A note on some tree similarity measures

Čulík, Karel; Wood, Derick

doi:10.1016/0020-0190(82)90083-7

Cited by 100 publications

(96 citation statements)

References 6 publications

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“…If there are no additional constraints, it is well known that (n) rotations are necessary and sufficient to rotate one tree into another in the worst case [2], [9] (the best-known upper bound is 2n − 6). However, in our setting there is additional structure to cope with.…”

Section: Tree Reconciliation Via Rotations-preliminariesmentioning

confidence: 99%

Morphing Simple Polygons

Guibas

Hershberger

Suri

2000

Discrete Comput Geom

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Abstract. In this paper we investigate the problem of morphing (i.e., continuously deforming) one simple polygon into another. We assume that our two initial polygons have the same number of sides n, and that corresponding sides are parallel. We show that a morph is always possible through an interpolating simple polygon whose n sides vary but stay parallel to those of the two original ones. If we consider a uniform scaling or translation of part of the polygon as an atomic morphing step, then we show that O(n log n) such steps are sufficient for the morph. Furthermore, the sequence of steps can be computed in O(n log n) time.

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Section: Tree Reconciliation Via Rotations-preliminariesmentioning

confidence: 99%

Morphing Simple Polygons

Guibas

Hershberger

Suri

2000

Discrete Comput Geom

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“…It follows that given any two triangulations " and '2 we can convert " into '2 in 2n-6-deg,(x)-deg 2 (x) flips, where x is any vertex and the degree of x is deg, (x) in " and deg 2 The following lemma about sequences of diagonal flips shows that in some situations it is easy to find the first flip in an optimal sequence of flips.…”

Section: Definitions and Equivalencesmentioning

confidence: 99%

“…Any triangulation , can be normalized with respect to the diagonal (l, r) to create a new triangulation N (,). The diagonals of N (,) are of three types: (1) N (,) contains the diagonal (l, r) , (2) …”

Section: S=t O (=")Tt 2 T K (='2)·mentioning

confidence: 99%

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Rotation distance, triangulations, and hyperbolic geometry

Sleator¹,

Tarjan²,

Thurston³

1988

J. Amer. Math. Soc.

212

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A rotation in a binary tree is a local restructuring that changes the tree into another tree. Rotations are useful in the design of tree-based data structures. The rotation distance between a pair of trees is the minimum number of rotations needed to convert one tree into the other. In this paper we establish a tight bound of 2 n − 6 2n - 6 on the maximum rotation distance between two n n -node trees for all large n n . The hard and novel part of the proof is the lower bound, which makes use of volumetric arguments in hyperbolic 3 3 -space. Our proof also gives a tight bound on the minimum number of tetrahedra needed to dissect a polyhedron in the worst case and reveals connections among binary trees, triangulations, polyhedra, and hyperbolic geometry.

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“…This can be seen as a further step towards settling the complexity of deciding the flip distance between triangulations of convex polygons or, equivalently, the rotation distance between binary trees. This variant of the problem was probably first addressed by Culik and Wood [7] in 1982 (showing a flip distance of 2n − 6) in the context of similarity measures between trees.…”

Section: Introductionmentioning

confidence: 99%

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

Aichholzer

Mulzer

Pilz

2015

Discrete Comput Geom

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Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study.We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set.

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A note on some tree similarity measures

Cited by 100 publications

References 6 publications

Morphing Simple Polygons

Morphing Simple Polygons

Rotation distance, triangulations, and hyperbolic geometry

Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

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