The Minimum Perimeter Convex Hull of
a Given Set of Disjoint Segments
Yiyang Jia and Bo Jiang
Dalian Maritime University
Linghai Road 1, Dalian, China
2017.5.15
C
ONTENTS
ONTENTS
01
Abstract
04
Construction of the minimum
perimeter convex hull
02
Introduction
05
Our Algorithm
03
Preliminaries
06
Conclusion
01
Abstract
In this paper, we present an algorithm to compute the minimum perimeter
convex hull of a given disjoint segment set, so that each segment is
contained in the hull completely or intersects with the boundary of the hull.
The problem discussed in this paper is a novel transformation of TSP and
MPSP. To solve this problem, we use a contraction strategy to contract the
convex hull from a larger one which contains all endpoints of given segments
to the direction of a smaller one which only contains some necessary points.
We also assess the spatial relationships between outside segments and its
convex hull, and add necessary segments into the convex hull successively
by finding the shortest path of a sequence of segments. As a result, we
present an O(n5) algorithm for computing the minimum perimeter convex hull
of a given set of disjoint segments.
Introduction
02
1. The Problems of Stabbing a given set of disjoint segments (MPSP)
(1) It is an open problem posed by Löffler and van Kreveld [1].
(2) Rappaport presented an O(3mn+nlogn) time algorithm [2].
(3) For parallel segments, Goodrich et al. gave an O(nlogn) time algorithm [3].
(4) Several approximation algorithms are presented by Dumitrescu [4],Hassanzadeh [5] et al.
(5) José Miguel Díaz-Báñez et al. proved that if S is a set of pairwise disjoint segments, this
problem can be solved in polynomial time; for general segments, this problem is NP-hard
[6]
2. Traveling Salesman problems (TSP)
(1) In 1979 Garey M.R and Johnson D.S[7] proved the NP- Completeness of TSP.
(2) Related problems: finding simple paths; finding the shortest paths of ordered or partially
ordered line segments have been presented and solved in [8,9].
02
Introduction
Computing the minimum perimeter convex hull of a given set of
disjoint segments is one of the most important extensions, different from
traditional convex problems and MPSP, it asks for the minimum perimeter
convex hull that each segment in the given set either is contained in it or
intersects with its boundary. In other words, it defines a region with a
minimum perimeter, each given segment is either completely contained in
this region or only has a part in it.
The solution can be used to deal with many practical problems such as:
finding the containing region of some necessary elements with a minimum
perimeter, determining some spatial extent limited by some fixed objects.
02
Introduction
l2
l1
Copt
l3
l5
l4
03
Preliminaries and Notation
Notation 1: Let Φ denote
a set with n disjoint
segments in a plane. Each
line segment in Φ is
denoted by li (1≤i≤n). In
addition, the set of all
endpoints of all the given
segments in Φ is denoted
by P, each of them is
denoted by pk (1≤k≤2n).
Our objective is to find
the minimum perimeter
convex hull of Φ, its
interior or boundary
contains at least one point
of each li, and denote it as
Copt.。
Preliminaries: A convex polygon is a simple polygon (not self-intersecting)
with the property that each of its internal angles is strictly smaller than π. A
convex hull of a given set of points is a convex polygon that contains all the
points in the point set. In a similar fashion, Copt is defined as a minimum
perimeter convex polygon which satisfies the following two conditions:
(1) Each segment in Φ either is contained in it or intersects with the
boundary of it;
(2) It has the minimum perimeter among the convex hulls satisfying
condition (1).
Notation 2: In this paper, we define two kinds of convex hulls:
Ω is the convex hull of P ;
Ψ is a lower bound of Copt that only contains some necessary points in P.
We denote the convex hull got in the j-th contraction as Ωj, and the
convex hull got in the k-th expansion as Ψk (j, k∈N+).
03
Preliminaries and Notation
Ω
p0
p5
l2
l1
Copt=Ψ
p7
p6
l3
p4
B
pA1
p8
l5
p2
p9
l4
p3
04
Construction of the minimum perimeter convex hull
(1) Construction of Ω
Ω is constructed based on P with the Graham Scanning Algorithm
p1
p0
p7
p6
p2
p3
p4
p5
04
Construction of the minimum perimeter convex hull
(2) Construction of Ψ
Construct Ψ with the method below:
(1) Put the segments which have points on
the boundary of Ω into three sets based
on the number of points. L0 : one endpoint;
L1 : two endpoints; L2 : whole segment (L0
= {l1, l3, l4, l5}, L1 = {l2}, L2 = {l6}).
(2) If L0 has less than two line segments, we
consider that Ψ does not exist. Otherwise, p2
compute the convex hull of the endpoints
inside Ω of the segments in L0, Ψ is
obtained by discarding its vertices whose
corresponding line segments have parts
inside this convex hull.
p0
p1
l1
l6
p7
l2
l3
p3
p6
l5
l4
p4
p5
Construction of the minimum perimeter convex hull
04
C0 is a precursor of Copt, which is a result of
contraction from Ω to Ψ without dealing with the
p2
segments in L2.
