Route bundling in polygonal domains using Differential Evolution
 Victor Parque^{1, 2}Email authorView ORCID ID profile,
 Satoshi Miura^{1} and
 Tomoyuki Miyashita^{1}
Received: 28 September 2017
Accepted: 25 November 2017
Published: 8 December 2017
Abstract
Route bundling implies compounding multiple routes in a way that anchoring points at intermediate locations minimize a global distance metric to obtain a treelike structure where the roots of the tree (anchoring points) serve as coordinating locus for the joint transport of information, goods and people. Route bundling is a relevant conceptual construct in a number of pathplanning scenarios where the resources and means of transport are scarce/expensive, or where the environments are inherently hard to navigate due to limited space. In this paper we propose a method for searching optimal route bundles based on a selfadaptive class of Differential Evolution using a convex representation. Rigorous computational experiments in scenarios with and without convex obstacles show the feasibility and efficiency of our approach.
Keywords
Background
In this paper we tackle the route bundling problem which consists of compounding multiple routes in a way that intermediate points minimize a global distance metric of multiple origin–destinations pairs. In this context, the ultimate goal in route bundling is to construct treelike graph structures where the anchoring points, being roots of the tree structure, serve as coordinating locus for the joint transport of information, goods and people.
Application background
Compared to the pathplanning problem with single origin–destination nodes, the route bundling problem is a generalized formulation in the sense that the latter considers multiple origin–destination pairs. Naturally, in path planning with single origin–destination pairs, the anchoring point (root of minimal tree) coincides with the origin–destination nodes. In the literature, the path planning is a wellstudied topic [1–6]; yet, the route bundling problem is an emergent research topic having potential applications in problem scenarios involving compounded path planning with multiple origin–destination pairs. Here, designing the optimal network is relevant for the efficient use of resources while integrating and coordinating transport/communication needs.
Concretely speaking, route bundling has specific applications in environments where resources or means of transport/communication are scarce or expensive. For instance, consider the design of a network for transporting goods from/to multiple areas in an environment covered with obstacles; naturally, since onetoone transport would cause unwanted traffic or excessive cost in network construction, one is interested in designing a network where intermediate nodes serve as coordinating locus for source/destination locations.
Also, consider the design of optimal wire harness topologies for machines (e.g., cars and ships). Here, free space for electrical wiring is scarce and onetoone links are rather undesirable; thus, wire harness becomes essential to build treelike structures aiming to minimize global connectivity while ensuring minimal use of space.
Furthermore, consider the deployment of sensor networks for manytomany robots in the presence of attenuating obstacles (e.g., disaster areas). Here, attenuating obstacles induce in data loss or deterioration of the ability to communicate (e.g., concrete floors, steel reinforced floors, ceilings, elevators, walls, rock, and reinforced materials). Thus, it becomes imperative to design networks allowing to get the sensor signal around the obstructing materials. In disaster areas, route bundling becomes the building block to enable energyefficient networks.
Generally speaking, the presence of obstacles and holes in the environment induces on limited space and navigability, thus making route bundling relevant when either transport and communication means are scarce and expensive, or when optimal networking is a goal in manytomany origin–destination settings.
Related works
Basically, the algorithmic foundations of the route bundling have been laid out in two different fields: wireless sensor networks and network visualization.
In one hand, the study of wireless sensor networks [7] has rendered methodologies for network protocol and topology construction. Examples include the construction of a connected network to ensure complete coverage of an area of interest with the minimum number of nodes as possible [8], the degreeconstrained minimumweight connected dominated set for energyefficient topology control in wireless sensor networks [9], the topological optimization for consensusbased clock synchronization protocols [10], the tree topology construction for heterogeneous wireless sensor networks [11], the selfstabilizing algorithms to construct rooted trees under the assumption of node disconnection [12] and a number of topology optimization for network coverage, connectivity, energy savings, delay minimization, optimal routing and broadcasting [13–17].
In other domains, the route bundling problem has its closest foundations in edge bundling for network visualization field. Here, the basic aim is to compound edges in complex networks to ease the visualization or rendering of largescale networks. In particular, the conventional works have focused on the geometrybased edge clustering, in which the edges in the graph are forced to pass through points in a control mesh [18]. Also, the forcebased edge bundling where edges are modeled as springs being able to attract to each other [19, 20]. Furthermore, hierarchical clustering approaches have emerged. Here, in [21], the authors describe an approach based on attraction to the skeleton of the adjacent edges. And, in [22], the authors describe a kdtreebased optimization of the centroid points of close edges in the graph.
Our contribution
Although the above described algorithms for topology optimization in wireless sensor networks aim at finding an optimal hierarchy given a number of nodes, the existing algorithms are irrelevant to our scope since route bundling does not require centralized communication and compliance with network coverage (e.g., hoping diameter).
On the other hand, existing algorithms for edge bundling have a different scope: network bundles aim at rendering aesthetically pleasing and topologically compact drawings; yet, the existing algorithms do not necessarily aim at minimizing a global distance metric. Furthermore, it is nontrivial for the existing edge bundling algorithms to obtain minimallength networks in the presence of obstacles and holes in the environment.

