# Natural Computing Applied to the Underground System: A Synergistic Approach for Smart Cities
## 5 Algorithms Involved
### Fireworks Algorithm  
As history shows, mathematical optimization has always been a field under investigation ([PERSON], 2004). More concretely, and due to its vital importance on Computer Science, the search processes involved in computing problems have been widely investigated. These processes are highly related to sorting algorithms too. A search algorithm is a set of mathematical instructions aiming to place a custom order between the elements within a collection. Optimization techniques, as well as efficient sorting, has been the main objective when it comes to these kinds of algorithms since the first investigations on the field ([PERSON], 2004). In a similar way, Fireworks Algorithm (FWA onwards) belongs to the Swarm Intelligence spectrum and takes its inspiration by observing fireworks explosion ([PERSON], 2004; [PERSON], 2004). This algorithm is proposed for the  
Figure 6: Particle swarm optimization model: position update using a star topology (\(g_{best}\)) and an inertia \(w\) term. Please note that \(g_{best}\), \(p_{best}^{1}\) and \(p_{best}^{2}\) are fixed in time \(t\) in order to make the figure as simple as possible, their position could change (or not) depending on the fitness value of particles \(x^{1}\) and \(x^{2}\) along time in a real example. optimization of complex functions, and its implemented following an accurate simulation of fireworks explosion process. FWA takes special attention to keep the diversity of sparks as it will be explained, by maintaining two different search processes. In general, terms, FWA constitutes an approach to explore a massive search space. This search is based on the search of random points confined by a certain distance measure that hopes one or more of the points of interest will yield promising results. Once these points give interesting results in terms of the mathematical function to be optimized, a more concentrated search will be spawned in the near points, iterating through the algorithm until an optimal solution is found. Regarding FWA operating mode, the algorithm starts with the selection of a concrete number, \(N\), of initial locations. \(N\) fireworks will be thrown at these locations, and the fireworks will consequently throw sparks. The location of these sparks will be retrieved to evaluate its quality, and in case the optimal solution has been found, the algorithm will finish. Otherwise, FWA will set another \(N\) fireworks at \(N\) locations, spawning and iterating repeatedly over the aforementioned process until the optimal solution is found. To validate the convergence curves of FWA, Clonal Particle Swarm Optimization (CPSO) and Standard Particle Swarm Optimization (SPSO) are taken [33]. A set of eight different benchmark functions (Sphere, Rosenbrock, Ellipse, Cigar, Rastrigin, Griewank, Tablet and Schwefel) averaged over 20 independent runs are thrown on these algorithms, reaching the conclusion that FWA presents a higher speed compared to CPSO and SPSO, see Figure 7.  
The information shown in the last figure, with the addition of Ackley benchmark function, can be analyzed in the following figure, where the Statistical Mean and Standard Deviation for FWA are far more agile compared to CPSO and SPSO, see Table 1.  
Figure 7: Convergence curves of the FA, the CPSO and the SPSO on eight benchmark functions. The function fitness is averaged over 20 independent runs. Reprinted from [35], with kind permission from Springer Science + Business Media. Copyright © Springer-Verlag Berlin Heidelberg 2010. To sum up, FWA finds brilliant solutions with 1000 times of function evaluations, reflecting the quick convergence speed of the algorithm against CPSO and SPSO. This algorithm is therefore a strong ally when it comes to Smart Cities environment applications in a computing-related way, being currently investigated to create a system capable of searching lost people in the Smart City rural surroundings. FWA has multiple applications, as [34] shows with a system for data mining low-rank applications such web search.