The naive approach is obvious: for each k, sort the prefix or check a set.
Track the maximum value seen so far in the prefix and compare it to the prefix length. If the maximum equals the length, then the prefix contains exactly the numbers 1 through k; otherwise, it does not. This yields an O(n) time and O(1) extra space solution.
Pro tip: Mention that since the array is a permutation, the condition max(prefix) == k is both necessary and sufficient. This avoids unnecessary data structures like sets or hash maps, showing you understand the problem's constraints.
Clarify that the array is a permutation of 1..n, so all numbers are distinct and within range. The goal is to check for each prefix whether it contains exactly the set {1,2,...,k}.
Realize that a prefix of length k contains exactly 1..k if and only if the maximum value in that prefix is k. Since all numbers are distinct, if the max is k, then the prefix must contain all numbers from 1 to k.
Iterate through the array while maintaining the maximum value seen so far. For each index i (0-based), update the max and set result[i] = (max == i+1).
The algorithm runs in O(n) time and uses O(1) extra space (besides the output array). This is optimal because we must read the entire input.
Walk through a small example, e.g., [1,3,2,4], to verify the logic: prefixes [1] -> true, [1,3] -> false, [1,3,2] -> true, [1,3,2,4] -> true.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.