Shortest Path Tree Research Articles

Network Creation Games with Traceroute-Based Strategies

Network creation games have been extensively used as mathematical models to capture the key aspects of the decentralized process that leads to the formation of interconnected communication networks by selfish agents. In these games, each user of the network is identified by a node and selects which link to activate by strategically balancing his/her building cost with his/her usage cost (which is a function of the distances towards the other player in the network to be built). In these games, a widespread assumption is that players have a common and complete information about the evolving network topology. This is only realistic for small-scale networks as, when the network size grows, it quickly becomes impractical for the single users to gather such a global and fine-grained knowledge of the network in which they are embedded. In this work, we weaken this assumption, by only allowing players to have a partial view of the network. To this aim, we borrow three popular traceroute-based knowledge models used in network discovery: (i) distance vector, (ii) shortest-path tree view, and (iii) layered view. We settle many of the classical game theoretic questions in all of the above models. More precisely, we introduce a suitable (and unifying) equilibrium concept which we then use to study the convergence of improving and best response dynamics, the computational complexity of computing a best response, and to provide matching upper and lower bounds to the price of anarchy.

Path Optimization along Buoys Based on the Shortest Path Tree with Uncertain Atmospheric and Oceanographic Data.

In order to design an optimized route for ships in line with economic benefits; avoid bad weather; reduce unnecessary detours; shorten the navigation time; and achieve the purpose of safety, fuel saving, and punctual arrival, this paper takes the navigation mark as the node of the tree, takes the connection of the adjacent navigation marks as the tree path, and divides the distance of the adjacent buoys by the ship's speed as the path cost. The speed calculation collects the current hydrometeorological data such as wind and wave data, uses Aertssen's deceleration formula to adjust the speed, and improves Dijkstra to find the shortest path. In the experiment, two routes from Dongdu to Xiamen Gang Kou are compared under bad weather conditions. Route 1 is with 5.877 m/s average wind speed, 0.860 m/s wave speed, and total distance 34717 m. Route 2 is with 8.503 m/s average wind speed, 1.429 m/s wave speed, and total distance 30223 m. The calculated ship speed travelling in route 1 is 12.243 km, and its travelling time is 1.53 h. The calculated ship speed travelling in route 2 is 10.523 km, and its travelling time is 1.55 h. Although the total distance of route 1 is longer, it takes less time for ships to travel in route 1. The experimental results verify the effectiveness of the navigation algorithm based on the shortest path tree of uncertain weather maps.

A Priority-Based Multicast Flow Scheduling Method for a Collaborative Edge Storage Datacenter Network

Edge storage, as a supplement to cloud storage, reduces latency by providing services in a timely and efficient manner near the source. In a collaborative edge storage datacenter network (CESN), not only does the edge storage datacenter (ESDC) that is closest to the user provide services, but multiple ESDCs work together to provide better services. In this collaborative work mechanism, different application session requests create large persistent multicast flows with diverse performance requirements. Existing multicast scheduling methods such as unicast shortest path (USP) and static single tree (SST) do not consider flow characteristics or performance requirements. In this paper, we first modeled the multicast flow scheduling problem in a CESN. The model is based on different types of flows with diverse network requirements. Then, we tailored a multicast flow scheduling method based on multiple-attribute decision-making and a genetic algorithm (MDGA). MDGA selects appropriate multicast routing paths for flows in a CESN by considering the requested flow types and network status. The experimental results show that the proposed MDGA method can balance network loads and reduce the average transmission delay for high-priority flows better than USP and SST.

