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Implement multiplication without using the multiplication operator

OtherData Scientist · Technical Screen
MediumData ScientistTechnical ScreenCoding & Algorithms

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This question evaluates proficiency in bitwise arithmetic, integer overflow semantics, sign handling, and algorithmic efficiency for implementing multiplication without using multiplication or division operators.

Requirements:

  • Handle negatives, zero, and the corner case INT_MIN * −1.
  • Achieve O(log |b|) additions via shift‑and‑add (Russian peasant), not naive O(|b|) repeated addition.
  • Assume 32‑bit signed ints. First implement unchecked overflow (two’s‑complement wrap), then a safe version that saturates to INT_MAX/INT_MIN on overflow without using 64‑bit types.
  • Analyze time and space complexity and prove correctness for all sign combinations.
  • Provide unit tests: 0×x, 1×x, −1×x, powers of two, random pairs, INT_MAX×2, INT_MIN×2, INT_MIN×−1. Follow‑ups: how would you extend to big integers and compare with Karatsuba? What changes if the platform lacks arithmetic right shift?