Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 345049, 23 pages

http://dx.doi.org/10.1155/2015/345049

## A Universal Concept for Robust Solving of Shortest Path Problems in Dynamically Reconfigurable Graphs

Institute of Smart Systems Technologies, Transportation Informatics Group (TIG), Universität Klagenfurt, Klagenfurt, Austria

Received 27 May 2015; Accepted 4 November 2015

Academic Editor: John D. Clayton

Copyright © 2015 Jean Chamberlain Chedjou and Kyandoghere Kyamakya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper develops a flexible analytical concept for robust shortest path detection in dynamically reconfigurable graphs. The concept is expressed by a mathematical model representing the shortest path problem solver. The proposed mathematical model is characterized by three fundamental parameters expressing (a) the graph topology (through the “incidence matrix”), (b) the edge weights (with dynamic external weights’ setting capability), and (c) the dynamic reconfigurability through external input(s) of the source-destination nodes pair. In order to demonstrate the universality of the developed concept, a general algorithm is proposed to determine the three fundamental parameters (of the mathematical model developed) for all types of graphs regardless of their topology, magnitude, and size. It is demonstrated that the main advantage of the developed concept is that arc costs, the origin-destination pair setting, and the graph topology are dynamically provided by external commands, which are inputs of the shortest path solver model. This enables high flexibility and full reconfigurability of the developed concept, without any retraining need. To validate the concept developed, benchmarking is performed leading to a comparison of its performance with the performances of two well-known concepts based on neural networks.

#### 1. Introduction

The shortest path problem (SPP) is one of the classical combinatorial optimization problems [1–3]. This problem relates to finding the minimum cost path in a predefined source-destination node pair in a weighted graph network [3]. The interest devoted to shortest path finding can be explained by the various potential applications of the SPP in science and engineering. Indeed, the SPP is essentially involved in many use cases, in, for example, the following: vehicle routing in transportation systems [4], traffic routing in communication networks [5], path planning in robotic systems [6, 7] and scheduling [8], and video image analysis [9]. Further applications of SP are found in electronics (e.g., for VLSI physical design) [10], medical imagery (e.g., for virtual endoscopy) [11], and image processing (e.g., for energy minimization in vision) [12], just to name a few.

In intelligent transportation systems, the SPP can be considered as a subproblem for many broader problems such as route guidance [4], vehicle dispatching [13], real-time traffic information sensing [4], and production systems planning. These specific problems require real-time (e.g., ultrafast) processing in order to achieve results in extreme short time deadlines, examples of results being the efficient schedules and identification of new routes (or paths) in transportation networks.

In computer science SP algorithms can be used in the automatic search of directions between physical locations (e.g., driving directions on web mapping websites like in the American free online web mapping service (MapQuest) and also in the desktop web mapping service (Google Maps)) [14]. SP can also be used in applications like network management (e.g., finding the most vital node of a shortest path) [15] and graph structure in the web [16].

Thus, there is an explicit necessity of enriching the state of the art by developing new and efficient (i.e., extremely fast) shortest path problem (SPP) solver concepts. The efficiency in this context relates to some key performance metrics: (a) fast computing, (b) low memory consumption, (c) robustness, (d) accuracy, and (f) dynamic/runtime reconfigurability.

In the aforementioned applications, the shortest path detection problem is an important problem to be addressed. This problem has been studied extensively in the fields of computer science, operations research, and transportation engineering [4–9]. The well-known polynomial-time algorithms for solving shortest path problems include Bellman’s dynamic programming algorithm [17], Dijkstra’s algorithm [18], and Bellman-Ford successive algorithm [19, 20]. However, polynomial-time algorithms are suitable only for nonnegative costs of edges and for cases of additive linear path cost models. Thus, these algorithms cannot solve the SPP in the presence of negative costs [21] or in the presence of nonlinear/nonadditive path cost models [22]. Further, polynomial algorithms may appear to be too time consuming (slow) specifically when dealing with real-time applications in real traffic networks. These algorithms can also fail to provide the correct solution in certain cases such as when multiple shortest paths of equal total cost but with differences in the number of hops (i.e., number of involved edges) exist in a given graph.

A dynamic algorithm was developed as an extension of polynomial Dijkstra’s and Bellman-Ford algorithms in order to analyze shortest path tree problems [23]. Despite the fact that the dynamic algorithm can easily adapt to changes in the graph topology, it is however very time consuming and the convergence is an issue specifically for huge size problems [23]. Further methods have been developed to address SP problems while assuming integer values of the costs of edges such as the scaling technique [24], the integer matrix multiplication technique [25], the fast integer sorting technique [26], and the component hierarchy technique [27]. These methods are not valid for noninteger values of edge weights and therefore are not applicable to real-life or technical systems.

A first basic mathematical approach for SPP solving consists of modeling the shortest path detection problem into a linear programming constrained problem, which is further solved using Dantzig’s simplex method [28]. This method is prone to failures and the accuracy is poor [29]. Several interesting contributions for SPP solving presented in the related literature do involve selected artificial intelligence concepts. Genetic algorithm (GA) [29] and particle swarm optimization (PSO) [30] can solve both deterministic and stochastic shortest path problems. In both cases, the solutions converge towards the optimal paths. However, GA is computationally very expensive compared to PSO [30] while the PSO also needs additional heuristics. Likewise, evolutionary algorithms (EA) are prone to invalid paths detection [30]. Heuristic search algorithms (e.g., ) [21] are developed to make use of additional knowledge in order to reduce the search efforts; however, heuristic search algorithms depend on the quality of the heuristic function used. This dependency affects the accuracy of results. Further, when the search area increases, more computation effort is required [21]. Hence, the solutions provided by heuristic algorithms are possibly less accurate and cost inefficient.