Algebraic Algorithms for Betweenness and Percolation Centrality

In this paper, we explored different ways to write the algebraic version of betweenness centrality algorithm. Particularly, we focused on Brandes' algorithm [Brandes, Journal of Mathematical Sociology, 2001]. We aimed for algebraic betweenness centrality that can be parallelized easily. We proposed 3-tuple geodetic semiring as an extension to the usual geodetic semiring with 2-tuples [Batagelj, Journal of Mathematical Sociology, 1994]. Using the 3-tuple geodetic semiring, Dijkstra's and Brandes' algorithm, we wrote more concise and general algebraic betweenness centrality (ABC) algorithm which is valid for weighted and directed graphs. We also proposed an alternative version of ABC using the usual geodetic semiring with 2-tuple where we used a simple way to construct shortest path tree after computing shortest path distances in the usual geodetic semiring. This allows us to avoid computational complexity of ABC implementation using 3-tuple geodetic semiring. We used numba [Lam et al., LLVM'15, 2015] to optimize and parallelize ABC. We evaluated the performance of ABC using 2-tuple geodetic semiring as compared to NetworkX [Hagberg et al., SciPy 2008, 2008], a common python package for graph algorithms. We did scalability experiments on parallel ABC and showed its total speedup. We also showed that with small modification, ABC can be adapted to algebraicly compute other centrality measures such as percolation centrality.

An Effective Genetic Algorithm for Solving the Clustered Shortest-Path Tree Problem

The clustered shortest-path tree problem (CluSPTP) is an extension of the classical single-source shortest-path problem, in which, given a graph with the set of nodes partitioned into a predefined, mutually exclusive and exhaustive set of clusters, we are looking for a shortest-path spanning tree from a given source to all the other nodes of the graph, with the property that each cluster should induce a connected subtree. CluSPTP belongs to the class of generalized combinatorial optimization problems, and, in general, is proved to be a non-deterministic polynomial time hard (NP-hard) problem. In this paper, we propose a novel genetic algorithm (GA), which is designed to fit the challenges of the investigated problem. The main features of our GA are: the use of an innovative representation scheme that allows us to define meaningful genetic operators and the use of a hybrid initial population. Extensive computational results are reported and discussed for two sets of instances: euclidean and non-euclidean. The performance of the proposed algorithm was evaluated on six types of benchmark euclidean instances available in the literature and on six types of non-euclidean instances obtained from the corresponding euclidean ones. The obtained results show an improvement with respect to existing methods from the literature, both in terms of the quality of the achieved solutions and the computation times necessary to obtain them. They demonstrate that our genetic algorithm outperforms all the existing methods from the literature, providing for all the existing benchmark instances the optimal solutions in all 30 independent trials.

Algorithms for Weighted Matching Generalizations II: f-factors and the Special Case of Shortest Paths

For an undirected graph or multigraph $G=(V,E)$ and a function $f:V\to \mathbb{Z_+}$, an $f$-factor is a subgraph whose degree function is $f$. For integral edge weights of maximum magnitude $W$ our algorithm finds a maximum weight $f$-factor in time $\tilde{O}(Wf(V)^{\omega})$, where $f(V)=\sum_{v\in V} f(v)$ and $\omega$ is the exponent of matrix multiplication. The algorithm is randomized and has two versions. For worst-case time the algorithm is correct with high probability. For expected time the algorithm is Las Vegas. The algorithm is based on a detailed analysis of the structure of the optimum blossoms. A special case gives a representation for single-source shortest-paths in conservative undirected graphs, generalizing the standard shortest-path tree to a “tree of cycles”. The representation can be constructed by a randomized algorithm with the same time bound as above, or deterministically by an algorithm for maximum weight matching, achieving time $O(n(m + n \log n))$ or $O(\sqrt{n }\ m \log (nW))$.

Hardness of an Asymmetric 2-Player Stackelberg Network Pricing Game

Consider a communication network represented by a directed graph G=(V,E) of n nodes and m edges. Assume that edges in E are partitioned into two sets: a set C of edges with a fixed non-negative real cost, and a set P of edges whose costs are instead priced by a leader. This is done with the final intent of maximizing a revenue that will be returned for their use by a follower, whose goal in turn is to select for his communication purposes a subnetwork of Gminimizing a given objective function of the edge costs. In this paper, we study the natural setting in which the follower computes a single-source shortest paths tree of G, and then returns to the leader a payment equal to the sum of the selected priceable edges. Thus, the problem can be modeled as a one-round two-player Stackelberg Network Pricing Game, but with the novelty that the objective functions of the two players are asymmetric, in that the revenue returned to the leader for any of her selected edges is not equal to the cost of such an edge in the follower’s solution. As is shown, for any ϵ>0 and unless P=NP, the leader’s problem of finding an optimal pricing is not approximable within n1/2−ϵ, while, if G is unweighted and the leader can only decide which of her edges enter in the solution, then the problem is not approximable within n1/3−ϵ. On the positive side, we devise a strongly polynomial-time O(n)-approximation algorithm, which favorably compares against the classic approach based on a single-price algorithm. Finally, motivated by practical applications, we consider the special cases in which edges in C are unweighted and happen to form two popular network topologies, namely stars and chains, and we provide a comprehensive characterization of their computational tractability.

