Algorithm Analysis and Optimization Questions

Assess the ability to analyze, compare, and optimize algorithmic solutions with respect to time and space resources. Candidates should be fluent in Big O notation and able to identify dominant operations, reason about worst case, average case, and amortized complexity, and calculate precise time and space bounds for algorithms and data structure operations. The topic includes recognizing complexity classes such as constant time, logarithmic time, linear time, linearithmic time, quadratic time, and exponential time, and understanding when constant factors and lower order terms affect practical performance. Candidates should know and apply common algorithmic patterns and techniques, including two pointers, sliding window, divide and conquer, recursion, binary search, dynamic programming, greedy strategies, and common graph algorithms, and demonstrate how to transform brute force approaches into efficient implementations. Coverage also includes trade offs between time and space and when to trade memory for speed, amortized analysis, optimization tactics such as memoization, caching, pruning, iterative versus recursive approaches, and data layout considerations. Candidates must be able to reason about correctness, invariants, and edge cases, identify performance bottlenecks, and explain practical implications such as cache behavior and memory access patterns. For senior roles, be prepared to justify precise complexity claims and discuss optimization choices in system level and constrained environment contexts.

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What is the worst-case time complexity of quicksort? Explain why pivot selection matters for average-case performance. In practice, what small-array optimization is commonly used inside quicksort implementations to improve performance and why?

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You compute pairwise distances between m query points and n reference points in d dimensions. Analyze time and space complexity of the naive approach. Then describe algorithmic strategies to reduce computation for large n and d: indexing structures, approximate methods, dimensionality reduction. When is approximate NN preferable in real systems?

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Implement in Python a function prefix_sums(arr) that returns a list where prefix[i] is the sum of arr[0..i]. Provide an O(n) time, O(n) space implementation and then explain how you would modify it to run in O(1) extra space if modifying input is allowed. Discuss complexity and edge cases (empty input, large integers).

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Consider the Python loop:

for i in range(n): x = arr[i] * 2

Explain the time and space complexity of this snippet when arr is (a) a Python list and (b) a NumPy ndarray. Discuss practical performance differences (constant factors, memory access patterns) important for large n in data pipelines.

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Define vectorization in the context of NumPy and pandas. Compare the complexity and practical performance of a vectorized sum over a large column vs a Python loop summing each row. Provide tips for when vectorization increases memory usage and how to handle memory-pressure (chunking, in-place ops).

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