Computing Path Bundles in Bipartite Networks

Victor Parque

1,2

, Satoshi Miura

and Tomoyuki Miyashita

Dept. of Modern Mechanical Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, Japan

Dept. of Mechatronics and Robotics, Egypt-Japan University of Science and Technology, Borg Al Arab, Alexandria, Egypt

Keywords:

Path Bundling, Optimization, Bipartite Networks.

Abstract:

Path bundling, a class of path planning problem, consists of compounding multiple routes to minimize a

global distance metric. Naturally, a tree-like structure is obtained as a result wherein roots play the role of

coordinating the joint transport of information, goods, and people. In this paper we tackle the path bundling

problem in bipartite networks by using gradient-free optimization and a convex representation. Then, by using

7,500 computational experiments in diverse scenarios with and without obstacles, implying 7.5 billion shortest

path computations, show the feasibility and efﬁciency of the mesh adaptive search.

1 INTRODUCTION

Path bundling is the problem which consists on com-

pounding multiple paths and ﬁnding anchoring points

at intermediate joints in order to minimize a global

distance metric. Naturally, by using coordinate nodes,

the aim of computing path bundles is to coordinate the

transport and communication of goods, information

and people. Path bundling is mainly relevant in sce-

narios where (1) the resources for transport are scarce,

and (2) the environment is hard to navigate due to nar-

row space or limited navigability. Thus it becomes

necessary to join single paths into compounded ones

to ensure efﬁcient transport/communication. For ex-

ample consider the optimization of a the location of

coordinating nodes in a ZigBee network (and its IoT

applications), or consider the problem of building op-

timal wire harness of the electrical system of a ve-

hicle (or any complex mechanical system), or con-

sider the decentralized communication of multiagent

robotic systems over a large area (where the location

of the coordinating agents is to be optimized for efﬁ-

cient communication).

Research on path bundling has its origins in the

well-known developments of the shortest-path prob-

lem: how to search for the shortest route path over

polygonal domains? (Dijkstra, 1959; P.E. Hart, 1968;

Kallmann, 2005). Here, the main goal is to ﬁnd the

most optimal path between single origin-destination

pairs; and the widely-known algorithms are Dijsktra

(Dijkstra, 1959) and A* (P.E. Hart, 1968), and their

extensions are well-studied.

In practical domains, yet with a different scope,

the research of path bundling has attracted the atten-

tion of the following ﬁelds: optimization of sensor

and wireless networks (Falud, 2014; Wightman and

Labardor, 2011; Torkestani, 2013; Panigrahi and Khi-

lar, 2015; Szurley et al., 2015; Singh and Sharma,

2015; Parque et al., 2015), and network visualiza-

tion (Cui et al., 2008; Selassie et al., 2011; Ersoy

et al., 2011; Gansner et al., 2011; Holten and van

Wijk, 2009; R. Osada and Dobki, 2002; Parque et al.,

2014b). The closest developments to path bundling

regard the edge bundling problem in network visu-

alization. Here, the conventional works have fo-

cused on the geometry-based clustering of edges(Cui

et al., 2008; Parque et al., 2014b), the force-based

edge bundling where edges are able to attract to each

other(Cui et al., 2008; Selassie et al., 2011), the clus-

tering and attraction to the skeleton of adjacent edges

(Ersoy et al., 2011), and the kd-tree based optimiza-

tion of the centroid points of close edges(Gansner

et al., 2011). The above solutions for route bundling

render compounded networks which are aesthetically

pleasing, topologically compact and locally optimal.

However, the study of route bundling under global op-

timization, in the sense of minimizing a global dis-

tance metric, has been elusive.

In this paper, in order to ﬁll the above gap, we fur-

ther advance our previous work(Parque et al., 2017)

by designing globally optimal path bundles through

sample-based global optimization algorithms over a

convex representation of polygonal domains and bi-

partite networks. The unique point of our approach

422

Parque, V., Miura, S. and Miyashita, T.

Computing Path Bundles in Bipartite Networks.

DOI: 10.5220/0006480604220427

In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 422-427

ISBN: 978-989-758-265-3

Figure 1: Basic idea of path bundling. Given a polygo-

nal map and edges representing desirable origin-destination

pairs, the goal is to ﬁnd optimal anchoring points minimiz-

ing the global distance metric of the bundled path.

is to explicitly avoid the computations of point in-

side polygons while sampling for globally optimal

path bundles in both convex and non-convex polyg-

onal domains. Computational experiments in scenar-

ios with a relevant set of polygonal domains and rel-

evant global optimization algorithms show the feasi-

bility and efﬁciency of our approach.

