5]. The minimum is 4. Further distribution would reduce the minimum relative to the others.

Example 2: Input: arr = [1, 1, 1], k = 2 Output: 1 Explanation: Adding the 2 units results in [2, 2, 1]. The minimum capacity remains 1.

Edge Cases

  • k = 0: The answer is simply the minimum element in arr.
  • Large k: The answer may exceed the maximum value in the original array.
  • Empty list: Though excluded by constraints, consider if the array size is 1.

Hints

Hint 1

The function that determines if a minimum value 'X' is achievable is monotonic; if 'X' is possible, any value less than 'X' is also possible.

Hint 2

Binary search for the answer in the range between the current minimum of the array and the maximum possible value (initial minimum + k).

Hint 3

To check if a target minimum 'T' is feasible, calculate the sum of differences `max(0, T - arr[i])` for all elements and ensure this sum is less than or equal to k.

Follow-up questions

  • How would the solution change if each vehicle had a specific maximum capacity limit?
  • What is the time complexity of your solution, and can you explain why binary search is more efficient than a greedy approach with a priority queue?
  • If 'k' is very large, could we use a mathematical formula to find the answer directly instead of binary search?