I went straight to ternary search, which was fine, but the interviewer pushed back on the number of F evaluations per iteration.
Use ternary search, which is optimal for finding the minimum of a convex function on an interval, requiring O(log(1/ε)) evaluations. Alternatively, if derivatives are available, use gradient descent or Newton's method, but since it's a black-box, ternary search is the standard. Discuss the trade-offs and mention that ternary search is essentially a derivative-free optimization method.
Pro tip: Mention that ternary search can be improved by using the golden section search, which reuses one evaluation per iteration, reducing the number of function evaluations by about 50% compared to naive ternary search. This shows awareness of practical efficiency.
Confirm that the function is convex, continuous, and unimodal on [a, b]. Ask about the desired precision ε and whether function evaluations are expensive.
Select ternary search or golden section search for derivative-free optimization. Explain why they are optimal for convex functions.
Outline the iterative process: evaluate at two interior points, compare values, and discard the subinterval that cannot contain the minimum. Repeat until interval is small enough.
State that the number of evaluations is O(log((b-a)/ε)) and discuss how to choose the number of iterations to achieve ε precision.
Mention golden section search for fewer evaluations, or if derivatives are available, use gradient-based methods. Also note that for high-dimensional problems, other methods are needed.
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Explain that for a strictly convex function on an integer interval, we can use binary search on the discrete derivative (difference) to find the minimum in O(log n) time, which is better than ternary search's O(log n) but with a smaller constant factor. Then implement the algorithm by comparing f(mid) and f(mid+1) to decide which half to discard.
Pro tip: Mention that ternary search does two evaluations per iteration, while binary search on the derivative does one, making it more efficient in practice. Also, handle edge cases like interval size 1 or 2 explicitly to avoid infinite loops.
Clarify that the function is strictly convex on an integer interval, meaning the discrete derivative is strictly increasing. The goal is to find the integer minimizer efficiently.
Use binary search on the discrete derivative: compare f(mid) and f(mid+1). If f(mid) < f(mid+1), the minimum is at or left of mid; otherwise, it is right of mid.
Initialize low and high bounds. While low < high, compute mid = low + (high - low) // 2. If f(mid) < f(mid+1), set high = mid; else set low = mid + 1. Return low as the minimizer.
Ensure the loop terminates correctly for intervals of size 1 or 2. Test with small intervals to verify correctness.
State that the algorithm runs in O(log n) time with O(1) space, and uses one function evaluation per iteration, improving over ternary search's two evaluations.
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Honestly the trickiest part wasn't the code, it was the assumption you have to state before writing anything.
Start by clarifying the problem constraints (rectangle size, function properties, evaluation cost) and then propose a solution that extends 1D ternary search to 2D. Explain that for strictly convex functions, you can alternate ternary searches on x and y until convergence, or use nested ternary search. Discuss trade-offs between these approaches and mention potential optimizations like gradient-based methods if derivatives are available.
Pro tip: Emphasize that the number of function evaluations matters in practice, especially if f is expensive to compute. Suggest using a more efficient method like coordinate descent with ternary search or even a 2D version of golden-section search to reduce evaluations.
Ask about the size of the rectangle, whether f is differentiable, and if function evaluations are costly. This determines the appropriate algorithm and its complexity.
Describe nested ternary search: for each x, find the optimal y via ternary search, then ternary search over x. Analyze its time complexity (O(log^2 N) evaluations).
Mention alternating ternary search (coordinate descent) which may converge faster in practice. If gradients are available, consider gradient descent with projection or Newton's method.
Compare methods in terms of number of function evaluations, convergence guarantees, and implementation complexity. Highlight that for strictly convex functions, any local minimum is global.
Discuss termination criteria (e.g., when interval size < 1), integer rounding, and ensuring the search stays within the rectangle bounds.
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