2 COMPUTING PATH BUNDLES

2.1 Basic Idea

The basic concept of path bundling is depicted

by Fig. 1 wherein the output is a tree structure

with compounded paths avoiding obstacle collision,

wherein roots of the tree denote coordinating points

to their leaves. A-priori knowledge of the follow-

ing is necessary: (1) a bipartite graph G = (V, E)

wherein every edge e ∈E represents the communica-

tion/transportation needs between origin-destination

pairs, and (2) obstacle geometry which denote unfea-

sible areas for navigation/transportation.

2.2 Representation of Bundled Paths

In our approach, any feasible point is represented by

the 3-element tuple:

P = (i, r

, r

) (1)

where i ∈ [n] and r

, r

∈ [0, 1]. In the above en-

coding, i is the index of i-th triangle t

∈T of the trian-

gulation of the free-space, and r

, r

are real numbers

in the interval [0, 1]. The unique feature of the above

convex representation lies in the ability to encode ar-

bitrary points which guarantee to be inside the navi-

gable space by using the tuple (i, r

, r

) for i ∈ [n], for

P ∈N

[n]

×R

[0,1]

×R

[0,1]

. Furthermore, the equivalent

2-dimensional cartesian coordinates can be computed

as follows (R. Osada and Dobki, 2002):

, P

) = (1 −r

√

(1 −r

√

(2)

where A

, B

, C

are the 2-dimensional coordinates

of the vertices of the i-th triangle t

∈ T . Then, by

using the encoding in Eq. 1, the bundled path can be

represented by the 6-element tuple:

x = (i

, r

, i

, r

) (3)

where i

, i

are natural numbers in the interval [n],

and r

, r

∈ [0, 1]. For simplicity and without

loss of generality, we denote the search space x ∈T:

T = N

[n]

×R

[0,1]

×R

[0,1]

×N

[n]

×R

[0,1]

×R

[0,1]

(4)

2.3 Optimization Problem

We solve the following equation:

Minimize

F(x)

subject to x ∈ T

(5)

where, x is the encoding (representation) of the

bundled path, F(x) is the global distance metric to

evaluate the quality of the bundled paths, and T is the

search space of feasible bundled paths. The main ra-

tionale of the above is as follows: once the search

space x ∈ T is constructed by the procedures de-

scribed in the previous subsection, our goal is to ﬁnd

anchoring points P and Q which minimize a distance

metric. For simplicity, we use the following function:

F(x) =

∑

e∈E

d(e

, P) + d(P, Q) +

∑

e∈E

d(Q, e

) (6)

where, d(a, b) is the Euclidean obstacle-free

shortest distance metric between points a and b, e

) is the coordinate of the origin (destination) node

of the edge e ∈ E, and P and Q are anchoring points

being closer to the origin e

and destination e

, re-

spectively. The 2-dimensional coordinates of P and Q

can be computed by combining Eq. (1)-(3). Solving

Eq. 5 is realized by:

• DE: Differential Evolution with Successful Parent

Selection/Best1(S-M Guo, 2015).

• NPSO: Particle Swarm Optimization with Nich-

ing Properties(B. Y. Qu and Suganthan, 2012).

• RBDE, Real-Binary Differential Evolution

(RBDE)(Sutton et al., 2007).

• SHADE, Success History Parameter Adaptation

for Differential Evolution(R. Tanabe, 2013).

Computing Path Bundles in Bipartite Networks

423

• DIRECT, Direct Global Optimization Algo-

rithm(Jones, 1999).

The main reason/motivation of using the above

algorithmic set is to rigorously tackle path bundling

problem by using a representative class of gradient-

free optimization algorithms. The above algorithms

are relevant in the literature due to the fact of con-

sidering multimodality, parameter adaptation, search

memory, selection pressure, search over neighbour-

hood concepts, and mesh partitioning. Parameters for

each algorithm are default and described in the refer-

ences. Fine tuning the respective parameters is out of

the scope of the paper.

3 COMPUTATIONAL

EXPERIMENTS

In order to evaluate the performance of our approach,

we used diverse polygonal domains with convex and

non-convex obstacles, as well as different conﬁgu-

rations of bipartite networks. This section describes

our experimental conditions, results and insights ob-

tained.

