The Sparsest Additive Spanner via Multiple Weighted BFS Trees

Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing (+2) and (+4)-all-pairs spanners, the sparsest case of (+6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a (+6)-spanner with the essentially-optimal sparsity of O~(n^{4/3}) edges. We then show a distributed implementation of our algorithm, answering an open problem in [Keren Censor{-}Hillel et al., 2016]. A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in |S|+D-1 rounds, where S is the set of sources and D is the diameter of the network graph

The idemetric property: when most distances are (almost) the same

We introduce the idemetric property, which formalizes the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst small-world network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that small-world network models such as the Watts-Strogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property. We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs, we observe, for example, that a single breadth-first search provides a solution to the all-pairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralized algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralized) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than in the worst case in order to achieve stretch less than 3 with high probability: while Ω(n 2) routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is O(nlog(n))

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}

Navigability of temporal networks in hyperbolic space

Information routing is one of the main tasks in many complex networks with a communication function. Maps produced by embedding the networks in hyperbolic space can assist this task enabling the implementation of efficient navigation strategies. However, only static maps have been considered so far, while navigation in more realistic situations, where the network structure may vary in time, remain largely unexplored. Here, we analyze the navigability of real networks by using greedy routing in hyperbolic space, where the nodes are subject to a stochastic activation-inactivation dynamics. We find that such dynamics enhances navigability with respect to the static case. Interestingly, there exists an optimal intermediate activation value, which ensures the best trade-off between the increase in the number of successful paths and a limited growth of their length. Contrary to expectations, the enhanced navigability is robust even when the most connected nodes inactivate with very high probability. Finally, our results indicate that some real networks are ultranavigable and remain highly navigable even if the network structure is extremely unsteady. These findings have important implications for the design and evaluation of efficient routing protocols that account for the temporal nature of real complex networks.Comment: 10 pages, 4 figures. Includes Supplemental Informatio

Compact routing on the Internet AS-graph

Compact routing algorithms have been presented as candidates for scalable routing in the future Internet, achieving near-shortest path routing with considerably less forwarding state than the Border Gateway Protocol. Prior analyses have shown strong performance on power-law random graphs, but to better understand the applicability of compact routing algorithms in the context of the Internet, they must be evaluated against real- world data. To this end, we present the first systematic analysis of the behaviour of the Thorup-Zwick (TZ) and Brady-Cowen (BC) compact routing algorithms on snapshots of the Internet Autonomous System graph spanning a 14 year period. Both algorithms are shown to offer consistently strong performance on the AS graph, producing small forwarding tables with low stretch for all snapshots tested. We find that the average stretch for the TZ algorithm increases slightly as the AS graph has grown, while previous results on synthetic data suggested the opposite would be true. We also present new results to show which features of the algorithms contribute to their strong performance on these graphs