Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism

We present implications from the known conjectures like APSP, 3SUM and ETH in a form of a negated containment of a linear-time with a non-deterministic logarithmic-bit oracle in a respective deterministic bounded-time class They are different for different conjectures and they exhibit in particular the dependency on the input range parameters.Comment: The section on range reduction in the previous version contained a flaw in a proof and therefore it has been remove

All-Pairs LCA in DAGs: Breaking through the O(n2.5)O(n^{2.5}) barrier

Let $G=(V,E)$ be an $n$-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices $u$ and $v$ is a common ancestor $w$ of $u$ and $v$ such that no descendant of $w$ has the same property. In this paper, we consider the problem of computing an LCA, if any, for all pairs of vertices in a DAG. The fastest known algorithms for this problem exploit fast matrix multiplication subroutines and have running times ranging from $O(n^{2.687})$ [Bender et al.~SODA'01] down to $O(n^{2.615})$ [Kowaluk and Lingas~ICALP'05] and $O(n^{2.569})$ [Czumaj et al.~TCS'07]. Somewhat surprisingly, all those bounds would still be $\Omega(n^{2.5})$ even if matrix multiplication could be solved optimally (i.e., $\omega=2$). This appears to be an inherent barrier for all the currently known approaches, which raises the natural question on whether one could break through the $O(n^{2.5})$ barrier for this problem. In this paper, we answer this question affirmatively: in particular, we present an $\tilde O(n^{2.447})$ ($\tilde O(n^{7/3})$ for $\omega=2$) algorithm for finding an LCA for all pairs of vertices in a DAG, which represents the first improvement on the running times for this problem in the last 13 years. A key tool in our approach is a fast algorithm to partition the vertex set of the transitive closure of $G$ into a collection of $O(\ell)$ chains and $O(n/\ell)$ antichains, for a given parameter $\ell$. As usual, a chain is a path while an antichain is an independent set. We then find, for all pairs of vertices, a \emph{candidate} LCA among the chain and antichain vertices, separately. The first set is obtained via a reduction to min-max matrix multiplication. The computation of the second set can be reduced to Boolean matrix multiplication similarly to previous results on this problem. We finally combine the two solutions together in a careful (non-obvious) manner

Equivalences between triangle and range query problems

Copyright © 2020 by SIAM 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(m2ω/(ω+1)) ≤ O(m1.41)time. Moreover, if ω = 2, the O(m2ω/(ω+1)) running time is known to be tight (within mo(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

Equivalences between triangle and range query problems

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