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... Filtser and Le [FL22] generalized Chan et al. [CHJ20] result to doubling spaces, 2 showing that every metric space with doubling dimension d admits a −O(d) , -LSO. Furthermore, they generalized the concept of LSO to other metric spaces, defining the two related notions of triangle-LSO (which turn to be useful for general metric spaces), and left-sided LSO (which turn to be useful for topologically restricted graphs). ...

... Furthermore, they generalized the concept of LSO to other metric spaces, defining the two related notions of triangle-LSO (which turn to be useful for general metric spaces), and left-sided LSO (which turn to be useful for topologically restricted graphs). Here, instead of presenting the left-sided LSO's of [FL22], we introduce the closely related notion of rooted-LSO, which has some additional structure. All the results and constructions for left-sided LSO in [FL22] hold for rooted LSO as well. ...

... Here, instead of presenting the left-sided LSO's of [FL22], we introduce the closely related notion of rooted-LSO, which has some additional structure. All the results and constructions for left-sided LSO in [FL22] hold for rooted LSO as well. We refer to [FL22] for a comparison between the different notions, and to Figure 1 for an illustration. ...

Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A $(\tau,\rho)$-LSO is a collection $\Sigma$ of orderings such that for every $x,y\in\mathbb{R}^d$ there is an ordering $\sigma\in\Sigma$, where all the points between $x$ and $y$ w.r.t. $\sigma$ are in the $\rho$-neighborhood of either $x$ or $y$. In essence, LSO allow one to reduce problems to the $1$-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, $\ell_p$, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model.

... Another related problem is h-hop t-spanners. Here we are given a metric space (X, d), and the goal is to construct a graph G over X such that ∀x, y ∈ X, d X (x, y) ≤ d Sol13,HIS13] for Euclidean metrics, [FN18] for different metric spaces, [FL21b] for reliable 2-hop spanners, and [ASZ20] for low-hop emulators (where d (O(log log n)) G respects the triangle inequality). The idea of one-to-many embedding of graphs was originated by Bartal and Mendel [BM04], who for k ≥ 1 constructed embedding into ultrametric with O(n 1+ 1 k ) nodes and path distortion O(k⋅log n⋅ log log n) (see Definition 8, and ignore all hop constrains). ...

In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,\beta,h)$ (here $t,\beta,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $\beta$ is called the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops. Haeupler {\em et al.} constructed embedding where $M$ contains $1-\epsilon$ fraction of the vertices and $\beta=t=O(\frac{\log^2 n}{\epsilon})$. They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems. In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation. In other words, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y∈X has a low distortion path in one of the trees. If it has the stronger property that every point x∈X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minor-free graphs of diameter $D$ into graphs of treewidth $O_{\epsilon}(\log n)$ with expected additive distortion $+\epsilon D$. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to $O_{\epsilon}(\log\log n)^2$. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from $n^{O_{\epsilon}(\log(n))}$ to $n^{O_{\epsilon}(\log\log(n))^3}$. Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter $D$ into graphs of treewidth $O((\log\log n)^2)/\epsilon)$ and additive distortion $+\epsilon D$ that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time. A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth $O(\log\log n)$ and hop-diameter $O(\log \log n)$. This result may be of independent interest.

This survey provides a guiding reference to researchers seeking an overview of the large body of literature about graph spanners. It surveys the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.

The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [29] devised the so-called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems.
In this article, we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees . We ask for the largest subset S ⊆ V of a given graph G =( V , E ), such that there exists a spanning tree of G that has small stretch for S . Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices . The union of these trees serves as a special type of spanner, a tree-padding spanner . We use this spanner to devise the first compact stateless routing scheme with O (1) routing decision time, and labels that are much shorter than in all currently existing schemes.
We first revisit the metric Ramsey problem and provide a new deterministic construction. We prove that for every k , any n -point metric space has a subset S of size at least n 1−1/ k that embeds into an ultrametric with distortion 8 k . We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k , any n -vertex graph G =( V , E ) has a subset S of size at least n 1−1/ k , and a spanning tree of G , that has stretch O ( k log log n ) between any point in S and any point in V .

