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Path Planning in a Weighted Planar Subdivision Under the Manhattan Metric

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In this paper, we consider the problem of path planning in a weighted polygonal planar subdivision. Each polygon has an associated positive weight which shows the cost of path per unit distance of movement in that polygon. The goal is finding a minimum cost path under the Manhattan metric for two given start and destination points. We propose an O(n^2) time and space algorithm to solve this problem, where n is the total number of vertices in the subdivision. We also study the case of rectilinear regions in three dimensions, and generalize the proposed algorithm to find a minimum cost path under the Manhattan metric in O(n^3 log n) time and O(n^3) space.
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CCCG 2020, Saskatoon, Canada, August 5–7, 2020
Path Planning in a Weighted Planar Subdivision
Under the Manhattan Metric
Mansoor Davoodi Hosein Enamzadeh Ashkan Safari
Abstract
In this paper, we consider the problem of path planning
in a weighted polygonal planar subdivision. Each poly-
gon has an associated positive weight which shows the
cost of path per unit distance of movement in that poly-
gon. The goal is finding a minimum cost path under the
Manhattan metric for two given start and destination
points. We propose an O(n2) time and space algorithm
to solve this problem, where nis the total number of
vertices in the subdivision. We also study the case of
rectilinear regions in three dimensions, and generalize
the proposed algorithm to find a minimum cost path
under the Manhattan metric in O(n3log n) time and
O(n3) space.
1 Introduction
Path planning (PP) problem is one of the fundamen-
tal problems in motion planning whose objective is to
find an optimal path with minimum length between two
start and destination points sand tin a work space.
In the classical version of PP, the work space contains
some obstacles, and the path must avoid these obstacles
[7, 14]. However, in a general formulation of PP – called
Weighted Region Problem (WRP) – which was first in-
troduced by Mitchell and Papadimitriou [17], each ob-
stacle has an associated weight and a path is allowed to
enter them at extra costs. In fact, these weights rep-
resent the cost per unit distance of movement in the
obstacles (or say weighted regions). This generalization
of PP has a lot of applications, e.g., it can be used in
self-driving cars navigation, robot motion planning [6],
military purposes [16], crowd simulation [13], and gam-
ing applications [13]. An important theoretical result
on WRP [9] has shown that this problem cannot be
solved in the algebraic computation model over the ra-
tional numbers under the Euclidean metric. Motivated
by this result, we investigate WRP under the Manhat-
tan metric and show that it can be solved efficiently in
polynomial time.
Department of Computer Science and Information Technol-
ogy, Institute for Advanced Studies in Basic Sciences (IASBS),
Zanjan, Iran
mdmonfared@iasbs.ac.ir
hosein.enamzadeh@iasbs.ac.ir
ashkan.safari@iasbs.ac.ir
Mitchell and Papadimitriou [17] introduced an -
optimal algorithm with running time of O(n8L), where
nis the total number of vertices of polygonal regions
and Lis the precision of problem’s instance. Precisely,
L=O(log(nNW/w)), where Nis the maximum inte-
ger coordinate of any vertex of the subdivision, Wand
ware the maximum non-infinite and minimum non-zero
integer weights assigned to the faces of the subdivision,
and  > 0 is a user-specified error tolerance. The output
is the shortest path from the starting point sto all ver-
tices of the polygons with an error tolerance under the
Euclidean metric. Mata and Mitchell [16] have proposed
an algorithm based on constructing a relatively sparse
graph – called pathnet – that can search for paths that
are close to optimal. They have proved that a path-
net of size O(nk) can be constructed in O(kn3) time.
As a matter of fact, the pathnet limits the paths that
can extend from vertices with kcones at each vertex.
Searching for a path on the constructed pathnet yields
a path whose weighted length is at most (1 + ) of op-
timal path. Precisely, =W/w
kΘmin , where W/w is the
ratio of the maximum non-infinite weight to the mini-
mum non-zero weight, and θmin is the minimum internal
face angle of the subdivision. One of the common tech-
niques for obtaining approximate shortest paths is to
positioning Steiner points for discretizing the edges of
the triangular regions and then constructing a graph by
connecting them. Finally, by using graph search algo-
rithms such as Dijkstra, an approximate minimum cost
path can be computed [1, 2, 18].
