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 which previous job could have preceded the current one without overlap. Once sorted, I'd define a dynamic programming array where dp[i] stores the maximum profit achievable considering jobs from index i to the end of the list.
For each job i, we face two choices. First, we can skip the job entirely, taking the value from dp[i+1]. Second, we can accept the job. If we accept it, we must find the latest job that finishes before job i starts. Since our list is sorted by end time, we can use binary search to locate this compatible job index in logarithmic time. We then add the current job's profit to the maximum profit obtained from that compatible index.
The recurrence relation becomes: dp[i] = max(dp[i+1], profit[i] + dp[nextNonOverlappingIndex]). By iterating backwards from the last job to the first, we fill our table. Finally, dp[0] will contain the global maximum profit. This approach reduces the complexity from O(n^2) to O(n log n) due to the binary search step, which aligns well with Google's expectation for scalable algorithmic solutions handling large datasets efficiently.
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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