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I knew exponential search was the move but fumbled explaining why doubling the index works for bounding n.
First, find the upper bound of the array by exponentially increasing the index (1, 2, 4, 8, ...) until the API returns +infinity. Then perform a binary search on the range [0, high] to find the lower bound. If the target is greater than all elements, return -1.
Pro tip: Clarify the API's behavior: it returns +infinity for out-of-bounds, so you can safely compare with target. Also, discuss edge cases like empty array, target smaller than all elements, and target larger than all elements.
Restate the problem: sorted array, unknown length, API access, O(log n) time. Clarify that lower_bound returns the first index where value >= target, or -1 if none.
Use exponential search to find a high index such that the API returns +infinity. Start with index 1, double until out-of-bounds. This takes O(log n) time.
Perform binary search on the range [0, high] to find the smallest index where value >= target. If no such index (i.e., target > last element), return -1.
Check if the array is empty (API returns +infinity at index 0). If target is greater than all elements, return -1. Otherwise, return the found index.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Walk through each edge case systematically, explaining how your algorithm handles it without errors and maintains correctness. Emphasize the importance of defining clear termination conditions and invariants, and mention how you would test these cases.
Pro tip: Relate edge cases to real-world data scenarios (e.g., empty datasets, skewed distributions) to show practical awareness, and mention that handling edge cases is crucial for robust production code.
Briefly state the algorithm you are using (e.g., binary search for finding a target or insertion point) and its preconditions, such as sorted input.
Explain that an empty array should return a sentinel value (e.g., -1 or None) or raise an appropriate exception, and ensure the algorithm checks for this upfront.
Describe how the algorithm should return the insertion point (e.g., length of array) or indicate not found, depending on the problem, and ensure no out-of-bounds access.
Explain that the algorithm should return the first index (0) if all values are >= target, and ensure the search space is correctly narrowed.
Discuss how duplicates affect the result (e.g., finding first/last occurrence) and how to modify the algorithm to handle them, such as continuing the search after finding a match.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clearly state the algorithm: exponential search to find a range where the target lies, then binary search within that range. Then prove correctness by showing the exponential phase maintains the invariant that the target is within the current bounds, and the binary phase correctly narrows down to the target. Finally, derive the tight upper bound on API calls by analyzing the worst-case number of probes in each phase.
Pro tip: Emphasize that the bound is tight by constructing a worst-case input where the target is at the last possible position, and mention that the constant factors matter in practice for API call optimization.
Explain exponential search: start with bound=1, double it until the element at bound is >= target or bound exceeds array size. Then binary search in the range [bound/2, min(bound, n-1)].
Show that after each doubling, if the target exists, it lies within the current search interval. Use induction: initially interval [0,1] contains target if it's at index 0 or 1; if not, double bound and the target must be > previous bound, so it's in the new interval.
State that binary search on a sorted subarray correctly finds the target if it exists, using the standard loop invariant that the target is within the current low-high range.
Count calls: exponential phase makes O(log k) calls where k is the target index (or n if not found). Binary search makes O(log k) calls. Total O(log k). For worst-case, k = n, so O(log n).
Show that the total number of calls is at most 2*ceil(log2(k+1)) + ceil(log2(k+1)) + O(1) = 3*ceil(log2(k+1)) + O(1). For k = n, this is Θ(log n). Provide a specific worst-case example achieving this bound.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the problem: the array is sorted but rotated, and you need to search for a target. Explain how binary search adapts by identifying the sorted half, and then discuss how duplicates break the ability to determine which half is sorted, requiring a linear scan in the worst case. Emphasize the trade-off between time complexity and correctness, and mention that with duplicates the worst-case time becomes O(n) while without duplicates it remains O(log n).
Pro tip: Acknowledge that in real-world data science, duplicates are common, so you should always consider edge cases like all elements being the same. Also, mention that if the array is rotated multiple times, the same logic applies as long as it's sorted and rotated.
Confirm that the array is sorted and then rotated, and that we need to find a target or determine if it exists. Ask about duplicates and whether the rotation is exactly once.
Describe the modified binary search: at each step, determine which half is sorted by comparing mid with low and high. Then decide which half to search based on the target's value relative to the sorted half.
Show that when duplicates exist, comparing mid with low/high may not reveal the sorted half (e.g., all equal). In such cases, we cannot decide and must fall back to linear scan or shrink the search space by one.
State that without duplicates, the time complexity is O(log n). With duplicates, the worst-case time complexity degrades to O(n) because we may need to scan linearly when many duplicates are present.
Mention that if the array is known to have few duplicates, we can still achieve near O(log n) on average. Also, consider if preprocessing (like removing duplicates) is feasible, but note it changes the problem.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.