There are several variants of WRP due to the metric
and the shape of weighted regions. Lee et al. [15] have
solved the problem in the presence of isothetic obsta-
cles (the boundary edges of obstacles are either vertical
or horizontal line segments). They have presented two
algorithms for finding the shortest path under the Man-
hattan metric. The first algorithm runs in O(nlog2n)
time and O(nlog n) space, and the second one runs in
O(nlog3/2n) time and space. Gewali et al. [10] have
considered a special case of this problem in which there
are only three types of regions: regions with weight of
, regions with weight of 0, and regions with weight of
1. They have presented an algorithm in O(m+nlog n)
time, where mO(n2) is the number of visibility edges.
Furthermore, they have presented an algorithm for the
case that linear feathers are added. Precisely, edges of
32nd Canadian Conference on Computational Geometry, 2020
the subdivision are allowed to have arbitrary weights.
Their algorithm for this case takes O(n2) time for con-
structing a graph of size O(n2) for searching the short-
est path. In fact, it takes O(n2log n) time for finding
the shortest path. Gheibi et al. [11] have discussed the
problem in an arrangement of lines. Due to the fact that
this special case of the problem has unbounded regions,
they have presented a minimal region – called SP-Hull
– to bound the regions. This minimal region contains
the minimum cost path from sto t. They construct
SP-Hull in O(nlog n) time, where nis the number of
lines in the arrangement. After constructing SP-Hull,
an approximate minimum cost path can be obtained by
applying the existing approximation algorithms within
bounded regions. Jaklin et al. [13] have analyzed the
problem when the weighted regions are cells of a grid.
They have also presented a new hybrid method – called
vertex-based pruning which is able to compute paths
that are -optimal inside a pruned subset of the scene.
In this paper, we consider a planar subdivision with
arbitrary positive weights. We present an algorithm
which constructs a planar graph in O(n2) time with
O(n2) vertices and edges, where nis the total number of
vertices of the subdivision. The constructed graph con-
tains the minimum cost path between two points sand
tin the plane, where the distances are measured under
the weighted Manhattan metric – the length of a path is
the weighted sum of Manhattan lengths of the sub-paths
within each region. It has been shown that this prob-
lem is unsolvable over the rational numbers when the
distances are measured under the weighted Euclidean
metric [9]. To the best of our knowledge, this is the
first result that presents an exact algorithm for solv-
ing WRP under the Manhattan metric in a case where
the regions are arbitrary simple polygons with positive
weights. We propose an exact algorithm for finding the
minimum cost path under the weighted Manhattan met-
ric in O(n2) time which is also a 2approximation for
the Euclidean metric. Also, we show that the proposed
algorithm can be used for WRP with rectilinear subdivi-
sion in three dimensions in O(n3log n) time and O(n3)
space.
This paper is organized in five sections. In section 2,
we give some preliminaries and definitions. In section 3,
we present our algorithm for constructing a graph which
contains the minimum cost path in a two dimensional
work space, and prove that the shortest path is within
the constructed graph. In section 4, we generalize the
algorithm for the case of rectilinear regions in three di-
mensions, and in section 5, we draw a conclusion.
2 Preliminaries and Definitions
The problem of weighted region path planning, WRP,
considered in this paper is defined as follows: let Sbe
s
t
b1
b2
b3
b4
b5
Figure 1: A path from sto twith seven breakpoints.
a subdivision of the plane into polygonal regions with
nvertices, and s, t ∈ S be two start and destination
points in the plane. Each region of Shas an associated
positive weight. The weight of an edge e∈ S (boundary
of regions) is assumed to be min{wr, wr0}, where wrand
wr0are the weights of regions incident to e. The goal is
to find a minimum cost path between sand t, where the
distances are measured under the weighted Manhattan
metric – the length of a path is the weighted sum of
Manhattan lengths of the sub-paths within each region.
Let πst denote a path between sand twhich consists
of some sub-paths between consecutive breakpoints. A
breakpoint is a point on the path in which the path
turns. We also consider sand tas breakpoints (see
Fig. 1). Let ρ1, ρ2, ..., ρkbe sub-paths between consec-
utive breakpoints of a path πst in which each ρi, for
i= 1,2,...,klies completely within one region. If a
part of a path πst does not lie totally in one of the re-
gions, we decompose it to some sub-paths. We denote
d(ρi) as the Manhattan distance between two endpoints
of ρi. The weighted length of a path πst under the Man-
hattan metric, denoted by dw(πst), is defined as:
dw(πst) = Pk
i=1 d(ρi)×wi,
where wiis the weight of the region in which ρilies.