Spatio-Temporal Routing in Episodically Connected Vehicular Networks

The spatio-temporal routing in episodically connected vehicular networks es a communication pattern where the destination nodes is a set of vehicles that transit the destination region during the specified destination region. The spatio-temporal routing is well suited to support Smart City and Intelligent Transportation Systems applications (i.e. road safety, crowdsensing, entertainment). In this paper, we present Oportunistic Greedy Routing over Street-layout Graph (OGRoSG), a novel spatio-temporal routing protocol for episodically connected vehicular networks. Unlike previous works in the literature, OGRoSG takes advantage of information available in modern navigation systems, such as the streets map and the geographic position. The protocol performs greedy geographic routing, the notion of being closer to the destinationare a is based on the progress towards the destination region on the streets-map graph and the direction of the vehicles. In order to achieve this, the vehicles process the destination region and the street graph to generate a shortest-paths tree with an extended destination regionas root. This extended destination region guarantees that it covers the entire destination region and the paths that interconnect it. We evaluate the performance of the OGRoSG protocol using simulations in NS-3,using mobility traces generated in a map based in the city of Murcia, Spain. Our results show that OGRoSG outperforms a set of Spatio-Temporal variants of the Epidemic, Spray & Wait and Binary Spray & Wait routing protocols in terms of delivery ratio and overhead induced in the network.

Evolutionary algorithm and multifactorial evolutionary algorithm on clustered shortest-path tree problem

In literature, Clustered Shortest-Path Tree Problem (CluSPT) is an NP-hard problem. Previous studies focus on approximation algorithms which search for an optimal solution in relatively large space. Thus, these algorithms consume a large amount of computational resources while the quality of obtained results is lower than expected. In order to enhance the performance of the search process, this paper proposes two different approaches which are inspired by two perspectives of analyzing the CluSPT. The first approach intuition is to narrow down the search space by reducing the original graph into a multi-graph with fewer nodes while maintaining the ability to find the optimal solution. The problem is then solved by a proposed evolutionary algorithm. This approach performs well on those datasets having small number of edges between clusters. However, the increase in the size of the datasets would cause the excessive redundant edges in multi-graph that pressurize searching for potential solutions. The second approach overcomes this limitation by breaking down the multi-graph into a set of simple graphs. Every graph in this set is corresponding to a mutually exclusive search space. From this point of view, the problem could be modeled into a bi-level optimization problem in which the search space includes two nested search spaces. Accordingly, the Nested Local Search Evolutionary Algorithm (N-LSEA) is introduced to search for the optimal solution of glscluspt, the upper level uses a simple Local Search algorithm while the lower level uses the Genetic Algorithm. Due to the neighboring characteristics of the local search step in the upper level, the lower level reduced graphs share the common traits among each others. Thus, the Multi-tasking Local Search Evolutionary Algorithm (MLSEA) is proposed to take advantages of these underlying commonalities by exploiting the implicit transfer across similar tasks of multi-tasking schemes. The improvement in experimental results over N-LSEA via this multi-tasking scheme inspires the future works to apply M-LSEA in graph-based problems, especially for those could be modeled into bi-level optimization.

