An Optimal Dual Fault Tolerant Reachability Oracle

Let G=(V,E) be an n-vertices m-edges directed graph. Let s inV be any designated source vertex. We address the problem of reporting the reachability information from s under two vertex failures. We show that it is possible to compute in polynomial time an O(n) size data structure that for any query vertex v, and any pair of failed vertices f_1, f_2, answers in O(1) time whether or not there exists a path from s to v in G{f_1,f_2}. For the simpler case of single vertex failure such a data structure can be obtained using the dominator-tree from the celebrated work of Lengauer and Tarjan [TOPLAS 1979, Vol. 1]. However, no efficient data structure was known in the past for handling more than one failures. We, in addition, also present a labeling scheme with O(log^3(n))-bit size labels such that for any f_1, f_2, v in Vit is possible to determine in poly-logarithmic time if v is reachable from s in G{f_1,f_2} using only the labels of f1, f_2 and v. Our data structure can also be seen as an efficient mechanism for verifying double-dominators. For any given x, y, v in V we can determine in O(1) time if the pair (x,y) is a double-dominator of v. Earlier the best known method for this problem was using dominator chain from which verification of double-dominators of only a single vertex was possible

Distributed Dominating Set Approximations beyond Planar Graphs

The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, there has been much progress on computing local approximations on sparse graphs, and in particular planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm and (2) a local $\mathcal{O}(\log^*{n})$-time approximation scheme. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299

Sparse Weight Tolerant Subgraph for Single Source Shortest Path

In this paper we address the problem of computing a sparse subgraph of any weighted directed graph such that the exact distances from a designated source vertex to all other vertices are preserved under bounded weight increment. Finding a small sized subgraph that preserves distances between any pair of vertices is a well studied problem. Since in the real world any network is prone to failures, it is natural to study the fault tolerant version of the above problem. Unfortunately, it turns out that there may not always exist such a sparse subgraph even under single edge failure [Demetrescu et al. \u2708]. However in real applications it is not always the case that a link (edge) in a network becomes completely faulty. Instead, it can happen that some links become more congested which can be captured by increasing weight on the corresponding edges. Thus it makes sense to try to construct a sparse distance preserving subgraph under the above weight increment model where total increase in weight in the whole network (graph) is bounded by some parameter k. To the best of our knowledge this problem has not been studied so far. In this paper we show that given any weighted directed graph with n vertices and a source vertex, one can construct a subgraph of size at most e * (k-1)!2^kn such that it preserves distances between the source and all other vertices as long as the total weight increment is bounded by k and we are allowed to only have integer valued (can be negative) weight on edges and also weight of an edge can only be increased by some positive integer. Next we show a lower bound of c * 2^kn, for some constant c >= 5/4, on the size of such a subgraph. We further argue that the restrictions of integral weight and integral weight increment are actually essential by showing that if we remove any one of these two we may need to store Omega(n^2) edges to preserve the distances

Applications of matching theory in constraint programming

[no abstract

Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages

An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs

On 2-strong connectivity orientations of mixed graphs and related problems

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is either $(u,v)$ or $(v,u)$. The problem of finding an orientation of a mixed graph that makes it strongly connected is well understood and can be solved in linear time. Here we introduce the following orientation problem in mixed graphs. Given a mixed graph $G$, we wish to compute its maximal sets of vertices $C_1,C_2,\ldots,C_k$ with the property that by removing any edge $e$ from $G$ (directed or undirected), there is an orientation $R_i$ of $G\setminus{e}$ such that all vertices in $C_i$ are strongly connected in $R_i$. We discuss properties of those sets, and we show how to solve this problem in linear time by reducing it to the computation of the $2$-edge twinless strongly connected components of a directed graph. A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. A $2$-edge twinless strongly connected component (2eTSCC) of $G$ is a maximal subset of vertices $C$ such that any two vertices $u, v \in C$ are in the same twinless strongly connected component of $G \setminus e$, for any edge $e$. These concepts are motivated by several diverse applications, such as the design of road and telecommunication networks, and the structural stability of buildings