A path πst is called a horizontal (resp., vertical) path
if it consists of a horizontal (resp., vertical) sub-path
between only two consecutive breakpoints. Also, we say
two horizontal (resp., vertical) paths are consecutive if
and only if they have the same starting and termination
points. This definition is used in Lemma 1.
The basic idea behind the proposed algorithm is
reducing the problem to a graph searching problem.
Therefore, we provide an algorithm for constructing a
graph that contains the minimum cost path under the
weighted Manhattan metric. The constructed graph is
a planar graph with O(n2) vertices and edges, where n
is the total number of vertices of the subdivision. For
planar graphs with positive edge weights, Henzinger et
al. [12] have given a linear-time algorithm to compute
CCCG 2020, Saskatoon, Canada, August 5–7, 2020
single-source shortest paths. By running this algorithm
on the constructed graph, we obtain the minimum cost
path between sand tunder the Manhattan metric in
O(n2) time. Since a simple polygon with nvertices can
be triangulated in O(nlog n) time and O(n) space [8],
w.l.o.g. we assume all the regions to be triangular re-
gions in all parts of the paper.
3 The Graph Construction Algorithm
3.1 The Algorithm
Let G= (V, E ) be a graph. First, we initialize V
= Ø and E= Ø. Let HL(αi) and V L(αi) be hori-
zontal and vertical lines passing through point αi, for
i= 1,2,...,n. Precisely, αi, for i= 1,2,...,nare the
vertices of the subdivision which contain s,t, and the
vertices of the triangles. We add s,t, vertices of the tri-
angles, and the intersection points among HL(αi) and
V L(αj), for i, j = 1,2,...,nto V. We also add the
intersection points among HL(αi) (resp., V L(αi)), for
i= 1,2,...,nand the edges of the triangles to V. Next,
we add the line segments between two consecutive ver-
tices in Vthat lie on the considered horizontal lines,
vertical lines or the edges of the triangles as edges of G
to E. For an edge (u, v)Ewhere lies in a region with
wight wi, let d(u, v) denote the Manhattan distance be-
tween two endpoints of the edge. The weight of the edge
is equal to the product of d(u, v) and wi. Note that each
edge lies completely within one region.
The basic idea of our algorithm is to extend four rays
to the up, down, right and left directions (horizontal
and vertical lines) at every vertex of the subdivision.
This idea has similarity to vertical cell decomposition
(VCD) method [14]. In this method, the free space is
partitioned into a finite collection of one-dimensional
and two-dimensional cells by extending rays upward and
downward through free space. In this method, the rays
are not allowed to enter obstacles, however, in our al-
gorithm the rays are extended to all parts of the sub-
division since the paths are allowed to enter weighted
regions at extra costs. Also, we extend rays to the four
directions at every vertex, however, in the VCD method
the rays are extended only upward and downward. In
both methods, the motion planning problem is reduced
to a graph search problem. In VCD method, a roadmap
is constructed by selecting sample points from the cell
centroids, however, in our algorithm the graph is con-
structed by intersecting the rays with each other and
also by the edges of the triangles.
Some of the edges of Gwhich lie on an edge of a
triangle are oblique. These edges are useful when two
triangular regions are close to each other and the region
among them has a lower weight than these triangles. A
path which passes between these two triangles cannot
be completely horizontal or vertical since it will enter
s
t
Figure 2: The constructed graph of Fig. 1.
the triangles. So it will be oblique and lie on one of the
edges of the triangles (see the sub-path between b4and
b5on Fig. 1).
According to the construction of the graph, some ver-
tices and edges are added to the graph by vertical and
horizontal lines passing through vertices of the subdivi-
sion. We call the part of the work space which lies be-
tween two consecutive horizontal (resp., vertical) lines,
ahorizontal lane (resp., vertical lane) denoted by LH
(resp., LV ). So each LH (resp., LV ) is surrounded by
two consecutive horizontal (resp., vertical) lines. There-
fore, when we say the lines of an LH (resp., an LV ), we
mean these consecutive lines.
