Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Abstract

We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(n · k+ log n k · 2 k k 2 log n), where k is the treewidth of the graph. For every > 0, this bound is n 1+ exp O(k), which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log d n to d+ log n d , as originally observed by Monier (J. Alg. 1980).We also investigate the parameterization by vertex cover number. ACM Subject ClassificationTheory of computation → Shortest paths, Parameterized complexity and exact algorithms, Computational geometry. Mathematics of computing → Paths and connectivity problems.In fact, this holds under the potentially weaker Orthogonal Vectors conjecture, see [18] for an introduction to these arguments. Thus, under this assumption, the dependency on tw(G) in Theorem 1 cannot be significantly improved, even if the dependency on n is relaxed from just above linear to just below quadratic. Our analysis encompasses the Wiener index, an important structural graph parameter left unexplored by [1].Perhaps surprisingly, the main insight needed to establish Theorem 1 has nothing to do with graph distances or treewidth. Instead, we make-or re-discover-the following observation about the running time of d-dimensional range trees: Lemma 2 ([15]). A d-dimensional range tree over n points supporting orthogonal range queries for the aggregate value over a commutative monoid has query time O(2 d · B(n, d)) and can be built in time O (nd · B(n, d)This is a more careful statement than the standard textbook analysis, which gives the query time as O(log d n) and the construction time as O(n log d n). For many values of d, the asymptotic complexities of these bounds agree-in particular, this is true for constant d and for very large d, which are the main regimes of interest to computational geometers. But crucially, B(n, d) is always n exp O(d) for any > 0, while log d n is not.After Lemma 2 is realised, Theorem 1 follows via divide-and-conquer in decomposable graphs, closely following the idea of Cabello and Knauer [6] and augmented with known arguments [1,5]. We choose to give a careful presentation of the entire construction, as some of the analysis is quite fragile.Using known reductions, this implies that the following multivariate lower bound on orthogonal range searching is tight: K. Bringmann and T. Husfeldt and M. Magnusson 23:3Theorem 3 (Implicit in [1]). A data structure for the orthogonal range query problem for the monoid (Z, max) with construction time n · q (n, d) and query time q (n, d), where q (n, d) = n 1− exp o (d) for some > 0, refutes the Strong Exponential Time hypothesis.We a...

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

“…Bounded-treewidth graphs. More recently, quasi linear-time algorithms for diameter computation on bounded-treewidth graphs were presented in [1,13] with almost optimal dependency on the treewidth parameter. The cornerstone of these algorithms is the use of k-range trees in order to detect the furthest pairs that are disconnected by some small-cardinality separators.…”

“…We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time.…”

Fast Diameter Computation Within Split Graphs

“…Abboud et al proved that under the Strong Exponential-Time Hypothesis (SETH), for any ǫ > 0, there is no O(2 o(k) n 2−ǫ )-time algorithm for computing the diameter of n-vertex graphs of treewidth at most k [1]. An algorithm in O(2 O(k) n 1+o (1) ) time for this problem, thus matching the lower bound of Abboud et al, was proved recently in [11] by using the orthogonal range query framework of Cabello and Knauer [12]. For other aplications of this orthogonal range query framework to graph problems, see [25,26].…”

“…In order to prove Theorem 2, we will use a special instance of k-range tree. Such a data-structure stores a static set of k-dimensional points and it supports the following operation: The use of range queries for diameter computation dates back from [1] (see also [13,19] for some further applications). Roughly, if in a graph G we can find a separator S of size at most k that disconnects a diametral pair of G, then the idea is to compute a "distance profile" for every vertex v / ∈ S w.r.t.…”

“…[13]) For any k ≥ 2 and any n-set of k-dimensional points, we can construct a k-rangetree in time O(k k+1+⌈log n⌉ k+1 n) = O(k2 O(k) n 1+o(1) ) and answer any range query in time O(2 k k+1+⌈log n⌉ k+1 ) = O(2 O(k) n o(1) ).…”