It is long known that for every weighted undirected n-vertex m-edge graph G = (V, E, ω), and every integer k ⩾ 1, there exists a ((2k − 1) · (1 + ε))-spanner with O(n1 + 1/k) edges and weight O(k · n1/k · ω(MST(G)), for an arbitrarily small constant ε > 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O(m · (n1 + 1/k + n · log n)) time.
In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2k − 1) · (1 + ε))-spanners with O(k · n1 + 1/k) edges and weight O(k · n1/k) · ω(MST(G)), where ε > 0 is an arbitrarily small constant. The running time of our algorithm is O(k · m + min {n · log n, m · α(n)}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].

We present a streaming algorithm for constructing sparse spanners and show that our algorithm significantly outperforms the state-of-the-art algorithm for this task (due to Feigenbaum et al.). Specifically, the processing time per edge of our algorithm is O (1), and it is drastically smaller than that of the algorithm of Feigenbaum et al., and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters of the state-of-the-art algorithm.
We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has incremental update time of O (1), and a nontrivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides nontrivial bounds on both incremental and decremental update time for a wide range of stretch parameter t .

We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph
$\mathcal{G}$
on a point set P and a region F, we define
$\mathcal{G}\ominus F$
to be what remains of
$\mathcal{G}$
after the vertices and edges of
$\mathcal{G}$
intersecting F have been removed. A
$\mathcal{C}$
-fault tolerant
t-spanner is a geometric graph
$\mathcal{G}$
on P such that for any convex region F, the graph
$\mathcal{G}\ominus F$
is a t-spanner for
$\mathcal{G}_{c}(P)\ominus F$
, where
$\mathcal{G}_{c}(P)$
is the complete geometric graph on P. We prove that any set P of n points admits a
$\mathcal{C}$
-fault tolerant (1+ε)-spanner of size
$\mathcal{O}(n\log n)$
for any constant ε>0; if adding Steiner points is allowed, then the size of the spanner reduces to
$\mathcal{O}(n)$
, and for several special cases, we show how to obtain region-fault tolerant spanners of
$\mathcal{O}(n)$
size without using Steiner points. We also consider fault-tolerant geodesic
t
-spanners: this is a variant where, for any disk D, the distance in
$\mathcal{G}\ominus D$
between any two points u,v∈P∖D is at most t times the geodesic distance between u and v in ℝ2∖D. We prove that for any P, we can add
$\mathcal{O}(n)$
Steiner points to obtain a fault-tolerant geodesic (1+ε)-spanner of size
$\mathcal{O}(n)$
.

Let G=(V, E) be an n-vertex connected graph with positive edge weights. A subgraph G′=(V, E′) is a t-spanner of G if for all u, v∈V, the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight.
For an arbitrary positive edge-weighted graph G, for any t>1, and any ∈>0, we show that a t-spanner of G with weight [Formula: see text] can be constructed in polynomial time. We also show that (log ² n)-spanners of weight O(1) · wt(MST) can be constructed. We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the [Formula: see text] vector. We show that for these graphs, t-spanners with total weight O(log n) · wt(MST) can be constructed in polynomial time.

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ɛV, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

We define the notion of a well-separated pair decomposition of points in d-dimensional space. We then develop efficient sequential and parallel algorithms for computing such a decomposition. We apply the resulting decomposition to the efficient computation of k-nearest neighbors and n-body potential fields.

We study the problem of constructing universal Steiner trees for undirected
graphs. Given a graph $G$ and a root node $r$, we seek a single spanning tree
$T$ of minimum stretch, where the stretch of $T$ is defined to be the maximum
ratio, over all subsets of terminals $X$, of the ratio of the cost of the
sub-tree $T_X$ that connects $r$ to $X$ to the cost of an optimal Steiner tree
connecting $X$ to $r$. Universal Steiner trees (USTs) are important for data
aggregation problems where computing the Steiner tree from scratch for every
input instance of terminals is costly, as for example in low energy sensor
network applications.
We provide a polynomial time UST construction for general graphs with
$2^{O(\sqrt{\log n})}$-stretch. We also give a polynomial time
polylogarithmic-stretch construction for minor-free graphs. One basic building
block in our algorithm is a hierarchy of graph partitions, each of which
guarantees small strong cluster diameter and bounded local neighbourhood
intersections. Our partition hierarchy for minor-free graphs is based on the
solution to a cluster aggregation problem that may be of independent interest.
To our knowledge, this is the first sub-linear UST result for general graphs,
and the first polylogarithmic construction for minor-free graphs.

