Mathematical Foundations of Computer Science 2010

Volume 6281 of the series Lecture Notes in Computer Science pp 221-232

Toward a Deterministic Polynomial Time Algorithm with Optimal Additive Query Complexity

  • Nader H. BshoutyAffiliated withLancaster UniversityTechnion - Israel Institute of Technology
  • , Hanna MazzawiAffiliated withLancaster UniversityTechnion - Israel Institute of Technology

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In this paper, we study two combinatorial search problems: The coin weighing problem with a spring scale (also known as the vector reconstructing problem using additive queries) and the problem of reconstructing weighted graphs using additive queries. Suppose we are given n identical looking coins. Suppose that m out of the n coins are counterfeit and the rest are authentic. Assume that we are allowed to weigh subsets of coins with a spring scale. It is known that the optimal number of weighing for identifying the counterfeit coins and their weights is at least
$$\Omega\left(\frac{m\log n}{\log m}\right).$$
We give a deterministic polynomial time adaptive algorithm for identifying the counterfeit coins and their weights using
$$ O\left(\frac{m\log n}{\log m}+ m\log \log m\right) $$
weighings, assuming that the weight of the counterfeit coins are greater than the weight of the authentic coin. This algorithm is optimal when m ≤ n c/loglogn , where c is any constant. Also our weighing complexity is within loglogm times the optimal complexity for all m.

To obtain this result, our algorithm makes use of search matrices, the divide and conquer approach and the guess and check approach.

When combining these methods with the technique introduced in [Optimally Reconstructing Weighted Graphs Using Queries. SODA, 2010], we get a similar positive result for the problem of reconstructing a hidden weighted graph using additive queries.