The concept of cellular automata (CA) has been intensively used to address SP problems in complex graphs [31–33]. Despite the potential significant improvement of the computing speed due to the parallel nature of the CA concept (i.e., CA is suitable for parallel computing), the reconfigurability is however an issue due to the strong dependence of the CA concept on initial conditions/states [34].

Another interesting SPP solver concept does involve artificial neural networks (ANN). The ANN SPP solver has been given tremendous attention due to its capability of performing parallel computing as well as its easy hardware implementability [35]. However, the basic ANN approach is prone to limitations such as lack of adaptability to dynamic graph topological changes and poor accuracy of results [35–37]. The Hopfield neural networks were developed based on linear programming to provide approximate solutions (to SPP) faster than the aforementioned algorithmic solutions [35, 38–40]. Mehmet Ali and Kamoun [36] proposed a variation of Hopfield neural networks as a new concept that can adapt to external varying conditions. This method fails however to converge towards valid solutions. Further, the computing performance degrades with increasing magnitude and size of the graph. To address these last mentioned limitations, Park and Choi [37] proposed a concept that is capable of handling graphs with huge sizes. This last-named concept is however prone to convergence issues [41]. The dependent variable Hopfield neural network (DVHNN) [41] was also proposed as a new neural network concept capable of addressing the inherent limitations of the previously mentioned classical concepts involving neural networks. Specifically, it was demonstrated that the DVHNN can efficiently tackle issues related to accuracy, convergence, and reliability when dealing with SPP [41]. Despite these strong points of the DVHNN, the method cannot efficiently handle real-time SPP detection problems in reconfigurable graphs since new training or retraining (of the neural network) is needed for each new source-destination node-pair setting. Furthermore, the DVHNN does not consider negative cost of edges and the accuracy degrades with the increasing costs of edges (higher values of weight values). This is justified by the assumption in [41] which considered only small values of edge costs.

In view of the above underlined limitations which are inherent to most neural network based concepts, a new neural network approach (called dynamic neural network (DNN)) was introduced and some interesting related works do demonstrate the effectiveness and efficiency of the DNN approach (see [42] and the references therein). However, despite the very good features of the DNN for SPP solving, some crucial limitations are underscored in [42], which are specifically related to the convergence failure observed under specific parameter settings. Another interesting issue worth mentioning is the route to convergence. This issue, which is extremely sensitive to changes in both initial values and the parameter settings of the dynamic mathematical model, is characterized by the duration of the transient phase and the computing time needed to reach convergence.

The key/main objective of this paper is to contribute to the enrichment of the SPP related state of the art by proposing a new approach expressed in form of ordinary differential equations. These equations do in essence represent a novel NAOP SPP solver model. This new form of NAOP can/does efficiently overcome the limitations of the abovementioned DVHNN and DNN concepts in [41] and [42], respectively. For proof of concept and for stress-testing purposes of the novel NAOP SPP solver approach developed, extensive benchmarking is conducted whereby its performance is compared with that of the competing concepts presented in [41, 42]. The performance metrics involved in the benchmarking are the convergence under high values of edge costs, the convergence in the presence of negative edge-cost values, the convergence potential under various parameter settings, the transient phase cancellation/duration, and the necessary computing time until convergence.

The rest of the paper is organized as follows. Section 2 describes at an abstract level the new proposed NAOP SPP solver concept for dynamically reconfigurable network graphs. A synoptic representation of this concept is proposed and a description of the key parameters as well as the complete system model is presented. Section 3 presents the general methodology for finding shortest paths in graphs through NAOP. The BDMM application to the shortest path problem is then addressed. The resulting coupled ODE equations are derived. The comprehensive benchmarking of the novel NAOP SPP solver is presented in Section 4 whereby a series of selected examples published in [41] (i.e., DVHNN concept) and [42] (i.e., DNN concept) is systematically considered. SPP solving is carried out using the NAOP SPP simulator on the one hand and the concepts in [41, 42] on the other hand. Some graph scenarios addressed and published in [41, 42] are considered and systematic benchmarking of the new NAOP solver concept with the concepts in [41, 42] is performed. The performance metrics used for the diverse comparisons are hereby mainly the simulation duration until convergence (computational efficiency) and the robustness (convergence). Regarding reconfigurability, it is only valid for the novel NAOP based SPP solver; the other concepts cannot support it. The benchmarking results obtained are used to underscore both the effectiveness and the efficiency of the novel NAOP SPP solver concept in complex and dynamically reconfigurable network graphs while considering even negative as well as high values of edge costs. Lastly, Section 5 presents a series of concluding remarks. A summary of the core contributions of this paper is presented. Further, selected interesting open research questions (under investigation in some of our ongoing subsequent works) are listed in an outlook.

#### 2. General Methodology Based on Nonlinear Adaptive Optimization (NAOP) for Modeling Shortest Path Problems (SPP)

In some of our recent contributions/papers (see [43, 44]), the NAOP concept has been successfully used for solving differential equations (ODEs and/or PDEs). We now want to demonstrate that the NAOP concept can be efficiently used as a general and robust framework for modeling shortest path problems (SPP) even in dynamically reconfigurable graphs.

This section provides a full description of the proposed NAOP concept for finding shortest paths in complex and dynamically reconfigurable graphs/networks. The abstract input/out logical diagram of this concept is schematically presented in Figure 1. The proposed concept does take three basic inputs as illustrated in Figure 1.