Two levels approach based on multifactorial optimization to solve the clustered shortest path tree problem

The Clustered Shortest-Path Tree Problem (CluSPT) has a great meaning in theoretical research as well as a wide range of applications in everyday life, especially in the field of network optimization. Being able to solve the CluSPT will path the way for improving practical systems such as agricultural irrigation and product distribution. Multifactorial Evolutionary Algorithm (MFEA) is a variant of Evolutionary Algorithm (EA) aiming at simultaneously solving multiple factorial tasks which can be diverse in types, dimensionalities and search spaces. The applications of MFEA have yet to be fully exploited, but the realm has recently attracted much interest from the research community, especially those who are working on evolutionary multitasking. Considering these characteristics, this paper describes a new approach using the MFEA for solving the CluSPT. The MFEA has two tasks: the first is solving the CluSPT problem and the second is solving a new problem which is decomposed from the CluSPT problem. The goal of the first task is finding the fittest solution as possible for the CluSPT while the goal of the second task is determining the best tree (w.r.t. cost minimization) which envelops all vertices of the problem. This paper also introduces mutation, crossover and population initialization operators for a proposed MFEA to solve the CluSPT. Each of these operators deals with individuals in two phases, one of which ensures that the resulted individual is a spanning tree and the other phase ensures that each of its clustered sub-graphs is also a spanning tree. As the MFEA had to deal with these two tasks, a decoding method was also created to allow communication between the tasks. To assess the effectiveness of the proposed algorithm and methods, the authors implemented them on both Euclidean and Non-Euclidean instances. Experiment results show that the proposed MFEA outperformed the existing MFEA algorithms on two-thirds instances with average improvement value of 47%. Besides, this paper analysis the convergence trends of each task in generations and the affectation of the number of vertices in the largest cluster on the obtained results by using both the Spearman’s rank correlation coefficient and scatter graphs.

An Efficient Data Collection Protocol for Underwater Wireless Sensor Networks

This paper presents the design and evaluation of a new data collection protocol for Underwater Wireless Sensor Networks called the Data Collection Tree Protocol (DCTP). It uses an efficient distributed algorithm to proactively construct and maintain a data collection tree rooted at the sink node. The preconstructed and maintained data collection tree allows the efficient selection of a single forwarding node at each hop when routing a data packet. We prove the correctness of the constructed data collection tree and we show that under some stability conditions, the constructed tree converges to an optimal shortestpath tree. Results of extensive simulations show a big improvement in terms of packet delivery ratio, endto-end delay and energy consumption compared to the well-known VBF protocol. The simulated cases show increases in the packet delivery ratio between 20% and 122%, reductions in the average end-to-enddelay between 15% and 55% and reductions in the energy consumption between 20% and 50%. These results clearly demonstrate the attractiveness of the proposed DCTP protocol.

A novel genetic algorithm for solving the clustered shortest-path tree problem

The clustered shortest-path tree problem is an extension of the classical single-source shortest-path problem, in which, given a graph with the set of nodes divided into a redefined, mutually exclusive and exhaustive set of clusters, we want to determine a shortest-path spanning tree from a given source to all the other nodes of the graph, with the property that each cluster should induce a connected subtree. The investigated problem proved to be NP-hard and therefore we proposed an efficient genetic algorithm in order to solve it. The preliminary computational results reported on a set of benchmark instances from the literature proved that our proposed solution approach yields high-quality solutions within reasonable running times.

Set Invariance and Optimal Set Stabilization of Boolean Control Networks: A Graphical Approach

This article investigates set invariance and optimal set stabilization of Boolean control networks (BCNs) from a novel graph-theoretical perspective. First, we establish the relationship between the control invariant sets and the strongly connected components of the state transition graph (STG), and propose a new algorithm to compute the largest control invariant subset. Second, both set stabilizability and time-optimal set stabilization via state feedback are characterized by a breadth-first tree of the STG. Though the prior two problems have been attempted by existing algebraic methods, the primary advantage of our alternative graphical approach is the enhanced computational efficiency. Third, we consider for the first time, the optimal set stabilization problem with general cost criteria, not limited to time optimality. A state feedback controller is constructed for general optimal set stabilization using a shortest-path tree of the STG. Finally, it is straightforward to extend the proposed approach to handle various constraints as well as other types of logical networks by slightly modifying the STG. Two biological networks are used to test the performance of our approach. Results show that in accordance with time complexity analysis, our algorithms can dramatically reduce the running time for the first two problems compared with existing work, and effectively solve the general optimal set stabilization problem.

