Linear Number Of Edges Research Articles

Cops and Robbers on Dynamic Graphs: Offline and Online Case

We examine the classic game of Cops and Robbers played on models of dynamic graphs, that is, graphs evolving over discrete time steps. At each time step, a graph instance is generated as a subgraph of the underlying graph of the model. The cops and the robber take their turns on the current graph instance. The cops win if they can capture the robber at some point in time. Otherwise, the robber wins. In the offline case, the players are fully aware of the evolution sequence, up to some finite time horizon T. We provide a O(n 2k+1 T) algorithm to decide whether a given evolution sequence for an underlying graph with n vertices is k-cop-win via a reduction to a reachability game. In the online case, there is no knowledge of the evolution sequence, and the game might go on forever. Also, each generated instance is required to be connected. We provide a nearly tight characterization for sparse underlying graphs, i.e., with at most linear number of edges. We prove λ + 1 cops suffice to capture the robber in any underlying graph with n − 1 + λ edges. Further, we define a family of underlying graphs with n−1+λ edges where λ−1 cops are necessary (and sufficient) for capture.

δ-Greedy t-spanner

We introduce a new geometric spanner, δ-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The δ-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong (1+ε)-spanner, for every ε>0. The δ-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of n points in the plane in O(n2log⁡n) time.The δ-Greedy spanner has an additional parameter δ, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For δ=t, the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear.Finally, we show that, for a set of n points placed independently at random in a unit square, the expected construction time of the δ-Greedy algorithm is O(nlog⁡n). Our analysis indicates that the δ-Greedy spanner gives the best results among the known spanners of expected O(nlog⁡n) time for random point sets. Moreover, analysis implies that by setting δ=t, the δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected O(nlog⁡n) time.

Local routing in a tree metric 1-spanner

Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight \((1+\epsilon )\)-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after \(O(\log n)\) hops and requires \(O(\varDelta \log n)\) bits of storage per vertex where \(\varDelta \) is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.

Counting connected graphs with large excess

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.

Uniform Generation of Spanning Regular Subgraphs of a Dense Graph

Let $H_n$ be a graph on $n$ vertices and let $\overline{H_n}$ denote the complement of $H_n$. Suppose that $\Delta = \Delta(n)$ is the maximum degree of $\overline{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\overline{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$.
 Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $\Delta$. The first algorithm applies when $(d+\Delta)d\Delta = o(n)$. This improves on an earlier algorithm by the first author, which required constant $d$ and at most a linear number of edges in $\overline{H_n}$. The second algorithm applies when $H_n$ is regular and $d^2+\Delta^2 = o(n)$, adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform $d$-factor of $H_n$ in time $O(dn)$. Here the output distribution differs from uniform by $o(1)$ in total variation distance, provided that $d^2+\Delta^2 = o(n)$.

Braess's paradox in expanders

AbstractExpander graphs are known to facilitate effective routing and most real‐world networks have expansion properties. At the other extreme, it has been shown that in some special graphs, removing certain edges can lead to moreefficient routing. This phenomenon is known as Braess's paradox and is usually regarded as a rare event. In contrast to what one might expect, we show that Braess's paradox is ubiquitous in expander graphs. Specifically, we prove that Braess's paradox occurs in a large class of expander graphs with continuous convex latency functions. Our results extend previous work which held only when the graph was both denser and random and for random linear latency functions. We identify deterministic sufficient conditions for a graph with as few as a linear number of edges, such that Braess's Paradox almost always occurs, with respect to a general family of random latency functions. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Stable Roommates Spanner

We introduce a new geometric spanner whose construction is based on a generalization of the known Stable Roommates problem. The Stable Roommates Spanner combines the most desirable properties of geometric spanners: a natural definition, small degree, linear number of edges, strong (1+ϵ)-spanner for every ϵ>0, and an efficient construction algorithm. It is an improvement over the well-known Yao graph and Θ-graph and their variants. We show how to construct such a spanner for a set of points in the plane in O(nlog10n) expected time. We introduce a variant of the Stable Roommates Spanner called the Stable Roommates Θ-Spanner which we can generalize to higher dimensions and construct more efficiently in O(nlogdn) time. This variant possesses all the properties of the Stable Roommates Spanner except that it is no longer a strong spanner.

