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This logarithmic cost ( O(\log n) ) is the core efficiency feature.

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Consider two parties, Alice and Bob, each possessing a set of data blocks. They wish to find differing blocks with minimal communication. Using Merkle trees, they compare root hashes. If roots differ, they recursively compare children. The number of hash exchanges is proportional to the number of differing blocks times ( \log n ). This is exponentially better than sending all blocks. This logarithmic cost ( O(\log n) ) is