For constructing the graph, we can use one of the line
segments intersections algorithms [3, 5] which computes
all kintersections among nline segments in the plane
in O(nlog n+k) time. These intersection points are
vertices of G. After specifying the set of vertices of G,
the set of edges of Gcan be specified. It takes O(n2)
time to construct Gsince the graph has O(n2) vertices
and edges. The constructed graph of the work space
on Fig. 1 is shown on Fig. 2. For simplicity, we do
not triangulate the white regions with weight 1 in these
figures. Precisely, we can apply the proposed algorithm
in a polygonal subdivision in which the regions are not
triangular. The triangulation of the regions just helps
us for showing that Gcontains the minimum cost path
between sand t.
For computing the minimum cost path under the
Manhattan metric between sand t, we can apply Di-
jkstra’s algorithm to G. In this case, the minimum cost
path is obtained in O(n2log n) time. However, since
Gis a planar graph with positive edge weights, we can
apply the algorithm presented by Henzinger et al. [12],
which is a linear-time algorithm, to G. Therefore, the
minimum cost path is obtained in O(n2) time.
3.2 Correctness Proof
Now, we show that the constructed graph contains the
minimum cost path between sand tunder the Manhat-
32nd Canadian Conference on Computational Geometry, 2020
tan metric. Since our metric for measuring the distance
is Manhattan, we can convert any path between sand t
to a path which consists of vertical and horizontal line
segments. In other words, when a sub-path between
two consecutive breakpoints is oblique, we can replace
it by two horizontal and vertical line segments where
the cost of movement on these horizontal and vertical
line segments is equal to the cost of movement along
the oblique line segment. In a case where a sub-path
lies between two close triangular regions and the region
between these two triangular regions has lower weight
than these triangles, by applying this conversion, some
parts of the horizontal and vertical line segments may
lie in the triangular region with higher weight. In this
case, we can replace the part which lies in a triangular
region with higher cost with a line segment which lies
on an edge of the triangles (see the sub-path between
b4and b5on Fig. 1). Since the weight of each of the
edges of the work space is equal to the minimum weight
of the regions that are incident to that edge, the cost
of movement between two breakpoints on the replaced
line segments is equal to the cost of movement along the
oblique line segment. Therefore, a path between sand
tcan only consist of horizontal, vertical, and oblique
line segments, the latter of which are located on the
edges of the triangles. As a result, all the paths that we
consider in the following lemmas consist of the above
mentioned line segments. Our first objective is to prove
the following lemma.
Lemma 1 Let π1,π2, and π3be three consecutive hor-
izontal (or vertical) sub-paths from s0to t0which lie
inside an LH (resp., an LV) and pass through k > 0
triangular regions. If dw(π2)< dw(π1), then dw(π3)<
dw(π2).
Proof. We consider the case k= 2, the proof is similar
for any k > 0. For simple comparison among the sub-
paths, let the points s0and t0lie on the same horizontal
line segment. Assume w.l.o.g. that both triangles have
vertical edges (see Fig. 3). The weighted lengths of π1,
π2and π3are defined as follows (refer to Fig. 3 for the
notations):
dw(π1)=(w1×a1)+(w2×a2) + z2+x2+L,
dw(π2) = (2 ×h) + x1+ (w1×b1) + (w2×b2) + x2+L,
dw(π3) = (2 ×h) + x1+ (2 ×h0) + z1+ (w1×c1)
+ (w2×c2) + L.
According to Fig. 3, a1=b1+x1and a2=b2z2. Due
to the assumption that dw(π2)< dw(π1), we have the
following inequality:
(2 ×h)< x1×(w11) + z2×(1 w2),
and due to the triangle similarity theorems we have the
following equations:
π3
π2
π1
h
h
x1
z1c1
b1
a1
L
t′′
t
z2
x2
s
s′′
w1
c2
b2
a2
w2
HL(αi)
HL(αj)
Figure 3: Three consecutive horizontal sub-paths from
s0to t0through two triangular regions.
x1
h=z1
h0,z2
h=x2
h0.
By applying the triangle similarity equations in the
mentioned inequality and adding (w1×c1) + (w2×b2)
to both sides of the inequality we get:
(2 ×h0) + z1+ (w1×c1)+(w2×c2)<
(w1×b1)+(w2×b2) + x2=dw(π3)< dw(π2).