We propose a natureinspired algorithm for searching tree bundles aiming at minimizing global length in a polygonal domain. In our approach, we use Differential Evolution and study the performance of our proposed algorithmic variants including the following:

DENC, Differential Evolution with Neighborhood and Convex Representation.

DEN, Differential Evolution with Neighborhood.

DEC, Differential Evolution with Convex Representation.

DE, Differential Evolution without Neighborhood nor Convex Representation.


We perform more than 12,000 experimental evaluations to confirm the feasibility, efficiency and robustness of our approach by considering diverse number of edges in the input bipartite network, diverse complexity configurations of polygonal obstacles with convex and nonconvex geometry (number of edges in polygon up to 10), and parametric comparisons considering population size and neighborhood size. Furthermore, we compare the convergence performance of the above described algorithmic variants. Based on these computational experiments, we provide insights on how to design optimal tree bundles by using our natureinspired approach.
Methods
This section describes the basic ideas as well as the algorithmic foundations in our proposed approach for route bundling.
Basic framework
In the above definition, the encoding implicitly represents a tree structure whose edges are free of overlaps with obstacles.
Also, note that the constraint \(x \in {\mathbf {T}}\) makes explicit the requirement that optimization is realized within the space \({\mathbf {T}}\) of feasible route bundles. In latter sections, we describe the representation which allows sampling of feasible route bundles.

Definition of a bipartite graph \(G = (V,E)\) wherein the edge \(e \in E\) represents the origin–destination pair (implying needs for communication/transportation between two points). Here, the number of edges and the locations of the source–destination pairs in the graph G are defined by the characteristics of the environment and/or by the needs of the user or network designer.

Definition of locations and geometry of the obstacles in the environment (which denote unfeasible areas for navigation/transportation). For simplicity and without loss of generality, we use polygonal obstacles with/without convexity properties which are reminiscent of indoor environments.

First, the geometry of the free space is computed given information of the set of obstacles (map) and bipartite network.

Then, the free space is triangulated using the Delaunay approach.

Finally, the locations of the anchoring points are optimized within the triangulated free space. Here, during optimization, fitness is defined by the distance metric F(x).
Representation of bundled routes
This subsection describes the mechanism used to represent bundled routes.
The above representation is simple; yet, it has a fundamental problem: it is unable to encode feasible route bundles since the condition \(P,Q \in \mathbb {R}^2\), implying \(P = (P_x, P_y)\) and \(Q = (Q_x, Q_y)\), does not ensure that coordinates are outside the nonnavigable space (overlapping with obstacles).

The free space of the polygonal map is triangulated using the Delaunay approach [23], as Fig. 2 exemplifies, in which a set \(T = \{t_1, t_2, \ldots , t_i, \ldots , t_n\}\) of n triangles are obtained.

Then, the anchoring points can be represented by 3element tuples, as follows:where \(i \in [n]\) and \(r_1, r_2 \in [0,1]\). In the above encoding, i is the index of the ith triangle \(t_i \in T\), and \(r_1, r_2\) are real numbers in the interval [0, 1].$$\begin{aligned} P = {(i, r_1, r_2) } \end{aligned}$$(3)

Bijection to Cartesian coordinates is possible in O(1) by using Eq. 4.

Route bundles are guaranteed to avoid overlaps with obstacles, and

Explicit computation of point inside polygon is avoided, implying the efficiency in scalability while sampling a very large number of points in the free navigable space.
Cost function
In this subsection, we describe the distance metric used to measure the quality/fitness of route bundles.

the distance of the shortest paths between origin nodes to anchoring point P,

the distance of the shortest paths between the anchoring points P and Q, and

the distance of the shortest paths between the anchoring points and the destination nodes.
Further extensions are possible. For example, instead of using the above euclidean metric, it is possible to use Manhattan distance (useful for designing networks for integrated circuits and tubular networks inside buildings). Also, it is possible to include weights in the above metric to balance the relevance of the distance to the origins compared to the distance to destinations.
Differential Evolution
This subsection describes the optimization algorithm used to compute the optimal route bundles.
We use Differential Evolution [25, 26] considering global and local interpolation vectors in order to tackle the problem of dealing with multimodal search space (the reader may note that Eq. 6 is multimodal in the case of polygonal maps with nonconvex obstacles).