3.1 Experimental Settings

The computing environment used is Intel i7-4930K @

3.4GHz with Windows 8.1, and computational exper-

iments were performed using Matlab 2016a. In order

to enable a meaningful evaluation of our proposed ap-

proach, we consider the following environmental set-

tings:

• No. of edges in the input bipartite graph,

|E| = {5, 10, 15, 20, 25},

• Number of Polygonal Obstacles: {1, 2, 3, 4, 5},

• No. Sides in each Polygonal Obstacle: {5, 10}.

• For each combination of the above, 30 indepen-

dent experiments were performed to solve Eq. 5,

• For each independent experiment, the maximum

number of functions evaluations is set as 10

• In each independent experiment, the initial solu-

tions of route bundles x

∈ T are initialized ran-

domly and independently.

In order to show the kind of environments and

bipartite networks used in our experiments, Fig. 2

shows the rendering of the polygonal domains and bi-

partite networks with obstacles, each of which has (a)

5 sides and (b) 10 sides. In this ﬁgure, we show a

matrix-like conﬁguration, where the x-axis denote the

number of edges in the bipartite network, and the y-

axis denote the number of obstacles in the environ-

ment. The conﬁguration of edges in the bipartite net-

work (origin and destination pairs) are arbitrarily gen-

erated to allow exhaustive evaluation of the optimiza-

tion algorithms.

Furthermore, the main reason of using values of

the number of edges |E| up to 25 is due to our inter-

est in evaluating the performance close to the number

of transport needs in indoor environments, where the

complexity of the environment is controlled by

• the number of obstacles in the polygonal map, and

• the number of sides for each obstacle.

In the above, complex polygonal environments in-

duce in large number of triangles, thus representing

a challenging search space for any search algorithm.

Our future work aims at using conﬁgurations consid-

ering large scenarios and being close to outdoor envi-

ronments.

The use of 30 independent runs in each exper-

imental setting allows to evaluate the gradient-free

optimization algorithms under arbitrary initialization

conditions, thus avoiding random luckiness. Also, the

key rationale of using 10

function evaluations as up-

per bound of computational budget is due to our in-

terest of evaluating the effectiveness and efﬁciency

of the heuristics under restrictive computational re-

sources. Note that the use of function evaluations as

a surrogate metric for efﬁciency is relevant to avoid

bias in hardware or algorithmic implementation.

In line of the above, as a result, 7500 experimental

conditions were evaluated

, and 7.5 ×10

functions

evaluations were computed

, assuming a single opti-

mization using population size |¶| = 100.

3.2 Results and Discussion

In order to show the kind of tree structures obtained,

as well as to evaluate the efﬁciency in path bundling,

Fig. 3 shows the optimized path bundles; and Fig. 4

- 8 show the convergence behaviour. Note that our

results follows the same organization of Fig. 2, that

is x-axis show the number of edges in the bipartite

network, while y-axis show the number of obstacles

in the environment.

In regards to the obtained path bundles, by observ-

ing Fig. 3 we can conﬁrm the following facts:

• Regardless of polygonal domain, location of

origin-destination pairs in the bipartite network,

and evaluated optimization algorithm, it is pos-

sible to generate tree structures representing the

5 ×5 ×5 ×2 ×30

5 ×1500 ×10

×|¶|

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

424

Figure 2: Bipartite networks in a polygonal domain with

obstacles of (a) 5 sides and (b) 10 sides.

bundled paths which aim at minimizing the global

distance metric.

• The location of the anchoring points of the bun-

dled paths are close to, but not necessarily at, the

center of the origin and destination pairs of the bi-

partite graph.

Regarding convergence behaviour, we conﬁrmed

the following facts (representative examples in Fig. 4

and Fig. 5):

• Regardless of polygonal domain, location of

origin-destination pairs in the bipartite network,

and evaluated optimization algorithm, it is possi-

ble to converge to the bundled paths minimizing

the global distance metric within 2000 function

evaluations.

• DIRECT is the most efﬁcient algorithm achiev-

ing convergence to the path bundles with mini-

Figure 3: Path Bundles in a polygonal domain with obsta-

cles of (a) 5 sides and (b) 10 sides.

mum global distance within 100 function evalu-

ations, in 90% of the experimental cases. Out of

50 experimental cases, there exists 5 experimen-

tal cases wherein DIRECT achieves convergence

in more than 2000 function evaluations.