We give a short and easy upper bound on the worst-case size of fault tolerant spanners, which improves on all prior work and is fully optimal at least in the setting of vertex faults.

This paper makes two main contributions: a construction of a near-minimum spanning tree with constant average distortion, and a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion. Scaling distortion provides improved distortion for 1−ϵ fractions of the pairs, for all ϵ simultaneously. A stronger version called coarse scaling distortion, has improved distortion guarantees for the furthest pairs. Prioritized distortion allows to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation. This equivalence is used to construct the near-minimum spanning tree with constant average distortion, and has many further implications to metric embeddings theory. Among other results, we obtain a strengthening of Bourgain's theorem on embedding arbitrary metrics into Euclidean space, possessing optimal prioritized distortion.

A spanner H of a weighted undirected graph G is a “sparse” subgraph that approximately preserves distances between every pair of vertices in G. We refer to H as a δ-spanner of G for some parameter δ ≥ 1 if the distance in H between every vertex pair is at most a factor δ bigger than in G. In this case, we say that H has stretch δ. Two main measures of the sparseness of a spanner are the size (number of edges) and the total weight (the sum of weights of the edges in the spanner).
It is well-known that for any positive integer k, one can efficiently construct a (2k − 1)-spanner of G with O(n1+1/k) edges where n is the number of vertices [2]. This size-stretch tradeoff is conjectured to be optimal based on a girth conjecture of Erdős [17]. However, the current state of the art for the second measure is not yet optimal.
Recently Elkin, Neiman and Solomon [ICALP 14] presented an improved analysis of the greedy algorithm, proving that the greedy algorithm admits (2k − 1) · (1 + ε) stretch and total edge weight of Oε ((k/ log k) · ω (MST(G)) · n1/k), where ω(MST(G)) is the weight of a MST of G. The previous analysis by Chandra et al. [SOCG 92] admitted (2k − 1) · (1 + ε) stretch and total edge weight of Oε(kω(MST(G))n1/k). Hence, Elkin et al. improved the weight of the spanner by a log k factor.
In this article, we completely remove the k factor from the weight, presenting a spanner with (2k − 1) · (1 + ε) stretch, Oε(ω(MST(G))n1/k) total weight, and O(n1+1/k) edges. Up to a (1 + ε) factor in the stretch this matches the girth conjecture of Erdős [17].

We study the problem of routing in doubling metrics and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with a small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(<i<X</i<) at most α if every ball can be covered by 2α balls of half its radius. (A doubling metric is one whose doubling dimension dim(<i<X</i<) is a constant.) We consider the metric space induced by the shortest-path distance in an underlying undirected graph G. We show how to perform (1 + τ)-stretch routing on such a metric for any 0 < τ ≤ 1 with routing tables of size at most (α/τ)O(α)log Δlog δ bits with only (α/τ)O(α)log Δ entries, where Δ is the diameter of the graph, and δ is the maximum degree of the graph G; hence, the number of routing table entries is just τ − O(1)log Δ for doubling metrics. These results extend and improve on those of Talwar (2004).
We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables earlier; for τ > 0, we give algorithms to construct (1 + τ)-stretch spanners for a metric (X, d) with maximum degree at most (2 + 1/τ)O(dim(X)), matching the results of Das et al. for Euclidean metrics.

In this paper we describe a simple and efficient algorithm to construct FRT embeddings (optimal probabilistic tree embeddings) on graphs. For a graph with $n$ vertices and $m$ edges the algorithm runs in $O(m\log n+n\log^2 n)$ time. With the efficient FRT construction, we show that FRT trees are Ramsey partitions with asymptotically tight bound, and we give tighter bounds on coefficient than previous Ramsey partitions, hence improving other results on a series of distance oracles.

A classical result in metric geometry asserts that any n-point metric admits a linear-size spanner of dilation O(log n) [PS89]. More generally, for any c > 1, any metric space admits a spanner of size O(n[superscript 1+1/c]), and dilation at most c. This bound is tight assuming the well-known girth conjecture of Erdős [Erd63].
We show that for a metric induced by a set of n points in high-dimensional Euclidean space, it is possible to obtain improved dilation/size trade-offs. More specifically, we show that any n-point Euclidean metric admits a near-linear size spanner of dilation O(√log n). Using the LSH scheme of Andoni and Indyk [AI06] we further show that for any c > 1, there exist spanners of size roughly O(n[superscript1+1/c[superscript 2]]) and dilation O(c). Finally, we also exhibit super-linear lower bounds on the size of spanners with constant dilation.

Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing space-efficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM'12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketch-based algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use ~O(n1+1/k) bits of space and have stretch 2k. While this stretch is larger than the conjectured optimal 2k-1 for this amount of space, we show for an appropriate k that it implies the first 2-pass spectral sparsifier with n1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n1+c bits of space for a constant c > 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff.

A spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree is called a shallow-light tree (shortly, SLT). More specifically, an (α, β)-SLT of a weighted undirected graph G = (V, E, w) with respect to a designated vertex rt ∈ V is a spanning tree of G with:
(1) root-stretch α -- it preserves all distances between rt and the other vertices up to a factor of α.
(2) lightness β -- it has weight at most β times the weight of a minimum spanning tree MST(G) of G.
Tight tradeoffs between the parameters of SLTs were established by Awerbuch et al. in PODC'90 and by Khuller et al. in SODA'93. They showed that for any &epsis; > 0, any graph admits a (1 + &epsis;, O(1/&epsis;))-SLT with respect to any root vertex, and complemented this result with a matching lower bound.
Khuller et al. asked if the upper bound β = O(1/&epsis;) on the lightness of SLTs can be improved in constant-dimensional Euclidean spaces. In FOCS'11 Elkin and this author gave a negative answer to this question, showing a lower bound of β = Ω(1/&epsis;) that applies to 2-dimensional Euclidean spaces.
In this paper we show that Steiner points lead to a quadratic improvement in Euclidean SLTs, by presenting a construction of Euclidean Steiner (1 + &epsis;, O(√1/&epsis;))-SLTs. While the lightness bound β O(√1/&epsis;) of our construction applies to Euclidean spaces of any constant dimension, there is a matching lower bound of β Ω(√1/&epsis;) even in 2-dimensional Euclidean spaces. The runtime of our construction, and thus the number of Steiner points used, are bounded by O(n).

In STOC’95, Arya et al. [1995] showed that for any set of n points in Rd, a (1 + ε)-spanner with diameter at most 2 (respectively, 3) and O(n log n) edges (respectively, O(n log log n) edges) can be built in O(n log n) time. Moreover, it was shown in Arya et al. [1995] and Narasimhan and Smid [2007] that for any k ≥ 4, one can build in O(n(log n)2kαk(n)) time a (1 + ε)-spanner with diameter at most 2k and O(n2kαk(n)) edges. The function αk is the inverse of a certain function at the k/2 th level of the primitive recursive hierarchy, where α0(n) = n/2 , α1(n) = √n , α2(n) = log n , α3(n) = log log n , α4(n) = log* n, α5(n) = 12 log* n , ..., etc. It is also known [Narasimhan and Smid 2007] that if one allows quadratic time, then these bounds can be improved. Specifically, for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges can be constructed in O(n²) time [Narasimhan and Smid 2007].
A major open question in this area is whether one can construct within time O(n log n + nkαk(n)) a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges. In this article, we answer this question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nαk(n)) edges can be built in optimal time O(n log n).

Spanner of an undirected graph G = (V,E) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretch t ∈ ℕ is a subgraph (V,ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner that has as small number of edges as possible.
We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner.
Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t-spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a t-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.

When processing massive data sets, a core task is to construct synopses of the data. To be useful, a synopsis data structure should be easy to construct while also yielding good approximations of the relevant properties of the data set. A particularly useful class of synopses are sketches, i.e., those based on linear projections of the data. These are applicable in many models including various parallel, stream, and compressed sensing settings. A rich body of analytic and empirical work exists for sketching numerical data such as the frequencies of a set of entities. Our work investigates graph sketching where the graphs of interest encode the relationships between these entities. The main challenge is to capture this richer structure and build the necessary synopses with only linear measurements.
In this paper we consider properties of graphs including the size of the cuts, the distances between nodes, and the prevalence of dense sub-graphs. Our main result is a sketch-based sparsifier construction: we show that O̅(nε-2) random linear projections of a graph on n nodes suffice to (1 + ε) approximate all cut values. Similarly, we show that O(ε-2) linear projections suffice for (additively) approximating the fraction of induced sub-graphs that match a given pattern such as a small clique. Finally, for distance estimation we present sketch-based spanner constructions. In this last result the sketches are adaptive, i.e., the linear projections are performed in a small number of batches where each projection may be chosen dependent on the outcome of earlier sketches. All of the above results immediately give rise to data stream algorithms that also apply to dynamic graph streams where edges are both inserted and deleted. The non-adaptive sketches, such as those for sparsification and subgraphs, give us single-pass algorithms for distributed data streams with insertion and deletions. The adaptive sketches can be used to analyze MapReduce algorithms that use a small number of rounds.