\(\circ\) is the Hadamard product (elementwise).

\(x^c\) is the crossover individual at iteration t.

\(v_t\) is the mutant individual at iteration t.

\(m_t\) is a vector of masks containing zeros and ones.

\(\circ\) is the Hadamard product (elementwise).

\({\mathbf {r}}_{t,j}\) and jrand are random numbers uniformly distributed in \(\mathbb {R}^{[0,1]}\) and \(\mathbb {N}^{[D]}\) respectively.

CR is the probability of crossover.

\(D = 6\) is the dimensionality of the route bundling problem, Eq. (5).

\(g_t\) is the global donor individual.

\(l_t\) is the local donor individual.

\(\{x^1_t, x^2_t\} \subset {\mathbf P}\), are random individuals sampled from \({\mathbf P}\) for \(x^1 \ne x^2 \ne x_t\).

\(x_{gbest}\) is the global best in the population at iteration t.

\(x_{nbest_x}\) is the local best in the neighborhood \({\mathbf N} (x_t)\) of individual \(x_t\) at iteration t.

the neighborhood \({\mathbf N} (x_t)\) of vector \(x_t\) is the set of individuals contiguous to \(x_t\) by radius \(\rho = \frac{{\mathbf P} .\eta }{2}\) in a ring topology (see Fig. 3 for a ring topology with radius \(\rho = 2\)).

\(\{x^p, x^q\} \subset {\mathbf N} (x_t)\), are random individuals for \(x^p \ne x^q \ne x_t\).

\(w^{x_t}\) denotes the coefficient of individual \(x_t\), in which any coefficient \(w^{x_t} \in U[0,1]\) is set randomly at initial iteration.

\(w_{gbest}\) is the coefficient associated to \(x_{gbest}\),

\(w^{1}, w^{2}\) are the coefficients associated to the vectors \(x^{1}, x^{2}\) respectively.
The global (local) interpolation vector \(g_t\) (\(l_t\)) represents the position to which the direction of the trial vector should aim at in the 6dimensional search space, considering information from the population (neighborhood). Thus, the global (local) interpolation vector is the result of the current solution being translated by a sum of two directional vectors, one vector representing the direction to the best solution in the population (neighborhood) and another vector representing an arbitrary direction computed from the difference of two solutions within the population (neighborhood).
The first perturbation vector (the one multiplying \(\alpha\)) is an arithmetical recombination operator, while the second perturbation vector (the one multiplying \(\beta\)) is a differential mutation. The parameters \(\alpha\) and \(\beta\) have a scaling role toward best and arbitrary directions, respectively. A priori knowledge of problem convexity, unimodality or multimodality eases the selection of good values of \(\alpha\) and \(\beta\). Unimodal and convex (multimodal and nonconvex) fitness landscapes would favor values of \(\alpha\) (\(\beta\)) being larger than \(\beta\) (\(\alpha\)) to induce in an exploitative (explorative) behavior in both the global and local neighborhood, and to ease the faster convergence. Without aprior knowledge of the fitness landscape, it is recommendable to use \(\alpha , \beta \in (0, 2)\) to avoid overshooting while sampling in the search space [25, 26]. Furthermore, note that when \(\alpha = \beta\) and \(w = 1\), the above Differential Evolution is equivalent to the conventional DE/targettobest/1 strategy [26]; thus, the above algorithm is a generalization in which it considers not only the global population, but also the local neighborhood.
In computing the local neighborhood we use the ring topology to ensure speciation of individuals while preserving efficiency in the computation of best individuals in the neighborhood. An alternative approach is to use a clustering approach in which the neighborhood is defined as a local cluster. Yet, compared to the clustering approach, the ring topology is more efficient since computing the best individuals in the neighborhood takes \(O({\mathbf P} )\), while the clustering approach takes \(O({\mathbf P} ^2)\) for \({\mathbf P}\) being the population size.
Finally, the use of Differential Evolution with global and local interpolation vectors is advantageous to balance both exploration and exploitation over the entire search space \(x \in {\mathbf {T}}\), wherein the tradeoff between the global and the local search is selfadapted throughout the iterations.
Computational experiments
This section discusses our experimental results as well as obtained insights after evaluating the performance of our proposed method by using exhaustive computational experiments in diverse polygonal maps with both convex and nonconvex topology.
Settings
Our computing environment was Intel i74930K @ 3.4GHz, MATLAB 2016a.
Table 1 shows the key parameters of Differential Evolution such as the probability of crossover CR and the scaling factors \(\alpha\) and \(\beta\).
The reason of using a crossover probability \(CR = 0.5\) is to give equal importance to the search directions obtained from historical search, and those obtained considering local and global interpolations.
Parameters in Differential Evolution
Parameter  Symbol  Value 

