Algorithms for dynamic and distributed networks : shortest paths, betweenness centrality and related problems

In this thesis we study the problem of computing Betweenness Centrality in dynamic and distributed networks. Betweenness Centrality (BC) is a well-known measure for the relative importance of a node in a social network. It is widely used in applications such as understanding lethality in biological networks, identifying key actors in terrorist networks, supply chain management processes and more. The necessity of computing BC in large networks, especially when they quickly change their topology over time, motivates the study of dynamic algorithms that can perform faster than static ones. Moreover, the current techniques for computing BC requires a deeper understanding of a classic problem in computer science: computing all pairs all shortest paths (APASP) in a graph. One of the main contributions of this thesis is a collection of dynamic algorithms for computing APASP and BC scores which are provably faster than static algorithms for several classes of graphs. We use n = |V| and m = |E| to indicate respectively the number of nodes and edges in a directed positively weighted graph G = (V,E). Our bounds depend on the parameter [nu]* that is defined as the maximum number of edges that lie on shortest paths through any single vertex. The main results in the first part of this thesis are listed below. - A decrease-only algorithm for computing BC and APASP running in time O([nu]* n) that is provably faster than recomputing from scratch in sparse graphs. - An increase-only algorithm for computing BC and APASP that runs in O([nu]*² log n) per update for a sequence of at least [Omega](m*/[nu]*) updates. Here m* is the number of edges in G that lie on shortest paths. This algorithm uses O(m* [nu]*) space. - An increase-only algorithm for computing BC and APASP that runs in O([nu]*² log n) but improves the computational space to O(m*n). - A fully dynamic algorithm for computing BC and APASP that runs in O([nu]*² log³ n) amortized time per update for a sequence of at least [Omega](n) updates. - A refinement of our fully dynamic algorithm that improves the amortized running time to O([nu]*² log² n), saving a logarithmic factor. In the second part of this thesis, we study the case when the input graph is a distributed network of machines and the BC score of each machine, considering its location within the network topology, needs to be computed. In this scenario, each node in the input graph is a self-contained machine with limited knowledge of the network and communication power. Each machine only knows the (virtual) location of the neighbors machines (adjacent nodes in the input graph). The messages, exchanged in each round between machines, cannot exceed a bounded size of at most O(log n) bits. In this distributed model, called CONGEST, we present algorithms for computing BC in near-optimal time for unweighted networks, and some classes of weighted networks. Specifically, our main results are: - A distributed BC algorithm for unweighted undirected graphs completing in at most min(2n+O(D[underscore u]); 4n) rounds, where D[underscore u] is the diameter of the undirected network. - A distributed BC algorithm for unweighted directed graphs completing in at most min(2n + O(D); 4n) rounds, where D is the diameter of the directed network. - A distributed APSP algorithm for unweighted directed graphs completing in at most min(n + O(D); 2n) rounds. - A distributed BC algorithm for weighted directed acyclic graphs (dag) completing in at most 2n + O(L) rounds, where L is the longest length of a path in the dag. - A distributed APSP algorithm for weighted dags completing in at most n + O(L) rounds.Computer Science

Distributed Approximation Algorithms for Weighted Shortest Paths

A distributed network is modeled by a graph having $n$ nodes (processors) and diameter $D$. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a $O(\log n)$ {\em bandwidth restriction} on edges (the standard synchronous \congest model). The question whether approximation algorithms help speed up the shortest paths (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (\sssp) and all-pairs shortest paths (\apsp) in the weighted case. Our main result is an algorithm for \sssp. Previous results are the classic $O(n)$-time Bellman-Ford algorithm and an $\tilde O(n^{1/2+1/2k}+D)$-time $(8k\lceil \log (k+1) \rceil -1)$-approximation algorithm, for any integer $k\geq 1$, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use $\tilde O(\cdot)$ to hide the O(\poly\log n) term.) We present an $\tilde O(n^{1/2}D^{1/4}+D)$-time $(1+o(1))$-approximation algorithm for \sssp. This algorithm is {\em sublinear-time} as long as $D$ is sublinear, thus yielding a sublinear-time algorithm with almost optimal solution. When $D$ is small, our running time matches the lower bound of $\tilde \Omega(n^{1/2}+D)$ by Das Sarma et al. (SICOMP 2012), which holds even when $D=\Theta(\log n)$, up to a \poly\log n factor.Comment: Full version of STOC 201

Energy management in communication networks: a journey through modelling and optimization glasses