• Over independent runs, all population-based algo-

rithms show variance in the rate of convergence.

This result is due to the fact of randomness in the

initialization process and the sampling behaviour

of the algorithms, whereas DIRECT is a deter-

ministic algorithm using the DIviding RECTan-

gles concept, which samples solutions vectors at

the center of hypercubes, and then subdivides po-

tentially optimal hypercubes recursively.

• Convergence to local optima in observed in all

population-based algorithms, except RBDE. We

believe this result is due to RBDE uses a select-

Computing Path Bundles in Bipartite Networks

425

Figure 4: Convergence in polygonal domain with obstacles having 5 sides and a bipartite network with 25 edges.

Figure 5: Convergence in polygonal domain with obstacles having 10 sides and a bipartite network with 25 edges.

ing mechanism which is more greedy compared

to other population-based heuristics. Thus, due

to the single-optima nature of the path bundling

problem, RBDE focuses more on exploitation,

rather than exploration of the search space.

• Increasing the number of edges in the bipartite

network has a direct effect on increasing the dis-

tance metric by some small factor smaller than 1.

We believe the above observations has important

implications to design effective algorithms that solve

the path bundling problem effectively and efﬁciently.

In line of the above, we provide the following propo-

sitions:

• Instead of using arbitrary initial solutions in the

optimization algorithm, it may be possible to

compute the initial solutions of x of path bundles

which are close to the center/centroid of the origin

and destination pairs of the bipartite networks,

• Whenever the number of edges in the bipartite

network is expected to change (as a result of

increasing/decreasing the number of agents and

both origin-destination pairs), it may be possible

to use pre-computed paths as initial solutions x of

path bundles, since the new paths are expected to

be structurally similar and close distance metric.

• Instead of sampling vectors close to potential so-

lution vectors, it may be possible to sample at

equally and locally distributed partition of the

search space. Furthermore, a convex search space

(as the one proposed in this paper), may be key for

effective and efﬁcient performance of partitioning

the search space.

The above results imply the feasibility and efﬁ-

ciency to obtain optimal path bundles in polygonal

maps with both convex and non-convex obstacles.

Further work remains on the agenda. Key limita-

tions of our approach lie in our environments and net-

works: generalization to dynamic environments, non-

bipartite networks, and networks having very large

number of nodes is still unclear. Further computa-

tional experiments using large number of edges and

diverse obstacle conﬁgurations reminiscent of out-

door environments are in our agenda.

4 CONCLUSION

In this paper, we have proposed an approach for de-

signing optimal path bundles based on the idea of

sampling over a convex search space to optimize a

global distance metric. The unique point of our pro-

posed approach is to compute feasible path bundles

efﬁciently since the convex search space ensures the

avoidance of overlapping (computation of point in-

side polygon is explicitly avoided) while sampling for

optimal solutions.

Exhaustive computational experiments using a di-

verse and representative class of polygonal domains,

bipartite networks and gradient-free optimization al-

gorithms, show that (1) it is possible to obtain bundled

paths with an optimized global distance metric via a

reasonable number of sample evaluations (100 in the

best case), and (2) the convergence is most efﬁcient

with the DIviding RECTangles concept.

We provided relevant insights to develop gradient-

free algorithms for the bundling problem which

regard (1) the use of initialization close to cen-

ter/centroid of origin/destination pairs in the bipar-

tite networks, (2) the use of pre-computed paths to

approximate optimal bundles whenever the bipartite

network varies, and the use of partitioning combined

with a convex representation of the search space.

In future work, we aim at exploiting our insights in

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

426

polygonal environments reminiscent of outdoor con-

ﬁgurations. Also, we aim at exploring the generaliza-

tion ability in dynamic and unknown environments.

Furthermore, we aim at extending our approach to

tackle the bundling of networks with different topolo-

gies, e.g. it may be possible to use the DIviding

RECTangles concept with a number-based represen-

tation of undirected networks (Parque et al., 2014a)

and directed networks (Parque and Miyashita, 2017),

where the partition is realized in number-space (rather

than a high-dimensional matrix-space).

We believe our approach opens new possibili-

ties to develop compounded and global path planning

algorithms via gradient-free sampled-based learning

and convex representations of the search space.

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