Probability of Crossover  CR  0.5 
Scaling factor  \(\displaystyle \alpha , \beta\)  \(\Big \frac{ln(U(0,1))}{2} \Big \) 
Experimental scenarios
Experimental scenarios
Variables  Symbol  Values 

Edges of bipartite graph  E  \(\{5, 10, 15, 20, 25\}\) 
Polygonal obstacles  \(\{1, 2, 3, 4, 5\}\)  
Sides in obstacles  S  \(\{5, 10\}\) 
Population size  \({\mathbf P} \)  \(\{25, 50, 100, 200\}\) 
Neighborhood scaling factor  \(\displaystyle \eta\)  \(\{0.1, 0.2, 0.4, 0.8\}\) 
In order to give a glimpse of the type of polygonal maps used in our study, Figs. 4 and 5 show the topology of the bipartite network and the polygonal maps. Note that these domains are organized in a grid in which the horizontal (vertical) axis portrays, in ascending order from left (bottom) to right (top), the number of routes (polygons) involved in route bundling.

the number of obstacles in the polygonal map, and/or

the number of sides for each obstacle.

For each combination of the above, 15 independent experiments were performed to solve Eq. 1 by using the optimization algorithm in Eq. 9, and

For each independent experiment, the maximum number of functions evaluations is set as \(10^4\) with initial solutions of route bundles \(x_o \in {\mathbf {T}}\) being initialized randomly and independently.
Convergence
In order to show the kind of tree structures obtained in the route bundling process, Figs. 6 and 7 show the obtained route bundles in polygonal domains with obstacles of 5 and 10 sides, respectively. In order to show the efficiency of the proposed method, Figs. 8 and 9 show the convergence characteristics. Note that these figures are arranged in a grid in which the horizontal axis shows the number of edges E in the bipartite graph, and the vertical axis shows the number of obstacles in the polygonal map. In these figures, for the sake of simplicity, the values of \({\mathbf P} = 25\) and \(\eta = 0.2\) are used; other values as denoted in Table 2 are discussed in subsequent sections.

Regardless of the configuration of the polygonal map and the structure of the bipartite graph, it is possible to generate tree structures representing the bundled routes which aim at minimizing the global distance metric.

The location of the anchoring points of the bundled routes is close to, but not necessarily at, the center of the origin and destination pairs of the bipartite graph.

The route between the anchoring points of the bundled routes is not necessarily a straight line, and, regardless of increasing the number of edges in the bipartite graph, the routes between the anchoring points and the topologies of route bundles are structurally similar, but not equivalent. This is due to the fact of having edges with close origin–destination pairs.

instead of using arbitrary initial solutions in the optimization algorithm, it may be possible to compute the initial solutions of x which are close to the center/centroid of the origin and destination pairs, and

it may be possible to use precomputed routes between the anchoring points as initial solutions whenever the number of edges is expected to increase, since these routes are expected to be structurally similar.

Regardless of the configuration of the polygonal maps and the structure of bipartite graphs, it is possible to converge to the bundled routes minimizing a global distance metric within 1000 function evaluations and 15 independent runs.

Increasing the number of edges has a natural effect on increasing the distance metric by some small factor smaller than 1. This observation is in line with our above insights on the structural similarity of route bundles when increasing the number of edges.

