Expected query complexity of symmetric boolean functions

Abstract

Abstract-The problem of finding optimal querying policy, for expected query complexity of symmetric boolean threshold functions was solved in [1] in the context of collocated networks. In this paper, instead of considering the optimal policy to compute the functions, we define the problem of verification of the function value. We use this idea to provide a simpler proof of the optimal querying policy for threshold functions. The method is more generic and is extended to delta and some other symmetric functions. We also provide some partial results for interval functions and finally address a question posed in [1]. Recently we have extended these results to any symmetric function of boolean inputs, which we mention at the end.

The Stochastic Score Classification Problem

“…It follows from the bound of Deshpande et al on Adaptive Greedy for Stochastic Submodular Cover, that this strategy is an O(log Q)-approximation to the optimal adaptive strategy for evaluating f [8]. 1 We construct g as follows. Let r(x) = a 1 x 2 + .…”

“…, α B+1 and the a i . 1 Golovin and Krause originally claimed an O(log Q) bound for Stochastic Submodular Cover [9], but the proof was recently found to have an error [16]. They have since posted a new proof with an O(log 2 Q) bound [10].…”

“…Let denote this level. In the first phase, we test both x l and x n−l , testing first the variable whose probability is closest to 1 2 . In the second phase, we abandon the round robin and instead continue down a single branch in the adaptive tree.…”

“…Recall that since c < 1/2, we have c < 1 − c. Also, the first phase ends if all bits have been tested, which implies that for all l in the first phase, l ≤ (n − 1)/2 so p n−l ≤ p l . We break the first phase into two parts: (1) The first part consists of all levels l where p n−l ≤ c < 1 − c ≤ p l . (2) The second part consists of all levels l where p l ∈ (c, 1 − c) or p n−l ∈ (c, 1 − c), or both.…”

“…The unweighted adaptive SSClass problem for arbitrary numbers of classes was studied in the information theory literature [7,1,15], but only for unit costs. The main novel contribution there was to establish an equivalence between verification and evaluation, which we discuss below.…”