Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form

Algebraic techniques have had an important impact on graph algorithms so far. Porting them, e.g., the matrix inverse, into the dynamic regime improved best-known bounds for various dynamic graph problems. In this paper, we develop new algorithms for another cornerstone algebraic primitive, the Frobenius normal form (FNF). We apply our developments to dynamic and fault-tolerant exact distance oracle problems on directed graphs. For generic matrices $A$ over a finite field accompanied by an FNF, we show (1) an efficient data structure for querying submatrices of the first $k\geq 1$ powers of $A$, and (2) a near-optimal algorithm updating the FNF explicitly under rank-1 updates. By representing an unweighted digraph using a generic matrix over a sufficiently large field (obtained by random sampling) and leveraging the developed FNF toolbox, we obtain: (a) a conditionally optimal distance sensitivity oracle (DSO) in the case of single-edge or single-vertex failures, providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a multiple-failures DSO improving upon the state of the art (vd. Brand and Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved dynamic distance oracles in the case of single-edge updates, and (d) a dynamic distance oracle supporting vertex updates, i.e., changing all edges incident to a single vertex, in $\tilde{O}(n^2)$ worst-case time and distance queries in $\tilde{O}(n)$ time.Comment: To appear at FOCS 202

Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

In the Steiner k-Forest problem we are given an edge weighted graph, a collection D of node pairs, and an integer k leq |D|. The goal is to find a minimum cost subgraph that connects at least k pairs. The best known ratio for this problem is min{O(sqrt{n}),O(sqrt{k})} [Gupta et al., 2008]. In [Gupta et al., 2008] it is also shown that ratio rho for Steiner k-Forest implies ratio O(rho log^2 n) for the Dial-a-Ride problem: given an edge weighted graph and a set of items with a source and a destination each, find a minimum length tour to move each object from its source to destination, but carrying at most k objects at a time. The only other algorithm known for Dial-a-Ride, besides the one resulting from [Gupta et al., 2008], has ratio O(sqrt{n}) [Charikar and Raghavachari, 1998]. We obtain ratio n^{0.448} for Steiner k-Forest and Dial-a-Ride with unit weights, breaking the O(sqrt{n}) ratio barrier for this natural special case. We also show that if the maximum weight of an edge is O(n^{epsilon}), then one can achieve ratio O(n^{(1+epsilon) 0.448}), which is less than sqrt{n} if epsilon is small enough. To prove our main result we consider the following generalization of the Minimum k-Edge Subgraph (Mk-ES) problem, which we call Min-Cost l-Edge-Profit Subgraph (MCl-EPS): Given a graph G=(V,E) with edge-profits p={p_e: e in E} and node-costs c={c_v: v in V}, and a lower profit bound l, find a minimum node-cost subgraph of G of edge profit at least l. The Mk-ES problem is a special case of MCl-EPS with unit node costs and unit edge profits. The currently best known ratio for Mk-ES is n^{3-2*sqrt{2} + epsilon} (note that 3-2*sqrt{2} < 0.1716). We extend this ratio to MCl-EPS for arbitrary node weights and edge profits that are polynomial in n, which may be of independent interest

Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

We present a truly subquadratic size distance oracle for reporting, in constant time, the exact shortest-path distance between any pair of vertices of an undirected, unweighted planar graph G. For any ? > 0, our distance oracle requires O(n^{5/3+?}) space and is capable of answering shortest-path distance queries exactly for any pair of vertices of G in worst-case time O(log (1/?)). Previously no truly sub-quadratic size distance oracles with constant query time for answering exact shortest paths distance queries existed

Online Directed Spanners and Steiner Forests

We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $\tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $\tilde{O}(n^{2/3+\epsilon})$-competitive algorithm, and an efficient deterministic $\tilde{O}(k^{1/2+\epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $\tilde{O}(n^{3/5 + \epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $\tilde{O}(n^{2/3 + \epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $\tilde{O}(k^{1/2 + \epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $\tilde{O}(n^{26/45 + \epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion

A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $\Pi$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $\Omega(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^\epsilon)$, for an arbitrarily small $\epsilon > 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(\beta(k))$, for a function $\beta(\cdot)$, space $O(n^{1+1/k} \cdot \beta)$, and query time $O(n^\epsilon)$, for an arbitrarily small constant $\epsilon >0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $\beta(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+\epsilon,\beta)$-spanners due to Elkin and Peleg. However, so far $(1+\epsilon,\beta)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}

Improved Approximate Distance Oracles: Bypassing the Thorup-Zwick Bound in Dense Graphs

Despite extensive research on distance oracles, there are still large gaps between the best constructions for spanners and distance oracles. Notably, there exist sparse spanners with a multiplicative stretch of $1+\varepsilon$ plus some additive stretch. A fundamental open problem is whether such a bound is achievable for distance oracles as well. Specifically, can we construct a distance oracle with multiplicative stretch better than 2, along with some additive stretch, while maintaining subquadratic space complexity? This question remains a crucial area of investigation, and finding a positive answer would be a significant step forward for distance oracles. Indeed, such oracles have been constructed for sparse graphs. However, in the more general case of dense graphs, it is currently unknown whether such oracles exist. In this paper, we contribute to the field by presenting the first distance oracles that achieve a multiplicative stretch of $1+\varepsilon$ along with a small additive stretch while maintaining subquadratic space complexity. Our results represent an advancement particularly for constructing efficient distance oracles for dense graphs. In addition, we present a whole family of oracles that, for any positive integer $k$, achieve a multiplicative stretch of $2k-1+\varepsilon$ using $o(n^{1+1/k})$ space