The widespread proliferation of Internet and wireless applications has produced a significant increase of ICT energy footprint. As a response, in the last five years, significant efforts have been undertaken to include energy-awareness into network management. Several green networking frameworks have been proposed by carefully managing the network routing and the power state of network devices. Even though approaches proposed differ based on network technologies and sleep modes of nodes and interfaces, they all aim at tailoring the active network resources to the varying traffic needs in order to minimize energy consumption. From a modeling point of view, this has several commonalities with classical network design and routing problems, even if with different objectives and in a dynamic context. With most researchers focused on addressing the complex and crucial technological aspects of green networking schemes, there has been so far little attention on understanding the modeling similarities and differences of proposed solutions. This paper fills the gap surveying the literature with optimization modeling glasses, following a tutorial approach that guides through the different components of the models with a unified symbolism. A detailed classification of the previous work based on the modeling issues included is also proposed

A distributed, compact routing protocol for the Internet

The Internet has grown in size at rapid rates since BGP records began, and continues to do so. This has raised concerns about the scalability of the current BGP routing system, as the routing state at each router in a shortest-path routing protocol will grow at a supra-linearly rate as the network grows. The concerns are that the memory capacity of routers will not be able to keep up with demands, and that the growth of the Internet will become ever more cramped as more and more of the world seeks the benefits of being connected. Compact routing schemes, where the routing state grows only sub-linearly relative to the growth of the network, could solve this problem and ensure that router memory would not be a bottleneck to Internet growth. These schemes trade away shortest-path routing for scalable memory state, by allowing some paths to have a certain amount of bounded “stretch”. The most promising such scheme is Cowen Routing, which can provide scalable, compact routing state for Internet routing, while still providing shortest-path routing to nearly all other nodes, with only slightly stretched paths to a very small subset of the network. Currently, there is no fully distributed form of Cowen Routing that would be practical for the Internet. This dissertation describes a fully distributed and compact protocol for Cowen routing, using the k-core graph decomposition. Previous compact routing work showed the k-core graph decomposition is useful for Cowen Routing on the Internet, but no distributed form existed. This dissertation gives a distributed k-core algorithm optimised to be efficient on dynamic graphs, along with with proofs of its correctness. The performance and efficiency of this distributed k-core algorithm is evaluated on large, Internet AS graphs, with excellent results. This dissertation then goes on to describe a fully distributed and compact Cowen Routing protocol. This protocol being comprised of a landmark selection process for Cowen Routing using the k-core algorithm, with mechanisms to ensure compact state at all times, including at bootstrap; a local cluster routing process, with mechanisms for policy application and control of cluster sizes, ensuring again that state can remain compact at all times; and a landmark routing process is described with a prioritisation mechanism for announcements that ensures compact state at all times

Pervasive intelligent routing in content centric delay tolerant networks

This paper introduces a Swarm-Intelligence based Routing protocol (SIR) that aims to efficiently route information in content centric Delay Tolerant Networks (CCDTN) also dubbed pocket switched networks. First, this paper formalizes the notion of optimal path in CCDTN and introduces an original and efficient algorithm to process these paths in dynamic graphs. The properties and some invariant features of these optimal paths are analyzed and derived from several real traces. Then, this paper shows how optimal path in CCDTN can be found and used from a fully distributed swarm-intelligence based approach of which the global intelligent behavior (i.e. shortest path discovery and use) emerges from simple peer to peer interactions applied during opportunistic contacts. This leads to the definition of the SIR routing protocol of which the consistency, efficiency and performances are demonstrated from intensive representative simulations

Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time

In the decremental single-source shortest paths (SSSP) problem we want to maintain the distances between a given source node $s$ and every other node in an $n$-node $m$-edge graph $G$ undergoing edge deletions. While its static counterpart can be solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic $O(mn)$ total update time of Even and Shiloach [JACM 1981] has been the fastest known algorithm for three decades. At the cost of a $(1+\epsilon)$-approximation factor, the running time was recently improved to $n^{2+o(1)}$ by Bernstein and Roditty [SODA 2011]. In this paper, we bring the running time down to near-linear: We give a $(1+\epsilon)$-approximation algorithm with $m^{1+o(1)}$ expected total update time, thus obtaining near-linear time. Moreover, we obtain $m^{1+o(1)} \log W$ time for the weighted case, where the edge weights are integers from $1$ to $W$. The only prior work on weighted graphs in $o(m n)$ time is the $m n^{0.9 + o(1)}$-time algorithm by Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with quasi-polynomial edge weights. The expected running time bound of our algorithm holds against an oblivious adversary. In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse $(h, \epsilon)$-hop set introduced by Cohen [JACM 2000] in the PRAM literature. An $(h, \epsilon)$-hop set of a graph $G=(V, E)$ is a set $F$ of weighted edges such that the distance between any pair of nodes in $G$ can be $(1+\epsilon)$-approximated by their $h$-hop distance (given by a path containing at most $h$ edges) on $G'=(V, E\cup F)$. Our algorithm can maintain an $(n^{o(1)}, \epsilon)$-hop set of near-linear size in near-linear time under edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper was presented at the 55th IEEE Symposium on Foundations of Computer Science (FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character