Observer of changes in the forest of the shortest paths on dynamic graphs of transport networks

The purpose of the work is the development of basic data structures, speed-efficient and memoryefficient algorithms for tracking changes in predefined decisions about sets of shortest paths on transport networks, notifications about which are received by autonomous coordinated transport agents with centralized or collective control. A characteristic feature of transport operations is the independence and asynchrony of the emergence of perturbations of optimal solutions, as well as the lack of global influence of individual perturbations on the set of all processes on the network. This clearly determines the feasibility of realizing the idea of reoptimizing existing solutions in real time as information is received about disturbances in the structure and parameters of the transport network, various restrictions on the use of existing shortest paths. In contrast to the classical problems of finding shortest paths on static or dynamic graphs, it is proposed to supplement the set of situations controlled by the observer by taking into account the associations of shortest path trees with agents that actually use such paths. This will improve the responsiveness of agent notification processes for timely switching to a new path. The space of search states is a dynamically generated bipartite sparse graph of the transport network, represented by a list of arcs. The basic algorithm for finding the shortest paths uses Dijkstra's scheme, but implements a bootstrapping method to generate the search result. The compactness of the representation of the observed forest of shortest paths is achieved by mapping individual trees of such a forest onto the projection of tree vertices in memory, where the position of each vertex corresponds to the distance from the tree root. The proposed version of the construction of the search procedure is based on the mechanisms existing in database management systems for creating different relational representations of the physical data model. This eliminates the need to solve technological problems of complexing heterogeneous models of dynamic transport networks, memory allocation. As a result, the specification of various rules for the logistics of transport operations is simplified, since such operations in terms of object-oriented models are easily determined by polymorphic classes of transitions between nodes of the transport network.

Dynamic Defense against Stealth Malware Propagation in Cyber-Physical Systems: A Game-Theoretical Framework.

Stealth malware is a representative tool of advanced persistent threat (APT) attacks, which poses an increased threat to cyber-physical systems (CPS) today. Due to the use of stealthy and evasive techniques, stealth malwares usually render conventional heavy-weight countermeasures inapplicable. Light-weight countermeasures, on the other hand, can help retard the spread of stealth malwares, but the ensuing side effects might violate the primary safety requirement of CPS. Hence, defenders need to find a balance between the gain and loss of deploying light-weight countermeasures, which normally is a challenging task. To address this challenge, we model the persistent anti-malware process as a shortest-path tree interdiction (SPTI) Stackelberg game with both static version (SSPTI) and multi-stage dynamic version (DSPTI), and safety requirements of CPS are introduced as constraints in the defender’s decision model. The attacker aims to stealthily penetrate the CPS at the lowest cost (e.g., time, effort) by selecting optimal network links to spread, while the defender aims to retard the malware epidemic as much as possible. Both games are modeled as bi-level integer programs and proved to be NP-hard. We then develop a Benders decomposition algorithm to achieve the Stackelberg equilibrium of SSPTI, and design a Model Predictive Control strategy to solve DSPTI approximately by sequentially solving an approximation of SSPTI. Extensive experiments have been conducted by comparing proposed algorithms and strategies with existing ones on both static and dynamic performance metrics. The evaluation results demonstrate the efficiency of proposed algorithms and strategies on both simulated and real-case-based CPS networks. Furthermore, the proposed dynamic defense framework shows its advantage of achieving a balance between fail-secure ability and fail-safe ability while retarding the stealth malware propagation in CPS.

The functional brain network based on the combination of shortest path tree and its application in fatigue driving state recognition and analysis of the neural mechanism of fatigue driving

Aimed at studying the method of constructing a functional brain network (FBN) that can effectively recognize the state of fatigue driving based on electroencephalogram (EEG), and analyzing which regions of the brain (electrode) are closely related to the occurrence of fatigue driving. A method based on the combination of shortest path tree (CSPT) for constructing a functional brain network (denoted as CSP-FBN) is proposed, which is applied to fatigue driving state recognition and neural mechanism analysis of fatigue driving. Through the comparison experiment of the classification accuracy in the same frequency band (beta band), the results show that the functional brain network constructed by the combined shortest path tree in fatigue state recognition is better than the functional brain network constructed by other methods, the accuracy of 10-fold cross validation reaches 99.17%. At the same time, we also find that Fz, F4, Fc3, Fcz, Fc4, C3, Cz4, Cp3, Cpz, Cp4, P3, Pz and P4 are important electrodes for fatigue driving state recognition, which reflects that the right central region and the central parietal region of the brain have a close relationship with the occurrence of fatigue driving.

