Linear Time Constructions of Some $$d$$-Restriction Problems

Abstract

We give new linear time globally explicit constructions of some drestriction problems that follows from the techniques used in [1,30,31].

Exact learning of juntas from membership queries

“…[14,17] There is a deterministic construction for an (n, d)-universal set of size s that runs in time T where…”

Exact learning from an honest teacher that answers membership queries

“…When the class is also embedding closed, Bshouty and Hellerstein, [57], show Lemma 10. [41,57] Let C be a projection and embedding closed class. If C is adaptively learnable in time T (n) with Q(n) queries, then C r is adaptively learnable in time poly(T (n)) with…”

“…random vectors a ∈ {0, 1} n , where each a j is 1 with probability r/(s + r), is a (n, (s, r))-CFF with probability at least 1 − δ. It follows from [38,41,52,134] that there is a polynomial time (in the size of the CFF) deterministic construction of (n, (s, r))-CFF of size…”

“…where the o(1) is with respect to min(r, s). When r = o(s), the construction can be done in linear time [41,52].…”

“…Let H be a family of functions h : [n] → [q]. For d ≤ q we say that H is an (n, q, d)-perfect hash family ((n, q, d)-PHF) [24] if for every subset S ⊆ [n] of size |S| = d there is a hash function h ∈ H such that h| S is injective (one-to-one) on S, i.e., |h(S)| = d. In [41] it is shown that for q ≥ 2d 2 . There is a (n, q, d)-PHF of size O d 2 log n/log(q/d 2 ) that can be constructed in time O(qd 2 n log n/ log(q/d 2 )).…”