Equivalences between triangle and range query problems

Abstract

We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge $e$ in an $m$-edge graph the number of triangles through $e$. This natural triangle problem can be solved using the best known triangle counting algorithm, running in $O(m^{2\omega/(\omega+1)}) \leq O(m^{1.41})$ time. Moreover, if $\omega=2$, the $O(m^{2\omega/(\omega+1)})$ running time is known to be tight (within $m^{o(1)}$ factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound. To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the $(\min,\max)$-product.