Planar tree transformation: Results and counterexample

• Published in: Information processing letters Vol. 109; no. 1; pp. 61 - 67
• Main Authors: Akl, Selim G., Islam, Kamrul, Meijer, Henk
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16-12-2008; Elsevier; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Canonical tree; Combinatorics; Combinatorics. Ordered structures; Computational geometry; Computer science; control theory; systems; Exact sciences and technology; Flip; Geometry; Graph theory; Mathematics; Miscellaneous; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Planar tree; Sciences and techniques of general use; Studies; Theoretical computing; Transformation; Computational geometry; Transformation; Flip; Planar tree; Canonical tree; Enumeration; Computer theory; Plane; Mathematics; Algorithm; Counterexample; Spanning tree; Information processing; Two-dimensional calculations; Object; Algorithm analysis
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2008.09.005

We consider the problem of planar spanning tree transformation in a two-dimensional plane. Given two planar trees
T
1
and
T
2
drawn on a set
S of
n points in general position in the plane, the problem is to transform
T
1
into
T
2
by a sequence of simple changes called
edge-flips or just
flips. A flip is an operation by which one edge
e of a geometric object is removed and an edge
f (
f
≠
e
) is inserted such that the resulting object belongs to the same class as the original object. We present two algorithms for planar tree transformations. The first technique is an indirect approach which relies on some ‘canonical’ tree to obtain such transformation results. It is shown that it takes at most
2
n
−
m
−
s
−
2
flips (
m
,
s
>
0
) which is an improvement over the result in [D. Avis, K. Fukuda, Reverse search for enumeration, Discrete Applied Mathematics 65 (1996) 21–46]. Second, we present a direct approach which takes at most
n
−
1
+
k
flips
(
k
⩾
0
)
for such transformation when
S in convex position and also show results when the points are in general position. We provide cases where the second technique performs an optimal number of flips. A counterexample is given to show that if
|
T
1
∖
T
2
|
=
k
then they cannot be transformed to one another by
k flips.