Thus, the weighted length of π3is less than π2. In fact,
the proof is based on the following equation:
h
h0=x1
z1
=z2
x2
,
and since h
h0is constant, we can generalize the proof for
any k > 0 triangular regions between s0and t0. There-
fore, the lemma holds.
Note that inside an LH (resp., an LV ), we can con-
sider all the triangles to have vertical (resp., horizontal)
edges since vertical (resp., horizontal) lines are consid-
ered passing through vertices of the subdivision. The
result of this lemma helps us to show that there exists
a shortest path between sand tunder the Manhattan
metric such that all the horizontal (resp., vertical) sub-
paths between consecutive breakpoints in LHs (resp.,
LVs) lie on the lines of the LHs (resp., LVs). We call
such a path, a perfect shortest path between sand t,
denoted by πp
st. Note that according to the principle
of optimality, since πp
st is optimal in length, all of its
sub-paths in LHs and LVs are also optimal in length.
Lemma 2 There exists a shortest path between sand
tunder the Manhattan metric such that, for any sub-
path of the shortest path in an LH (resp., an LV), all
the horizontal (resp., vertical) sub-paths between consec-
utive breakpoints lie on the lines of the LH (resp., LV).
CCCG 2020, Saskatoon, Canada, August 5–7, 2020
According to Lemma 2, a path between the entrance
(s0) and exit point (t0) of an LH (resp., an LV ) is not
optimal in length, unless there exists an optimal path
in length such that all the horizontal (resp., vertical)
sub-paths between consecutive breakpoints lie on the
lines of the LH (resp., LV ). Precisely, there is always
a path πp
s0t0in an LH (resp., an LV ). According to the
construction of the graph, lines of an LH (resp., an LV )
are edges of Gand a horizontal (resp., vertical) sub-path
of a path πp
s0t0between two consecutive breakpoints in
an LH (resp., an LV ) lies on the edges of G.
Corollary 3 For any path πp
s0t0in an LH (resp., an
LV), the sub-paths between consecutive breakpoints can-
not be simultaneously horizontal (resp., vertical) and lie
between two lines of the LH (resp., LV).
Lemma 4 A breakpoint of a path πp
s0t0in an LH (resp.,
an LV) is located on a line of an LH or an LV or possibly
both.
Proof. We assume that bis a breakpoint in an LH
which is not located on a line of the LH or a LV. Ac-
cording to Corollary 3, the line segment that is incident
to bcannot be horizontal. Therefore, one of the line
segments is vertical and the other one is located on an
edge of a triangle. Since bis also located in an LV and
is not located on one of the lines of the LV, the vertical
line segment incident to b lies between the left and right
lines of the LV, which contradicts Corollary 3. Thus, the
lemma holds.
Lemma 4 shows that the breakpoints of the perfect
shortest paths in LHs (resp., LVs ) must lie on the lines
of the LHs and LVs, meaning that they lie on the edges
of G(since the lines of LHs and LVs are edges of G). The
next step is to show that these breakpoints are located
on the vertices of G.
Lemma 5 For a path πp
s0t0in an LH (resp., an LV),
the breakpoints of the path are located on the vertices of
G.
Proof. According to Lemma 4, a breakpoint of a path
πp
s0t0in an LH (resp., an LV ) is located on a line of
an LH or an LV or possibly both. If a breakpoint is
located on both a line of an LV and a line of an LH, it
is on the intersection point of these two lines. Thus, it
is on a vertex of G. If it is only located on a line of an
LH or an LV, and one of the incident line segments lies
on a triangle edge, then the breakpoint is located on a
vertex of G(since the intersection of an LH or LV line
with a triangle edge is a vertex of G). Therefore, the
breakpoints of a path πp
s0t0are on the vertices of G.
Lemma 5 shows that the breakpoints of a path πp
s0t0
in an LH (resp., an LV ) are located on the vertices of
G. The next step is to show that a path πp
s0t0under the
Manhattan metric in an LH (resp., an LV ) is on G. To
this end, we need to show that the edges of the path
πp
s0t0are on the edges of G.
Lemma 6 A path πp
s0t0in an LH (resp., an LV) is on
G.