In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour and M. Yannakakis [SIAM J. Comput. 23, 864–894 (1994; Zbl 0809.68075)] and is closely related to the metric labeling problem introduced by J. Kleinberg and E. Tardos [Proc. 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 14–23 (1999)]. We present approximation algorithms for 0-Extension. In arbitrary graphs, we present a O(logk)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by A. V. Karzanov [Eur. J. Comb. 19, 71–101 (1998; Zbl 0894.90147)]. It is similar in flavor to the linear programming relaxation of N. Garg, V. V. Vazirani, and M. Yannakakis [SIAM J. Comput. 25, 235–251 (1996; Zbl 0844.68061)] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least clgk for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations.

Given a graph G = (V, E), a subgraph Gapos; = (V, Eapos;) is a t-spanner of G if for every u, v ∈ V, the distance from u to v in Gapos; is at most t times longer than that distance in G. This paper presents some results concerning the existence and efficient constructability of sparse spanners for various classes of graphs, including general undirected graphs, undirected chordal graphs, and general directed graphs.

It is shown that anyn point metric space is up to logn lipeomorphic to a subset of Hilbert space. We also exhibit an example of ann point metric space which cannot be embedded in Hilbert space with distortion less than (logn)/(log logn), showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance,
to construct our example, we make use of random graphs.

We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ
p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the
constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn).

The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, d
Y) embeds (1 +ε)-isomorphically into the Hilbert spacel
2.

The main result is a metrical characterization of superreflexivity in Banach spaces. A Banach spaceX is not superreflexive if and only ifX contains hyperbolic trees as a metric space. The notion of non-linear cotype in discussed.

In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is . This improves upon the result of Bartal who gave a bound of . Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion -distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

We consider the problem of computing bounded-degree lightweight plane spanners of unit disk graphs in the local distributed
model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for
efficient unicasting and multicasting in wireless distributed systems. We present the first local distributed algorithm that computes a bounded-degree plane lightweight spanner of a given unit disk graph. The upper
bounds on the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply a
local distributed algorithm that computes a plane spanner of a given unit disk graph U, whose degree is at most 14, stretch factor at most 8.81, and weight at most 8.81 times the weight of a Euclidean Minimum
Spanning Tree of V(U).
We show a wider application of our techniques by giving an O(nlogn) time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include
Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight.

Aimed at an audience of researchers and graduate students in computational geometry and algorithm design, this book uses the Geometric Spanner Network Problem to showcase a number of useful algorithmic techniques, data structure strategies, and geometric analysis techniques with many applications, practical and theoretical. The authors present rigorous descriptions of the main algorithms and their analyses for different variations of the Geometric Spanner Network Problem. Though the basic ideas behind most of these algorithms are intuitive, very few are easy to describe and analyze. For most of the algorithms, nontrivial data structures need to be designed, and nontrivial techniques need to be developed in order for analysis to take place. Still, there are several basic principles and results that are used throughout the book. One of the most important is the powerful well-separated pair decomposition. This decomposition is used as a starting point for several of the spanner constructions. © Giri Narasimhan, Michiel Smid 2007 and Cambridge University Press, 2009. All rights reserved.

The problem of computing spanners of unweighted graphs in streaming model is presented. The streaming model has two characteristics, firstly the input data can be accessed only sequentially in the form of a stream, and secondly, the working memory is smaller than the size of the entire input stream. An algorithm in this model is allowed to make one or more passes over the input stream to solve a given computational problem. Single pass and linear time streaming algorithm for computing a spanner of size O(min(m, kn1+1/k)) for any unweighted graph. Ausiello and colleagues designed such an algorithm for spanners of stretch. Another interesting open problem is to design streaming algorithm for spanners of weighted graphs. The real challenge is to design a single pass streaming algorithm for weighted graphs without affecting the optimal bound on the spanner size and constant processing time per edge.