My first instinct was just to scan the range each query, which is obviously too slow at 10^7 elements and 10^5+ queries.
Use a prefix sum array where each element at index i stores the cumulative count of 1s from the start up to i. For a query [l, r], return prefix[r] - prefix[l-1] (or prefix[r] if l=0), achieving O(1) query time after O(n) preprocessing. Since the array is immutable, this is optimal and simple.
Pro tip: Mention that the prefix sum array can be built in a single pass and uses O(n) extra space, but if memory is a concern, you could use a bit-packed representation or a Fenwick tree for O(log n) queries with less space—though O(1) is usually preferred. Also, clarify that the array is 0-indexed or 1-indexed as per the problem statement.
Confirm the array size, whether indices are 0-based or 1-based, and the expected number of queries. This ensures the solution meets performance requirements.
Since the array is immutable and queries are frequent, a prefix sum array is ideal for O(1) query time. Discuss alternatives like Fenwick trees if updates were allowed, but emphasize immutability.
Describe how to build the prefix sum array in O(n) time: iterate through the array, maintaining a running sum of 1s, and store it at each index.
For a query [l, r], compute the count as prefix[r] - prefix[l-1] (with prefix[-1] = 0). Handle edge cases like l=0.
State that preprocessing is O(n), queries are O(1), and space is O(n). Mention that this is optimal for immutable arrays, but if memory is tight, consider a more space-efficient structure like a bit vector with rank/select, though it may have higher constant factors.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.