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This work analyzes effective-resistance sparsification using a pessimistic estimator that can itself be computed in catalytic logspace and shows that such a pessimistic estimator can be transformed into an algorithm that performs an in-place compression of a string with bad potential.
The main result is an algorithm that $(1+\epsilon)$-approximates distance to monotonicity in $\sqrt{n}^{1+o(1)}$ passes, and it is proved that this is the best pass-complexity one can hope for, for any $O(1)$-approximation, short of improving the state-of-the-art streaming algorithm for st-reachability.
We initiate the study of *unate distributions* over $\{\pm1\}^n$ -- a natural analogue of unate Boolean functions -- by considering two basic testing problems that parallel well-studied questions for monotone distributions: - Uniformity Testing of Unate Distributions: We show that $\widetilde{\Theta}(n^{3/2})$ samples...
The main result shows that the exact average-case complexity of this fundamental problem is $\Theta(nk)$ for every $k \leq n^{c'}$ and some $c'\in (0, 1)$, and reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input.
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