The convergence behavior of each simulation is different due to the heuristic nature of solution sampling in Differential Evolution, and the independent arbitrary initialization at each independent run. Note that it is imperative to use different arbitrary initializations in order to evaluate our approach exhaustively under diverse initialization conditions.
Furthermore, in order to show the convergence at finer scale under different number of obstacles, routes, complexity of map, population size and neighborhood ratio, Figs. 10 and 11 show the required number of evaluations to achieve convergence. In these figures, the achievement of convergence is computed from the comparison of (1) the average difference of the cost function within 5 units of the convergence time series, to (2) the tolerance for convergence, which is a userdefined value rather than an optimization variable since it depends on the granularity of the polygonal map (maps requiring higher granularity imply finer and lower values of convergence tolerance).
In Figs. 10 and 11, for each number of routes and number of obstacles in the map, the heatmaps represent the number of evaluations required to achieve convergence for route bundling. Here, for each heatmap, darker colors imply large number of evaluations (max. number of evaluations is provided in the right side of each heatmap). Also, the xaxis of each heatmap represents the population size \({\mathbf P}  = \{25, 50, 100, 200\}\) in Differential Evolution, whereas the yaxis of each heatmap denotes the neighborhood scaling factor \(\eta \in \{0.1, 0.2, 0.4, 0.8\}\). Heatmaps are arranged in a 2dimensional grid, in which the horizontal axis of the grid denotes the number of edges \(E \in \{5, 10, 15, 20, 25\}\) in the input bipartite network, and the vertical axis of the grid denotes the number of obstacles in the map \(\in \{1, 2, 3, 4, 5\}\). By this arrangement, heatmaps located at bottom/left grid imply simple route bundling scenarios, while heatmaps located at the top/right imply complex scenarios.
Then, by looking the results of Figs. 10 and 11, we observe that our proposed approach achieves faster convergence when using smaller populations (\({\mathbf P}  = 25\)) in most of the cases. We believe this is due to the fact of using a convex representation and a ring topology in Differential Evolution: whereas the convex representation helps sampling and evaluating unique solutions in the search space, the ring topology helps exploring the search space in areas close to the sampled solution. Thus, large populations or large neighborhood size has a detriment effect in widening the sampling space, which implies increasing the computational budget, and thus the required number of function evaluations to achieve convergence.
Population and neighborhood size
Based on the above observations, the reader may wonder: what are good values of population size \({\mathbf P} \) and neighborhood scaling factor \(\eta\)? In order to answer this question, Fig. 12 shows the histogram of (\({\mathbf P} \), \(\eta\)) for the fastest converged values. In this figure, the xaxis of the histogram denotes the population size \({\mathbf P} \) and the yaxis of the histogram denotes the neighborhood ratio \(\eta\). The histograms are based on the number of times in which the tuple (\({\mathbf P} \), \(\eta\)) achieved the smallest number of evaluations to achieve convergence. By observing Fig. 12, and in line of the above observations, for any value of evaluated tolerance for convergence in \(\{1, 10^{1}, 10^{2}, 10^{3}\}\) , smaller populations are always beneficial. In regard to the neighborhood size, for convergence tolerances being \(10^{1}\) or \(10^{2}\), the neighborhood scaling factor \(\eta = 0.2\) is always beneficial, whereas for convergence tolerances being \(10^{3}\), the neighborhood scaling factor \(\eta = 0.8\) is beneficial. These observations occur due to the fact of smaller tolerances implying the need to explore the search space at finer scale, thus higher neighborhood scaling factor \(\eta\) enables the effective sampling of the search space without the need to increase population size (which would induce in unwanted space memory overhead). These results show that our proposed approach performs the heuristic search efficiently by using small populations.
Algorithmic variants

DENC: Differential Evolution with Neighborhood and Convex Encoding. In this scheme, we use a neighborhood with \(\eta = 0.2\), based on our above observations, and the Convex Encoding denoted by the 6dimensional tuple in Eq. 5 (described by “Representation of bundled routes” section).

DEN: Differential Evolution with Neighborhood Only. In this scheme, we use a neighborhood with \(\eta = 0.2\), based on our above observations, and the encoding denoted by the 4dimensional tuple in Eq. 2, which is a simple representation yet computationally more expensive due to the fact of requiring checks of point inside polygon per every sampled solution to ensure feasible route bundles (coordinates are to be outside of obstacles). In this scenario, the cost function is computed as follows:$$\begin{aligned} F(x)= & {} {\left\{ \begin{array}{ll} G(x) &{}\, P, Q \in {\mathrm{free\,space}}\\ \infty &{} {\mathrm{otherwise}} \end{array}\right. } \end{aligned}$$(17)where d(a, b) is the Euclidean obstaclefree shortest distance metric between points a and b, \(e_o\) is the Cartesian coordinate of the origin node of the edge \(e \in E\), \(e_d\) is the Cartesian coordinate of the destination node of the edge \(e \in E\), and P and Q are the Cartesian coordinates of anchoring points being closer to the origin \(e_o\) and destination \(e_d\), respectively. The condition \(P, Q \in\) free space satisfies that both P and Q are outside of the polygonal obstacles. Note that in this scheme, the Delaunay triangulation is not a requirement since all points are in \(\mathbb {R}^2\).$$\begin{aligned} G(x)= & {} \sum _{e \in E}d(e_o,P) + d(P,Q) + \sum _{e \in E}d(Q,e_d) \end{aligned}$$(18)

