Range Sum Query - Immutable
Given an integer array `nums`, handle multiple queries of the following type: Calculate the sum of elements of `nums` between indices $i$ and $j$ inclusive. Use a Pre-calculated Sum array.
Why Interviewers Ask This
Microsoft interviewers use this problem to evaluate a candidate's ability to optimize time complexity through preprocessing. They specifically test if you recognize that repeated range queries on static data warrant an O(1) lookup strategy rather than recalculating sums, demonstrating foundational algorithmic thinking and memory-time trade-off awareness.
How to Answer This Question
1. Clarify Requirements: Confirm if the array is mutable or immutable and how many queries to expect. Microsoft values precise communication before coding.
2. Analyze Complexity: Discuss the naive O(N) per query approach versus the desired optimization for multiple queries.
3. Propose Prefix Sum: Explain the concept of storing cumulative sums in a pre-calculated array where index i holds the sum from start to i.
4. Derive Formula: Articulate the math: sum(i, j) equals prefixSum[j] minus prefixSum[i-1], handling edge cases where i is zero.
5. Implement & Test: Write clean code with proper initialization and validate against boundary conditions like single-element ranges or full-array sums.
Key Points to Cover
- Explicitly identifying the trade-off between preprocessing time and query speed
- Correctly deriving the formula: prefixSum[j] - prefixSum[i-1]
- Handling the edge case where the starting index i is zero
- Demonstrating understanding of O(1) query complexity after O(N) setup
- Maintaining clarity on the immutable nature of the input array
Sample Answer
I would start by acknowledging that while calculating a sum directly takes linear time relative to the range size, doing this repeatedly becomes inefficient. Since the problem specifies an immutable array, we can preprocess the data once to answer any query in constant time.
My approach involves creating a prefix sum array. I'll initialize a new array where each element at index k stores the sum of all elements from the original array's start up to index k. This preprocessing step takes O(N) time.
Once built, answering a query between indices i and j becomes a simple subtraction. The sum from i to j is simply the cumulative sum at j minus the cumulative sum just before i (at i-1). If i is zero, the result is just the value at j. This reduces the query time to O(1).
For example, given nums [1, 3, 5], the prefix array becomes [1, 4, 9]. A query for range [1, 2] returns prefix[2] - prefix[0], which is 9 - 1 = 8. This matches 3 + 5. This method aligns well with Microsoft's focus on scalable solutions, ensuring performance remains high even with thousands of subsequent queries without re-scanning the data.
Common Mistakes to Avoid
- Failing to handle the boundary condition when the left index is zero, leading to index out-of-bounds errors
- Implementing a loop inside the query function instead of using the pre-calculated array, missing the optimization goal
- Not discussing the space complexity cost of maintaining the additional prefix sum array
- Ignoring the constraint that the array is immutable, potentially suggesting unnecessary mutation strategies
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