Lower bounds for linear degeneracy testing

Lower Bounds for Linear Degeneracy Testing NIR AILON AND BERNARD CHAZELLE

In the late nineties, Erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry: given n numbers, do any r of them add up to 0? His lower bound of (n r/2 ), for any fixed r , is optimal if the polynomials at the nodes are linear and at most r -variate. We generalize his bound to s-variate polynomials for s > r . Erickson’s ...

Tight Lower Bounds for Testing Linear Isomorphism

We study lower bounds for testing membership in families of linear/affine-invariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} → {0, 1}} is linear/affine invariant if for any f ∈ P , it is the case that f ◦L ∈ P for any linear/affine transformation L of the domain. Motivated by the recent resurgence of attention to the permutation isomorphism problem, we first focus...

Improved Bounds for 3SUM, k-SUM, and Linear Degeneracy

Given a set of n real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Grønlund and Pettie [FOCS’14], a simple Θ(n2)-time deterministic algorithm for this problem was conjectured to be optimal. Over the years many algorithmic problems have been shown to be reducible from the 3SUM problem or its variants, including the more g...

Some Lower Bounds for Estrada Index

Notes on Lower-Bounds for Testing

Recall that we showed that there is an algorithm for testing triangle-freeness of dense graphs that had a dependence on 1/ that was quite high. While we cannot exactly match that dependence with a lower bound, we’ll show that a super-polynomial dependence is necessary. Here we’ll give the proof only for the one-sided error case, and note that it can be extended to two-sided error algorithms. Na...