DEC, Differential Evolution with Convex Encoding Only. In this scheme, we use the Convex Encoding denoted by the 6dimensional tuple in Eq. 5 (described by “Representation of bundled routes” section); yet, we avoid using neighborhood concepts, and thus the mutant vector \(v_t\) (denoted by Eq. 13) is computed without the local interpolation vector, as follows:In the above, we keep the weight to be multiplying the global interpolation vector in order to enable individual selfadaptation.$$\begin{aligned} v_t = w^{x_t}.g_t \end{aligned}$$(19)

DE: Differential Evolution without Neighborhood nor Convex Encoding. In this scheme, we avoid using neighborhood concepts as well as the Convex Encoding. Therefore, the mutant vector \(v_t\) is computed by Eq. 19, the encoding is depicted by the 4dimensional tuple in Eq. 2, and the cost function is computed by Eq. 18.
In the above described variants, we used the population size as \({\mathbf P}  = 25\), based on our above observations describing the superiority of smaller populations. All other parameters in Differential Evolution are kept constant as described by Table 2.

In all cases, DENC and DEC have better solutions during the initialization phase (when the number of evaluations is up to \({\mathbf P}  = 25\)) and the first number of evaluations (up to 200 in all cases). This occurs due to that fact of DENC and DEC using the convex representation which ensures sampling feasible points, always. Conversely, DE and DEN require additional number of evaluations (and checks of point inside polygon) in order to sample and evaluate feasible solutions.

In all cases, either DENC or DEN has better convergence performance compared to DE. And in 26 out of 50 cases, DE has issues in stagnation. These observations imply that Differential Evolution using the interpolation vectors in the global and local neighborhood alone, or embedded with the convex representation, is useful not only to allow faster convergence, but also to allow escaping from stagnation. This occurs due to the fact of generating feasible solutions (by the convex representation), and due to the fact of selfbalancing between directions toward the best in the population, and directions toward the best in the local neighborhood.

Furthermore, in 10 out of 50 cases, DEC has issues of stagnation, which is in line with the above insights. Since DEC uses no information of the local neighborhood, sampled solutions will stagnate in directions close to the global best. Then, without any explorative factor, DEC is likely to stagnate. Thus, as the above observations indicate, the use of neighborhood enables to add an explorative factor to avoid stagnation.

Delaunay triangulation in 3D takes in the worst case \(O(n^2)\), and in the expected case can be even O(n).

Differential Evolution can sample in an 8dimensional tuple (for a convex representation) or 6dimensional tuple (for a nonconvex representation).

Path planning in 3D is possible by either geometric or point cloud approaches.

Obstacle geometry have minor changes over small time intervals, thus, to obtain the next optimal bundle, instead of using arbitrary initialization over the entire search space, it is possible to initialize candidate solutions with small perturbations to the current converged solutions and run Differential Evolution over a small number of evaluations. In this way, it becomes possible to quickly update the topology of the route bundles when changes in obstacle geometry are expected occur.

The same strategy can be used whenever the locations of origin and destination in the bipartite graph are expected to occur.

