Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201

Fully Dynamic Approximate Distance Oracles for Planar Graphs via Forbidden-Set Distance Labels

This paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbidden-set labeling schemes for planar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M]andparameterε>0, our forbidden-set labeling scheme uses labels of length λ = O(ε −1 log 2 n log (nM) · (ε −1 +logn)). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1 + ε), in O(|F | 2 λ)time. We then present a general method to transform (1 + ε) forbidden-set labeling schemas into a fully dynamic (1 + ε) distance oracle. Our fully dynamic (1 + ε) distanceoracle is of size O(n log n · (ε −1 +logn)) and has Õ(n1/2)query and update time, both the query and the update time are worst case. This improves on the best previously known (1+ε) dynamic distance oracle for planar graphs, which has worst case query time Õ(n2/3) and amortized update time of Õ(n2/3). Our (1 + ε) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1 + ε)

Exact Distance Oracles for Planar Graphs with Failing Vertices

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex $u$, a target vertex $v$ and a set $X$ of $k$ failed vertices, such an oracle returns the length of a shortest $u$-to-$v$ path that avoids all vertices in $X$. We propose oracles that can handle any number $k$ of failures. More specifically, for a directed weighted planar graph with $n$ vertices, any constant $k$, and for any $q \in [1,\sqrt n]$, we propose an oracle of size $\tilde{\mathcal{O}}(\frac{n^{k+3/2}}{q^{2k+1}})$ that answers queries in $\tilde{\mathcal{O}}(q)$ time. In particular, we show an $\tilde{\mathcal{O}}(n)$-size, $\tilde{\mathcal{O}}(\sqrt{n})$-query-time oracle for any constant $k$. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures ($k=1$), our $\tilde{\mathcal{O}}(\frac{n^{5/2}}{q^3})$-size, $\tilde{\mathcal{O}}(q)$-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for $q = \Omega(n^t)$, $t \in (1/4,1/2]$. For multiple failures, no planarity exploiting results were previously known

Fully Dynamic All-pairs Shortest Paths with Worst-case Update-time revisited

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds

Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs

The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two original shortest paths. However, the lemma is tiebreaking-sensitive: if one selects a particular canonical shortest path for each node pair, it is no longer guaranteed that one can build replacement paths by concatenating two selected shortest paths. They left as an open problem whether a method of shortest path tiebreaking with this desirable property is generally possible. We settle this question affirmatively with the first general construction of restorable tiebreaking schemes. We then show applications to various problems in fault-tolerant network design. These include a faster algorithm for subset replacement paths, more efficient fault-tolerant (exact) distance labeling schemes, fault-tolerant subset distance preservers and $+4$ additive spanners with improved sparsity, and fast distributed algorithms that construct these objects. For example, an almost immediate corollary of our restorable tiebreaking scheme is the first nontrivial distributed construction of sparse fault-tolerant distance preservers resilient to three faults