Applications of dynamic programming tp problems of routing pipework and cables

Imperial Users onl

Topology Control Multi-Objective Optimisation in Wireless Sensor Networks: Connectivity-Based Range Assignment and Node Deployment

The distinguishing characteristic that sets topology control apart from other methods, whose motivation is to achieve effects of energy minimisation and an increased network capacity, is its network-wide perspective. In other words, local choices made at the node-level always have the goal in mind of achieving a certain global, network-wide property, while not excluding the possibility for consideration of more localised factors. As such, our approach is marked by being a centralised computation of the available location-based data and its reduction to a set of non-homogeneous transmitting range assignments, which elicit a certain network-wide property constituted as a whole, namely, strong connectedness and/or biconnectedness. As a means to effect, we propose a variety of GA which by design is multi-morphic, where dependent upon model parameters that can be dynamically set by the user, the algorithm, acting accordingly upon either single or multiple objective functions in response. In either case, leveraging the unique faculty of GAs for finding multiple optimal solutions in a single pass. Wherefore it is up to the designer to select the singular solution which best meets requirements. By means of simulation, we endeavour to establish its relative performance against an optimisation typifying a standard topology control technique in the literature in terms of the proportion of time the network exhibited the property of strong connectedness. As to which, an analysis of the results indicates that such is highly sensitive to factors of: the effective maximum transmitting range, node density, and mobility scenario under observation. We derive an estimate of the optimal constitution thereof taking into account the specific conditions within the domain of application in that of a WSN, thereby concluding that only GA optimising for the biconnected components in a network achieves the stated objective of a sustained connected status throughout the duration.fi=Opinnäytetyö kokotekstinä PDF-muodossa.|en=Thesis fulltext in PDF format.|sv=Lärdomsprov tillgängligt som fulltext i PDF-format

Topology Control, Routing Protocols and Performance Evaluation for Mobile Wireless Ad Hoc Networks

A mobile ad-hoc network (MANET) is a collection of wireless mobile nodes forming a temporary network without the support of any established infrastructure or centralized administration. There are many potential applications based the techniques of MANETs, such as disaster rescue, personal area networking, wireless conference, military applications, etc. MANETs face a number of challenges for designing a scalable routing protocol due to their natural characteristics. Guaranteeing delivery and the capability to handle dynamic connectivity are the most important issues for routing protocols in MANETs. In this dissertation, we will propose four algorithms that address different aspects of routing problems in MANETs. Firstly, in position based routing protocols to design a scalable location management scheme is inherently difficult. Enhanced Scalable Location management Service (EnSLS) is proposed to improve the scalability of existing location management services, and a mathematical model is proposed to compare the performance of the classical location service, GLS, and our protocol, EnSLS. The analytical model shows that EnSLS has better scalability compared with that of GLS. Secondly, virtual backbone routing can reduce communication overhead and speedup the routing process compared with many existing on-demand routing protocols for routing detection. In many studies, Minimum Connected Dominating Set (MCDS) is used to approximate virtual backbones in a unit-disk graph. However finding a MCDS is an NP-hard problem. In the dissertation, we develop two new pure localized protocols for calculating the CDS. One emphasizes forming a small size initial near-optimal CDS via marking process, and the other uses an iterative synchronized method to avoid illegal simultaneously removal of dominating nodes. Our new protocols largely reduce the number of nodes in CDS compared with existing methods. We show the efficiency of our approach through both theoretical analysis and simulation experiments. Finally, using multiple redundant paths for routing is a promising solution. However, selecting an optimal path set is an NP hard problem. We propose the Genetic Fuzzy Multi-path Routing Protocol (GFMRP), which is a multi-path routing protocol based on fuzzy set theory and evolutionary computing