An Open-Source Framework of Generating Network-Based Transit Catchment Areas by Walking

The transit catchment area is an important concept for public transport planning. This study proposes a methodological framework to generate network-based transit catchment areas by walking. Three components of the framework, namely subgraph construction, extended shortest path tree construction and contour generation are presented step by step. Methods on how to generalize the framework to the cases of the directed road network and non-point facilities are developed. The implementation of the framework is provided as an open-source project. Using metro stations in Shanghai as a case study, we illustrate the feasibility of the proposed framework. Experiments show that the proposed method generates catchment areas of high geospatial accuracy and significantly increases computational efficiency. The open-source program can be applied to support research related to transit catchment areas and has the potential to be extended to include more routing-related factors.

A shortest-path tree approach for routing in space networks

A new space era is bringing deployments of thousands of networked assets in near-Earth and deep-space. Delay-Tolerant Networking (DTN) protocols and algorithms such as Contact Graph Routing (CGR) are state-of-the-art technologies certainly capable of achieving this ambitious objective. Nevertheless, in this paper, we highlight and analyze the roots of current scalability limits of CGR, to then propose more efficient alternatives. In particular, we leverage a different graph modeling approach that enables optimizations such as priority queues and spanning-tree searches in the core route determination process. While such abstract data types are difficult to adapt to CGR, we incorporate them into a novel Shortest-Path Tree Approach for Routing in Space Networks (SPSN), which is presented as a compute-efficient alternative. This paper thoroughly describes the SPSN approach including route object modeling, capacity-oriented search, and potential route management techniques. Prototype evaluations analyzed in this paper present significant evidence that SPSN outperforms legacy CGR in more than an order of magnitude in contact plans comprising thousands of contacts.

Performance of routing and spectrum allocation approaches for multicast traffic in elastic optical networks

In this paper, we investigate routing and spectrum allocation (RSA) approaches in elastic optical networks for multicast traffic demands. We consider that a light-tree is constructed for each multicast traffic demand and spectrum is allocated to this light-tree using the first-fit spectrum allocation policy. We present an approximation based Steiner tree approach for RSA (STA-RSA) and compare it with the shortest path tree based routing and spectrum allocation (SPT-RSA) to show the benefits of STA-RSA over SPT-RSA. The numerical results show that the presented approach outperforms the traditional approaches in terms of bandwidth blocking probability.

A multifactorial optimization paradigm for linkage tree genetic algorithm

Linkage Tree Genetic Algorithm (LTGA) is an effective Evolutionary Algorithm (EA) to solve complex problems using the linkage information between problem variables. LTGA performs well in various kinds of single-task optimization and yields promising results in comparison with the canonical genetic algorithm. However, LTGA is an unsuitable method for dealing with multi-task optimization problems. On the other hand, Multifactorial Optimization (MFO) can simultaneously solve independent optimization problems, which are encoded in a unified representation to take advantage of the process of knowledge transfer. In this paper, we introduce Genetic Algorithm (MF-LTGA) by combining the main features of both LTGA and MFO. MF-LTGA is able to tackle multiple optimization tasks at the same time, each task learns the dependency between problem variables from the shared representation. This knowledge serves to determine the high-quality partial solutions for supporting other tasks in exploring the search space. Moreover, MF-LTGA speeds up convergence because of knowledge transfer of relevant problems. We demonstrate the effectiveness of the proposed algorithm on two benchmark problems: Clustered Shortest-Path Tree Problem and Deceptive Trap Function. In comparison to LTGA and existing methods, MF-LTGA outperforms in quality of the solution or in computation time.