The emergence of sparse spanners and well-separated pair decomposition under anarchy

A spanner graph on a set of points in R d provides shortest paths between any pair of points with lengths at most a constant factor of their Euclidean distance. A spanner with a sparse set of edges is thus a good candidate for network backbones, as desired in many practical scenarios such as the transportation network and peer-to-peer network overlays. In this paper we investigate new models and aim to interpret why good spanners ‘emerge’ in reality, when they are clearly built in pieces by agents with their own interests and the construction is not coordinated. We show that the following algorithm of constructing an edge pq , if and only if there is no existing edge p ′ q ′ with p ′ and q ′ at distances no more than (4(1+1/e)) -1 ·| pq | from p , q respectively, generates a (1+e)-spanner with a linear number of edges. The algorithm also implies a simple algorithm for constructing linear-size well-separated pair decompositions that may be of interest on their own. This new spanner construction algorithm has applications in the construction of nice network topologies for peer-to-peer systems, when peers join and leave the network and have only limited information about the rest of the network.

Long cycles in subgraphs of (pseudo)random directed graphs

AbstractWe study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c − o(1))n. Moreover, there is a subgraph G′′of G with (1/2 + γ − o(1))|E| edges that does not contain a cycle of length at least cn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012

Spanners of additively weighted point sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs ( p , r ) where p is a point in the plane and r is a real number. The distance between two points ( p i , r i ) and ( p j , r j ) is defined as | p i p j | − r i − r j . We show that in the case where all r i are positive numbers and | p i p j | ⩾ r i + r j for all i, j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a ( 1 + ϵ ) -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has a spanning ratio bounded by a constant. The straight-line embedding of the Additively Weighted Delaunay graph may not be a plane graph. Given the Additively Weighted Delaunay graph, we show how to compute a plane straight-line embedding that also has a spanning ratio bounded by a constant in O ( n log n ) time.

Small Hop-diameter Sparse Spanners for Doubling Metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+e)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k and any 0<e<1, one can find (1+e)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog * n+n e −O(k)) edges) and constant hop diameter; we can also obtain a (1+e)-spanner with linear number of edges (i.e., only n e −O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1–9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8 n edges for n vertices. Our best construction with 7 n − O ( 1 ) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5 n − O ( 1 ) , thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.

(QUASI) SPANNERS FOR MOBILE AD HOC NETWORKS

We introduce a notion of t-quasi-spanner as an alternative to classical t-spanners. Our motivation for quasi-spanner comes from a problem of computing a sparse backbone for ad hoc wireless networks with fixed transmission ranges. We study computational complexity of the problem of computing sparsest quasi-spanner. Then we concentrate on the case t = 2 and give a distributed algorithm for computing a 2-quasi-spanner with linear number of edges. Finally, we give a modification of our algorithm which performs well on a class of random ad hoc networks.

Some Results on Random Unsatisfiable k-Sat Instances and Approximation Algorithms Applied to Random Structures

It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs.Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.

Quasi-planar graphs have a linear number of edges

A graph is calledquasi-planar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasi-planar graph withn vertices isO(n).

Expanding graphs contain all small trees

The assertion of the title is formulated and proved. The result is then used to construct graphs with a linear number of edges that, even after the deletion of almost all of their edges or almost all of their vertices, continue to contain all small trees.

Limitations on Explicit Constructions of Expanding Graphs

Expanding graphs are the basic building blocks in constructions of many types of graphs with special connectivity properties which arise in a variety of applications including switching networks, sorting networks and establishing time-space trade-offs for numerous computational problems. Only one explicit method of constructing arbitrarily large expanding graphs with a linear number of edges is known (Margulis [13], Gabber and Galil [8]), but the number of edges used is much greater than the number known to be sufficient via probabilistic arguments. In this paper we show that various other constructions which have been proposed to obtain expanding graphs, including one-dimensional analogues of the Gabber–Galil construction and some pseudorandom constructions, cannot ever yield expanding graphs.

Determining the Stability Number of a Graph

We formalize certain rules for deriving upper bounds on the stability number of a graph. The resulting system is powerful enough to (i) encompass the algorithms of Tarjan’s type and (ii) provide very short proofs on graphs for which the stability number equals the clique-covering number. However, our main result shows that for almost all graphs with a (sufficiently large) linear number of edges, proofs within our system must have at least exponential length.