Contract Ω in the counterclockwise (clockwise) order
until no vertex on its boundary can be changed
again(Ψ is also updated if necessary), then we get C0.
p1
l1
l9
l10
p0
l6
l6
p7
l8
l2
p6
l3
l5
p3
l4
l1
l9
l10
p0
p5
p4
l6
p7
p7
l7
l8
l2
p6
p2
l8
l11
l3
p3
p0
l7
p1
l7
p2
l1 l9
p1
(3) Construction of C0
l10
l4
p4
l5
p5
l2
p6
l3
p3
l5
l4
p4
p5
04
Construction of the minimum perimeter convexp0 hull
(4) Construction of Copt
Copt is constructed by dealing with the
segments in L2.
(1) Initialize an access sequence of all the
segments fixed and ordered by the
boundary of C0, compute the shortest
path traversing them.
(2) For each line segment excluded by the
shortest path, insert it into its the subsequence by finding a right position in
this sequence which does not make
the shortest path intersect with itself.
(3) Remove right turns on the shortest
path.
l1 l9
p1
l7
l8
p2
l10
l6
p7
l2
p6
l11
l3
p3
l1
l8
l9
l10 l12
l5
l4
p5
p
l6
l13 l16
l14 l15
l7
l2
l3
l4
p
l5
l11
05
Algorithm
Algorithm MPCH
O(n5) time
Input: A set of segments Φ.
Output: The minimum perimeter convex hull Copt .
1 Construct Ω including all the endpoints of the line segments in Φ. (O(nlogn) time)
2 Construct Ψ: (O(nlogn) time)
2.1 Divide the given line segment involved in Ω’s boundary into sets L0, L1, L2.
2.2 Use the inside endpoints of the segments in L0 to construct Ψ.
3 Construct C0 by a contraction process from Ω to Ψ. (O(n2) time)
4 Construct Copt: (O(n5) time)
4.1 Initialize the sequence A of accessing line segments according to C0.
4.2 Find C by computing the shortest path of A.
(O(n3) time)
4.3 For each line segment outside C, insert it into its corresponding subsequence. (O(n5) time)
4.4 Remove the unreasonable segments that form right turns and get Copt. (O(n3) time)
06
Conclusions
In this paper, we first introduced a problem of computing the minimum perimeter convex hull of
a given set of disjoint segments, a given line segment either inside the convex hull or intersects
with its boundary. Then we proposed an O(n5) time solution of this problem. This method can be
further used to calculate the shortest path of traversing a set of potentially ordered segments with
the help of this convex hull we get. This is an interesting research area and we are now working
on it.
References
1. Maarten Loffler and Marc van Kreveld: Largest and Smallest Convex Hulls for Imprecise Points. Algorithmica,
2008.
2. Rappaport, D.: Minimum polygon transversals of line segments. Int. J. Comput. Geom. Appl., 1995, 5(3): 243–256.
3. Goodrich, M.T., Snoeyink, J.: Stabbing parallel segments with a convex polygon. Computer Vision, Graphics, and
Image Processing, 1990, 49(2): 152–170.
4. Dumitrescu, A., Jiang, M.: Minimum-perimeter intersecting polygons. Algorithmica, 2012, 63(3): 602–615.
5. Hassanzadeh, F., Rappaport, D.: Approximation algorithms for finding a minimum perimeter polygon intersecting a
set of line segments. In: Dehne, F., Gavrilova, M.,Sack, J.-R., Tóth, C.D. (eds.) WADS 2009, Springer, Heidelberg,
2009, LNCS 5664: 363–374.
6. José Miguel Díaz-Báñez, Matias Korman, Pablo Pérez-Lantero, Alexander Pilz, Carlos Seara, Rodrigo I. Silveira,
New results on stabbing segments with a polygon, Computational Geometry, 2015, 48(1): 14–29.
7. Garey M.R, Johnson D.S. Computers and intractability: A Guide to the Theory of NP-Completeness[M]. Freeman
San Francisco, California, USA, 1979.
8. Cheng, Q., Chrobak, M., Sundaram, G.: Computing simple paths among obstacles. Comput. Geom., 2000(16):
223–233.
9. Lijuan Wang, Bo Jiang, Qi Wei, Dandan He. Research on the Algorithm for Euclidean Shortest Paths of Visiting
Line Segments in the Plane[J], ICIC Express Letters, 2014, 8(6): 1683-1688.
10. RL Graham, An efficient algorithm for determining the convex hull of a finite planar set. In-formation Processing
Letters, 1972, 1(4): 132–133.
Thank You