Optimal Spanners for Unit Ball Graphs in Doubling Metrics

Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph $G$ with $n$ vertices and a positive constant $\epsilon < 1$ finds a $(1+\epsilon)$-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju, which runs in $\mathcal{O}(\log^*n)$ rounds in the LOCAL model, but has a $\mathcal{O}(\log \Delta)$ bound on its lightness, where $\Delta$ is the ratio of the length of the longest edge to the length of the shortest edge in the unit ball graph. Next, we adjust our algorithm to work in the CONGEST model, without changing its round complexity, hence proposing the first spanner construction for unit ball graphs in the CONGEST model of computation. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersections per node. Lastly, we provide experimental results that confirm our theoretical bounds, and show an efficient performance from our distributed algorithm compared to the best known centralized construction

On the Edge Crossings of the Greedy Spanner

The greedy t-spanner of a set of points in the plane is an undirected graph constructed by considering pairs of points in order by distance, and connecting a pair by an edge when there does not already exist a path connecting that pair with length at most t times the Euclidean distance. We prove that, for any t > 1, these graphs have at most a linear number of crossings, and more strongly that the intersection graph of edges in a greedy t-spanner has bounded degeneracy. As a consequence, we prove a separator theorem for greedy spanners: any k-vertex subgraph of a greedy spanner can be partitioned into sub-subgraphs of size a constant fraction smaller, by the removal of O(?k) vertices. A recursive separator hierarchy for these graphs can be constructed from their planarizations in linear time, or in near-linear time if the planarization is unknown

Distributed Construction of Lightweight Spanners for Unit Ball Graphs

Resolving an open question from 2006 [Damian et al., 2006], we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple ?(log^*n)-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph G with n vertices and a positive constant ? < 1 finds a (1+?)-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju [Damian et al., 2006], which runs in ?(log^*n) rounds in the LOCAL model, but has a ?(log ?) bound on its lightness, where ? is the ratio of the length of the longest edge to the length of the shortest edge in the unit ball graph. Next, we adjust our algorithm to work in the CONGEST model, without changing its round complexity, hence proposing the first spanner construction for unit ball graphs in the CONGEST model of computation. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersections per node. Lastly, we provide experimental results that confirm our theoretical bounds, and show an efficient performance from our distributed algorithm compared to the best known centralized construction

Online Spanners in Euclidean and General Metrics

Spanners are fundamental graph structures that preserve lengths of shortest paths in an input graph $G$, up to some multiplicative distortion.Given an edge-weighted graph $G = (V,E)$, a subgraph $H = (V, E_H)$ is a $t$-spanner of $G$, for $t\ge 1$, if for every $u,v \in V$, the distance between $u$ and $v$ in $H$ is at most $t$ times their distance than in $G$.In this thesis, we study the existing literature on offline and online spanners, and we introduce some new results on online spanners in metric spaces. Suppose that we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one-by-one, i.e., point $s_i$ is presented at the step~$i$, and $S_i = \{s_1, \ldots , s_i\}$ for $i=1,\ldots , n$.The objective of an online algorithm is to maintain a geometric $t$-spanner $G_i$ for $S_i$ for all $i$. The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the minimum weight of a $t$-spanner on $S_n$. Here the weight of a spanner is the sum of all its edge weights.Under the $L_2$-norm in $\mathbb{R}^d$ for arbitrary constant $d\in \mathbb{N}$,we present an online algorithm for (1+\eps)-spanner with competitive ratio O_d(\eps^{-d} \log n), improving the previous bound of O_d(\eps^{-(d+1)}\log n). Moreover, the spanner maintained by the algorithm has O_d(\eps^{1-d}\log \eps^{-1})\cdot n edges, almost matching the (offline) optimal bound of O_d(\eps^{1-d})\cdot n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(\eps^{-3/2}\log\eps^{-1}\log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a \Omega_d(\eps^{-d}) lower bound for the competitive ratio for online (1+\eps)-spanner algorithms in $\mathbb{R}^d$ under the $L_1$-norm.Then we turn our attention to online spanners in general metrics.Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline settings, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed \emph{ordered greedy}. With stretch factor t = (2k-1)(1+\eps) for $k\ge 2$ and \eps\in(0,1), we show that it maintains a spanner with O(\eps^{-1}\log\frac{k}{\eps}) \cdot n^{1+\frac{1}{k}} edges and O(\eps^{-1}n^{\frac{1}{k}}\log^2 n) lightness for a sequence of $n$ points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an $\Omega(\frac{1}{k}\cdot n^{1/k})$ competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+\eps)-spanner for ultrametrics with O(n\cdot\eps^{-1}\log\eps^{-1}) edges and O(\eps^{-2}) lightness

İslam içi çatışmalar ve Ortadoğu bölgesel güvenlik kompleksinin değişen dinamikleri: Irak savaşından Işid’in yükselişine kadar.

This thesis aims to analyse the impact of intra-Islamic conflicts on the security interactions in the Middle East regional subsystem in the context of the ongoing structural transformation of the region. This study will evaluate two main narratives concerning the role of the sectarian identities on the intra and inter-state security dynamics: 1) Deep Sunni-Shīʿī antagonism as a primordial conflictual factor in the Middle East 2) Sectarian strife has a secondary impact on the ongoing conflicts and is mainly a cover for geopolitical interests. Considering the chronological approach of the study, the author firstly examines the history of the transformation of the intra-Islamic relations from the intra-state conflict within the framework of the Arab and Turkic Muslim states to the inter-state conflicts following emergence of the Safavid state in Iran. Secondly, the research analyses the transformation of Iraq to the only Shīʿī Arab state, the rise of Sunni and Shīʿī Jihadist organizations in the context of the Syrian civil war and the Yemeni crisis in which has obtained an inter-state dimension. From this analysis, the research tries to portray the complex nature of the conflict through the integration of theological, sociological, historical and geopolitical aspects into one united vision.Thesis (M.S.) -- Graduate School of Social Sciences. Middle East Studies