In order to realize a fast response time, it is possible to use multicore computing since Differential Evolution supports parallelization inherently.
Conclusions
In this paper, we have proposed a method for searching optimal route bundles based on a selfadaptive class of Differential Evolution and a convex representation. The basic idea of our approach is to sample over a triangulated search space by using selfadaptive interpolation vectors. And the unique point of our proposed method is the possibility to balance exploration and exploitation while sampling arbitrary points in a convex search space of route bundles. Then, it becomes possible the rendering of feasible route bundles efficiently since: (1) absence of overlaps with obstacles is guaranteed, and (2) computation of point inside polygon is explicitly avoided. Computational experiments involving more than 12,000 route bundling cases and 11,250,000,000 evaluations of path planning in a diverse class of polygonal domains show that (1) it is possible to obtain bundled routes with an optimized global distance metric via a reasonable number of sample evaluations, (2) the convergence towards the optimal solutions is possible over independent runs, (3) smaller populations are always beneficial, and (4) the interpolation vectors in the global and local neighborhood and the convex representation are useful not only to allow faster convergence, but also to allow escaping from stagnation.
In our future work, we aim at using polygonal environments reminiscent of outdoor configurations in which the number of edges of the bipartite network is allowed to increase. Also, in our future endeavors, we aim at exploring the generalization ability in dynamic environments, where both the input bipartite graph and the polygonal obstacles are allowed to change. We believe our approach opens new frontiers to further develop compounded and global pathplanning algorithms via gradientfree optimization algorithms and convex representations of the search space.
considering population size, number of independent runs, number of max. evaluations for all experimental conditions.
Declarations
Authors' contributions
VP contributed to the ideas, the programming work, and the writing of the paper. SM and TM help to revise the manuscript. All authors read and approved the final manuscript.
Authors' information
Victor Parque is Assistant Professor at the Department of Modern Mechanical Engineering, Waseda University, as well as Associate Professor at EgyptJapan University of Science and Technology. He obtained the MBA degree from Esan University in 2009 and obtained the Doctoral degree from Waseda University in 2011. He was a Postdoctoral Fellow at Toyota Technological Institute from 2012 to 2014. His research interests include Learning and Intelligent Systems and its applications to Design Engineering and Control. Satoshi Miura is Research Associate at the Department of Modern Mechanical Engineering, Waseda University. He obtained the Master Degree in 2013, and the Doctoral degree in 2016 from Waseda University. His research interests span the intuitive operability in hand–eye coordination of master–slave robot using brain activity measurement, development of intuitive interface for locomotion robot, driving assist technology of automobile, sports training by realtime visual feedback, typing support device and development of interface for automatic driving of car. Tomoyuki Miyashita is Professor at the Department of Modern Mechanical Engineering, Waseda University. After joining Nippon Steel Co. Ltd. in 1992, he obtained the doctoral degree from Waseda University in 2000. He was Research Associate at Waseda University and Ibaraki University since 200 and 2002, respectively. He became Associate Professor and Professor at Waseda University in 2005 and 2007, respectively. His research interests are Design Process, Design Optimization, Organ Deformation.
Acknowlegements
The generous support from Waseda University to fund the publication in openaccess format is highly appreciated.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The dataset supporting the conclusions of this article is available in the GitHub repository: https://github.com/vparque/DEBundling.
Ethics approval and consent to participate
Not applicable.
Funding
Fund from Waseda University for OpenAccess Publications.
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Authors’ Affiliations
References
 Dijkstra EW. A note on two problems in connexion with graphs. Numer Math. 1959;1:269–71.MathSciNetView ArticleMATHGoogle Scholar
 Cormen T, Leiserson C, Rivest R. Introduction to algorithms. Cambridge: MIT Press; 1993.MATHGoogle Scholar
 Hart PE, Nilsson NJ, Raphael B. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans Syst Sci Cybern. 1968;4(2):100–7.View ArticleGoogle Scholar
 Kallmann M. Path planning in triangulations. In: Proceedings of the workshop on reasoning, representation, and learning in computer games, IJCAI, 2005. p 49–54Google Scholar
 Chazelle B. A Theorem on polygon cutting with applications. In: Proceedings of the 23rd IEEE symposium on foundations of computer science, 1982. p. 339–49Google Scholar
 Lee DT, Preparata FP. Euclidean shortest paths in the presence of rectilinear barriers. Networks. 1984;14(3):393–410.MathSciNetView ArticleMATHGoogle Scholar
