Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for f(x∧y) and f(x⊕y) with monotone functions f , where ∧ and ⊕ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions f which alternate values for a relatively small number of times on any monotone path from 0n to 1n. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.