Proof. According to Lemma 5, the breakpoints of a
path πp
s0t0in an LH (resp., an LV ) are on the vertices
of G. Let ebe an edge between two consecutive break-
points. If eis on an edge of a triangle, it is on G. Now
we assume that eis in an LH and is not on G. Accord-
ing to Corollary 3, ecannot be horizontal since it must
lie on one of the lines of the LH and the lines of LH s
are edges of G. Therefore, it is a vertical edge. Since it
is also located in an LV and is not on G, it is not on
a line of the LV. Therefore, it contradicts Corollary 3.
Thus, eis on G.
According to Lemma 6, perfect shortest paths in LHs
and LVs which are sub-paths of a path πp
st are on the
constructed graph. Note that in all the lemmas, a path
between sand tonly consists of horizontal, vertical, and
oblique line segments, the latter of which are located
on the edges of the triangles. In the continuous work
space, an arbitrary path between sand tconsists of line
segments which are not in the form of the mentioned line
segments. Finally, we prove that there exists a shortest
path between sand ton G.
Theorem 7 For a shortest path π1under the weighted
Manhattan metric in the continuous work space from s
to t, there exists a path π2from sto ton Gsuch that
dw(π2)dw(π1).
Proof. It is obvious that when the metric for measuring
the distance is Manhattan, any arbitrary path in the
continuous work space, can be converted to a path which
consists of the three mentioned line segments without
increment in the cost of the path. Thus, we convert π1
to π0
1such that the line segments in π0
1are in the form
of the mentioned line segments. Obviously, dw(π0
1) =
dw(π1). According to the principle of optimality, each
sub-path of an optimal path in length is also optimal.
Therefore, π0
1consists of optimal sub-paths in length in
LHs and LVs. According to Lemma 2, for any shortest
path in an LH (resp., an LV ), there exists a path πp
s0t0
and due to the Lemma 6, perfect shortest paths in LHs
and LVs are on G. Thus, π0
1can be converted to a
perfect shortest path (π2) without increment in the cost
of the path. Therefore, a path from sto ton Gexists
(π2) whose weighted length is not greater than π1.
According to Theorem 7, Gcontains a shortest path
from sto tunder the weighted Manhattan metric. Since
simple polygons can be triangulated in O(nlog n) time
32nd Canadian Conference on Computational Geometry, 2020
and O(n) space [8], work spaces with simple polygonal
regions can be discretized by using the mentioned graph
construction algorithm. Thus, the proposed algorithm
solves WRP under the Manhattan metric.
Theorem 8 The weighted region problem in a planar
polygonal subdivision with positive weights under the
Manhattan metric can be solved in O(n2)time and
space, where n is the total number of vertices of the
subdivision.
By using the triangular inequality, it is easy to see
that the length of a path under the Manhattan metric
is at most 2 times of the length of the path under the
Euclidean metric. Thus, the proposed algorithm is also
a2-approximation algorithm for solving WRP under
the Euclidean metric.
4 The Three-Dimensional Case
In this section, we consider WRP in three dimensions.
It has been shown that the problem of finding a shortest
path under any LPmetric in a three-dimensional poly-
hedral environment is NP-hard [4]. Here, we consider a
specific variation where the regions are rectilinear.
Since the metric for measuring the distance is Man-
hattan, any oblique path between two consecutive
breakpoints in three-dimensional space can be converted
to three parallel line segments to x,yand zaxes with-
out increment in the cost of the path. Thus, we consider
all the paths to be rectilinear.
Let nbe the total number of vertices of the subdivi-
sion and let (xi,yi,zi), for i= 1,2,...,nbe the coor-
dinates of the vertices of the regions (and of sand t).
Let Pbe the set of planes x=xi,y=yi,z=zi, for
i= 1,2,...,n. The set of vertices of the graph consists
of the intersection points among at least three planes in
P, and the set of edges of the graph consists of the line
segments between two consecutive vertices of the graph
which lie on the intersection lines between at least two
planes in P. The constructed graph has O(n3) vertices
and edges, and by applying Dijkstra’s algorithm to it,
the minimum cost path under the Manhattan metric
can be obtained in O(n3log n) time.
