Started fine with the basic search, but I fumbled when they asked me to be explicit about my invariant.
Start by clearly defining your interval convention (closed [lo, hi] or half-open [lo, hi)) and state the loop invariant that holds before and after each iteration. Implement iterative binary search with careful mid calculation and boundary updates, then prove termination by showing the interval strictly shrinks. Finally, generalize to a predicate-based search that finds the leftmost true by maintaining the invariant that the answer is always within the current range and adjusting based on the predicate.
Pro tip: Explicitly discuss the trade-offs between closed and half-open intervals—closed intervals are more intuitive but require careful handling of inclusive bounds, while half-open intervals avoid off-by-one errors and are preferred in production code. Mention that using mid = lo + (hi - lo) // 2 prevents overflow in languages with fixed-size integers.
Decide whether to use a closed interval [lo, hi] or half-open [lo, hi). Clearly state the loop invariant, e.g., 'the target, if present, lies within the current interval'.
Write the loop with correct termination condition (lo <= hi for closed, lo < hi for half-open), compute mid safely, and update lo/hi based on comparison with the target.
Show that each iteration strictly reduces the interval size, ensuring termination. Argue that the invariant is maintained and that upon exit, the result is correct.
Generalize by replacing the target comparison with a monotonic predicate. Maintain the invariant that the leftmost true is within the range, and adjust lo/hi to converge to the first true.
Mention testing with empty arrays, single elements, all true/false, and targets at boundaries to validate the implementation.
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