Balancing Degree, Diameter and Weight in Euclidean Spanners

In this paper we devise a novel \emph{unified} construction of Euclidean spanners that trades between the maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+eps)-spanner with maximum degree O(k), diameter O(log_k n + alpha(k)), weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al. \cite{ADMSS95}; in the corresponding result from \cite{ADMSS95} the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Also, for k = O(1) this gives rise to maximum degree O(1), diameter O(log n) and weight O(log^2 n) \cdot w(MST(S)), which reproves another classical result of Arya et al. \cite{ADMSS95}. Our bound of O(log_k n + alpha(k)) on the diameter is optimal under the constraints that the maximum degree is O(k) and the number of edges is O(n). Our bound on the weight is optimal up to a factor of log n. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log^2 n, but the diameter is allowed to grow beyond log n. For random point sets in the d-dimensional unit cube, we "shave" a factor of log n from the weight bound. Specifically, in this case our construction achieves maximum degree O(k), diameter O(log_k n + alpha(k)) and weight that is with high probability O(k \cdot log_k n) \cdot w(MST(S)). Finally, en route to these results we devise optimal constructions of 1-spanners for general tree metrics, which are of independent interest.Comment: 27 pages, 7 figures; a preliminary version of this paper appeared in ESA'1

Incubators vs Zombies: Fault-Tolerant, Short, Thin and Lanky Spanners for Doubling Metrics

Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin and Solomon's spanner construction is extremely complicated; we offer a simple alternative construction that is very intuitive and is based on the standard technique of net tree with cross edges. Indeed, our approach can be readily applied to our previous construction of k-fault tolerant spanners (ICALP 2012) to achieve k-fault tolerance, maximum degree O(k^2), hop-diameter O(log n) and lightness O(k^3 log n)

Dynamic tree shortcut with constant degree

LNCS v.9188 entitled: Computing and Combinatorics: 21st International Conference, COCOON 2015, Beijing, China, August 4-6, 2015, ProceedingsGiven a rooted tree with n nodes, the tree shortcut problem is to add a set of shortcut edges to the tree such that the shortest path from each node to any of its ancestors is of length O(log n) and the degree increment of each node is constant. We consider in this paper the dynamic version of the problem, which supports node insertion and deletion. For insertion, a node can be inserted as a leaf node or an internal node by sub-dividing an existing edge. For deletion, a leaf node can be deleted, or an internal node can be merged with its single child. We propose an algorithm that maintains a set of shortcut edges in O(log n) time for an insertion or deletion.postprin

Light Euclidean Spanners with Steiner Points

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any $d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}{\epsilon}$), and also shows the existence of point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$. Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with lightness $O(\epsilon^{-1} \log \Delta)$, where $\Delta$ is the spread of the point set. In the regime of $\Delta \ll 2^{1/\epsilon}$, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications $\epsilon$ often controls the precision, and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for spread polynomially bounded in $1/\epsilon$, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in $O_{\epsilon}(n)$ time for polynomially bounded spread, where $O_{\epsilon}$ hides a factor of $\mathrm{poly}(\frac{1}{\epsilon})$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(\epsilon^{-(d+1)/2} + \epsilon^{-2}\log \Delta)$ for any $3\leq d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$.Comment: 23 pages, 2 figures, to appear in ESA 2

New Doubling Spanners: Better and Simpler

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Optimal Euclidean spanners: really short, thin and lanky

In a seminal STOC'95 paper, titled "Euclidean spanners: short, thin and lanky", Arya et al. devised a construction of Euclidean (1+\eps)-spanners that achieves constant degree, diameter $O(\log n)$, and weight $O(\log^2 n) \cdot \omega(MST)$, and has running time $O(n \cdot \log n)$. This construction applies to $n$-point constant-dimensional Euclidean spaces. Moreover, Arya et al. conjectured that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time. This conjecture of Arya et al. became a central open problem in the area of Euclidean spanners. In this paper we resolve the long-standing conjecture of Arya et al. in the affirmative. Specifically, we present a construction of spanners with the same stretch, degree, diameter, and running time, as in Arya et al.'s result, but with optimal weight $O(\log n) \cdot \omega(MST)$. Moreover, our result is more general in three ways. First, we demonstrate that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics. Second, we provide a general tradeoff between the three involved parameters, which is tight in the entire range. Third, we devise a transformation that decreases the lightness of spanners in general metrics, while keeping all their other parameters in check. Our main result is obtained as a corollary of this transformation.Comment: A technical report of this paper was available online from April 4, 201

Forward-Secure Encryption with Fast Forwarding

Forward-secure encryption (FSE) allows communicating parties to refresh their keys across epochs, in a way that compromising the current secret key leaves all prior encrypted communication secure. We investigate a novel dimension in the design of FSE schemes: fast-forwarding (FF). This refers to the ability of a stale communication party, that is stuck in an old epoch, to efficiently catch up to the newest state, and frequently arises in practice. While this dimension was not explicitly considered in prior work, we observe that one can augment prior FSEs -- both in symmetric- and public-key settings -- to support fast-forwarding which is sublinear in the number of epochs. However, the resulting schemes have disadvantages: the symmetric-key scheme is a security parameter slower than any conventional stream cipher, while the public-key scheme inherits the inefficiencies of the HIBE-based forward-secure PKE. To address these inefficiencies, we look at the common real-life situation which we call the bulletin board model, where communicating parties rely on some infrastructure -- such as an application provider -- to help them store and deliver ciphertexts to each other. We then define and construct FF-FSE in the bulletin board model, which addresses the above-mentioned disadvantages. In particular, * Our FF-stream-cipher in the bulletin-board model has: (a) constant state size; (b) constant normal (no fast-forward) operation; and (c) logarithmic fast-forward property. This essentially matches the efficiency of non-fast-forwardable stream ciphers, at the cost of constant communication complexity with the bulletin board per update. * Our public-key FF-FSE avoids HIBE-based techniques by instead using so-called updatable public-key encryption (UPKE), introduced in several recent works (and more efficient than public-key FSEs). Our UPKE-based scheme uses a novel type of update graph that we construct in this work. Our graph has constant in-degree, logarithmic diameter, and logarithmic cut property which is essential for the efficiency of our schemes. Combined with recent UPKE schemes, we get two FF-FSEs in the bulletin board model, under the DDH and the LWE assumptions