Maximum Running Time of N Computers

Problem Description

You have n computers. You are given the integer n and a 0-indexed integer array batteries where the ith battery can run a computer for batteries[i] minutes. You are interested in running all n computers simultaneously using the given batteries. Initially, you can insert at most one battery into each computer. After that and at any integer time moment, you can remove a battery from a computer and insert another battery any number of times. The inserted battery can be a totally new battery or a battery from another computer. You may assume that the removing and inserting processes take no time. Note that the batteries cannot be recharged. Return the maximum number of minutes you can run all the n computers simultaneously.

Key Insights

• Each computer requires a battery to run, and we need to ensure that all n computers are powered simultaneously.
• We can swap batteries between computers at any point without any time cost.
• The total running time is limited by the total capacity of the batteries divided by the number of computers.
• A binary search approach can help to determine the maximum time efficiently.

Space and Time Complexity

Time Complexity: O(m log m), where m is the number of batteries. This is because we need to sort the batteries at least once and then do binary search. Space Complexity: O(1), as we are using a constant amount of space regardless of the input size.

Solution

To solve the problem, we can use a binary search approach to find the maximum time T for which all n computers can run simultaneously. The key steps are:

1. Calculate the total energy available from all batteries.
2. For a given T, check if we can sustain n computers for T minutes using the batteries.
3. Use binary search to find the maximum feasible T such that the total battery capacity used is greater than or equal to n * T.

Code Solutions