Similar to the definitions of LH and LV in the pla-
nar case, we define similar notations for the three-
dimensional case. Let XY C denote a part of the
work space which is surrounded by two consecutive
planes orthogonal to the x-axis and two consecutive
planes orthogonal to the y-axis in Pwhich is called an
XY container. Precisely, an XY C is not surrounded
along the z-axis. XZC and Y Z C notations are defined
similarly. Since all the paths are considered to be recti-
linear, for any path in an XY C , there exists an equiva-
lent path in length such that all the sub-paths between
consecutive breakpoints along the z-axis are located on
the planes surrounding XY C. Precisely, according to
the graph construction algorithm, each XY C consists
of some cuboids where the cost of movement in every
part of a cuboid is equal. Therefore, the sub-paths along
the z-axis in a cuboid have the same cost when they are
located either on the planes surrounding XY C or in-
side the cuboid. Similar results hold for an XZC and
aY ZC . Thus, an equivalent path in length between s
and texists where all the sub-paths between consecu-
tive breakpoints are located on the considered planes in
P. Arguments similar to the ones used in Theorem 7
show that the constructed graph contains the minimum
cost path between sand tunder the Manhattan metric.
Theorem 9 The weighted region problem in a three-
dimensional work space among rectilinear regions with
positive weights under the Manhattan metric can be
solved in O(n3log n)time and O(n3)space, where n is
the total number of vertices of the subdivision.
5 Conclusion
In this paper, we have considered a generalization of
path planning problem – called weighted region prob-
lem (WRP). While unsolvability of WRP over the ra-
tional numbers under the Euclidean metric has been
proved [9], we proposed an algorithm for solving WRP
under the Manhattan metric which is also a 2-
approximation solution for the Euclidean case. We also
considered the case of rectilinear regions in three dimen-
sions, and generalized our algorithm for it. Improving
the time complexity of the algorithm and providing a
better approximation factor for the Euclidean metric
remain open.
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Appendix
Proof of Lemma 2
Lemma 2 There exists a shortest path between sand tun-
der the Manhattan metric such that, for any sub-path of the
shortest path in an LH (resp., an LV), all the horizontal
(resp., vertical) sub-paths between consecutive breakpoints lie
on the lines of the LH (resp., LV).
e
c
a
f1fk
π3
π2
π1b
d
f
HL(αi)
HL(αj)
Figure 4: Three horizontal paths passing through ktri-
angular regions.
Proof. Suppose the lemma for the case of a horizontal lane.
Similarly, the lemma holds for a vertical lane. We consider
s0as the entrance point to the LH and t0as the exit point.
W.l.o.g. we consider that s0is on the left side of t0. Due to
the assumption that the path between sand tis optimal in
length, any sub-path of this path is also optimal in length.
Thus, the path between s0and t0is optimal in length. We
consider a path between s0and t0where a horizontal sub-
path between two consecutive breakpoints does not lie on
the lines of the LH. We show that there exists an equivalent
path in length between s0and t0such that all the horizontal
sub-paths between consecutive breakpoints lie on the lines of
the LH. We assume cand das two consecutive breakpoints
such that the horizontal sub-path between them does not lie
on the lines of the LH (see Fig. 4). There are ktriangular
regions between cand dand the sub-path between these two
breakpoints must pass all ktriangular regions (w.l.o.g. as-
sume cand dare located on the edges of the triangles). We
also assume that the path between s0and t0contains other
two breakpoints – we call them aand b– which are on the
lower line of the LH (these two breakpoints are also located
on the edges of the triangles). For passing these triangles, a
path can directly go from ato b. Since the path between s0
and t0is optimal in length, the path which contains cand d
(π2) has less than or equal length to the case in which it goes
directly from ato b(π1). If dw(π1) = dw(π2), an equivalent
path in length which does not contain the horizontal path
between cand dexists. If dw(π1)< dw(π2), it contradicts
our assumption that the path between sand tis optimal
in length. For the other case where dw(π2)< dw(π1), we
consider another path which goes from ato e(a breakpoint
on the upper line of the LH and on the edge of the left
most triangle) and then from eto f(a breakpoint on the
upper line of the LH and on the edge of the right most tri-
angle) and then to b(π3). According to Lemma 1, since
dw(π2)< dw(π1), therefore, dw(π3)< dw(π2) and this con-
tradicts our assumption that the path between sand tis
optimal in length. Thus, the lemma holds.
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