Maximum Profit in Job Scheduling
Given start time, end time, and profit for $n$ jobs, find the maximum profit you can achieve such that no two jobs overlap. Use DP with Binary Search.
Why Interviewers Ask This
Google asks this to evaluate a candidate's ability to combine dynamic programming with binary search for optimization. It tests whether you can recognize overlapping subproblems, sort data effectively, and optimize an O(n^2) solution to O(n log n). This specific combination reveals how well you handle resource-constrained scheduling scenarios common in Google's infrastructure.
How to Answer This Question
1. Clarify requirements: Confirm if jobs are inclusive or exclusive at endpoints and verify input constraints. 2. Sort strategy: Immediately propose sorting jobs by end time to enable efficient non-overlapping lookups using binary search. 3. Define DP state: Explain that dp[i] represents the maximum profit considering jobs from index i to the end. 4. Formulate recurrence: Detail the choice between skipping job i or taking it plus the best result from the next compatible job found via binary search. 5. Optimize space: Discuss converting the recursive memoization approach into an iterative bottom-up solution to save stack space. 6. Complexity analysis: Explicitly state why sorting takes O(n log n), the loop runs n times, and binary search adds another log n factor, resulting in O(n log n) total time.
Key Points to Cover
- Explicitly stating the decision to sort by end time rather than start time
- Demonstrating the exact logic for finding the next non-overlapping job via binary search
- Clearly defining the DP state as 'maximum profit from index i onwards'
- Analyzing the time complexity as O(n log n) and explaining the source of each term
- Discussing space optimization from recursion to iteration
Sample Answer
To solve the Maximum Profit in Job Scheduling problem efficiently, I would first sort all jobs based on their end times. This is crucial because it allows us to process jobs in a sequence where we can easily determine wh…
Common Mistakes to Avoid
- Sorting by start time instead of end time, which breaks the binary search logic for finding compatible previous jobs
- Using linear search to find the previous non-overlapping job, leading to an inefficient O(n^2) solution
- Failing to handle edge cases where no previous job exists (e.g., returning zero profit for the base case)
- Confusing the DP state definition, such as trying to store profit up to index i instead of from index i onwards
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