 Faludi R. Building Wireless Sensor Networks, O’Reilly Media. 2014Google Scholar
 Wightman P, Labardor M. A family of simple distributed minimum connected dominating setbased topology construction algorithms. J Netw Comput Appl. 2011;34:1997–2010.View ArticleGoogle Scholar
 Torkestani JA. An energyefficient topology construction algorithm for wireless sensor networks. Comput Netw. 2013;57:1714–25.View ArticleGoogle Scholar
 Panigrahi N, Khilar PM. An evolutionary based topological optimization strategy for consensus based clock synchronization protocols in wireless sensor network. Swarm Evol Comput. 2015;22:66–85.View ArticleGoogle Scholar
 Szurley J, Bertrand A, Moonen M. Distributed adaptive nodespecific signal estimation in heterogeneous and mixedtopology wireless sensor networks. Signal Process. 2015;117:44–60.View ArticleGoogle Scholar
 Christian G, Nicolas H, David I, Colette J. Disconnected components detection and rooted shortestpath tree maintenance in networks. In: The 16th international symposium on stabilization. safety, and security of distributed systems, Springer LNCS. 2014; 8736(SSS14): p. 120–34.Google Scholar
 He J, Ji S, Pan Y, Cai Z. Approximation algorithms for loadbalanced virtual backbone construction in wireless sensor networks. Theor Comput Sci. 2013;507:2–16.MathSciNetView ArticleMATHGoogle Scholar
 Chu S, Wei P, Zhong X, Wang X, Zhou Y. Deployment of a connected reinforced backbone network with a limited number of backbone nodes. IEEE Trans Mob Comput. 2013;12(6):1188–200.View ArticleGoogle Scholar
 Zou Y, Chakrabarty K. A distributed coverage and connectivitycentric technique for selecting active nodes in wireless sensor networks. IEEE Trans Comput. 2005;54(8):978–91.View ArticleGoogle Scholar
 Zhou H, Liang T, Xu C, Xie J. Multiobjective coverage control strategy for energyefficient wireless sensor networks. Int J Distrib Sensor Netw. 2012;2012:1–10.Google Scholar
 Park YK, Lee MG, Jung KK, Yoo JJ, Lee SH, Kim HS. Optimum sensor nodes deployment using fuzzy cmeans algorithm. In: International symposium on computer science and society (ISCCS), 2011. p. 38992Google Scholar
 Cui W, Zhou H, Qu H, Wong PC, Li X. Geometrybased edge clustering for graph visualization. IEEE Trans Vis Comput Graph. 2008;14:1277–84.View ArticleGoogle Scholar
 Selassie D, Heller B, Heer J. Divided edge bundling for directional network data. IEEE Trans Vis Comput Graph. 2011;17(12):2354–63.View ArticleGoogle Scholar
 Holten D, van Wijk JJ. Forcedirected edge bundling for graph visualization. In: IEEEVGTC symposium on visualization: eurographics; 2009.Google Scholar
 Ersoy O, Hurther C, Paulovich F, Cabtareiro G, Telea A. Skeletonbased edge bundling for graph visualization. IEEE Trans Vis Comput Graph. 2011;17(12):2364–73.View ArticleGoogle Scholar
 Gansner ER, Hu Y, North S, Scheidegger C. Multilevel agglomerative edge bundling for visualizing large graphs. In: Pacific visualization symposium, 2011. p. 187–94Google Scholar
 Chew LP. Constrained Delaunay triangulations In: Proceedings of the annual symposium on computational geometry. ACM; 1987. p. 215–22Google Scholar
 Osada R, Funkhouser T, Chazelle B, Dobki D. Shape distributions. ACM Trans Graph Eurographics. 2002;21(4):807–32.MathSciNetView ArticleMATHGoogle Scholar
 Das S, Abraham A, Chakraborty UK, Konar A. Differential evolution using a neighborhoodbased mutation operator. IEEE Trans Evol Comput. 2009;13(3):526–53.View ArticleGoogle Scholar
 Storn R, Price K. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim. 1997;11:341–59.MathSciNetView ArticleMATHGoogle Scholar
 Parque V, Kobayashi M, Higashi M. Bijections for the numeric representation of labeled graphs. In: IEEE international conference on systems, man and cybernetics, 2014. p. 447–52Google Scholar
 Parque V, Miyashita T. On succinct representation of directed graphs. In: IEEE international conference on big data and smart computing, 2017. p. 199–205Google Scholar
 Parque V, Miyashita T. On ksubset sum using enumerative encoding. In: International symposium on signal processing and information technology, 2016. p. 81–6Google Scholar
 Parque V, Kobayashi M, Higashi M. Searching for machine modularity using explorit. In: IEEE international conference on systems, man and cybernetics, 2014. p. 2599–604Google Scholar
 Parque V, Kobayashi M, Higashi M. Neural computing with concurrent synchrony. In: International conference on neural information processing, 2014. p. 304–11Google Scholar
 Parque V, Kobayashi M, Higashi M. Reinforced explorit on optimizing vehicle powertrains. In: International conference on neural information processing, 2013. p. 579–86Google Scholar
 Liu S, Watterson M, Mohta K, Sun K, Bhattacharya S, Taylor CJ, Kumar V. Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3D complex environments. IEEE Robot Autom Lett. 2017;2(3):1688–95.View ArticleGoogle Scholar
 Deits R, Tedrake R. Computing large convex regions of obstaclefree space through semidefinite programming. In: Akin H, Amato N, Isler V, van der Stappen A, editors. Algorithmic foundations of robotics XI, vol 107. Cham: Springer; 2015. p. 109–24.Google Scholar
 Deits R, Tedrake R. Efficient mixedinteger planning for UAVs in cluttered environments. In: IEEE international conference on robotics and automation, 2015. p. 42–9Google Scholar