21,475 research outputs found
Bounding right-arm rotation distances
Rotation distance measures the difference in shape between binary trees of
the same size by counting the minimum number of rotations needed to transform
one tree to the other. We describe several types of rotation distance where
restrictions are put on the locations where rotations are permitted, and
provide upper bounds on distances between trees with a fixed number of nodes
with respect to several families of these restrictions. These bounds are sharp
in a certain asymptotic sense and are obtained by relating each restricted
rotation distance to the word length of elements of Thompson's group F with
respect to different generating sets, including both finite and infinite
generating sets.Comment: 30 pages, 11 figures. This revised version corrects some typos and
has some clearer proofs of the results for the lower bounds and better
figure
Bounding right-arm rotation distances
Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets
On Rotation Distance of Rank Bounded Trees
Computing the rotation distance between two binary trees with internal
nodes efficiently (in time) is a long standing open question in the
study of height balancing in tree data structures. In this paper, we initiate
the study of this problem bounding the rank of the trees given at the input
(defined by Ehrenfeucht and Haussler (1989) in the context of decision trees).
We define the rank-bounded rotation distance between two given binary trees
and (with internal nodes) of rank at most , denoted by
, as the length of the shortest sequence of rotations that
transforms to with the restriction that the intermediate trees must
be of rank at most . We show that the rotation distance problem reduces in
polynomial time to the rank bounded rotation distance problem. This motivates
the study of the problem in the combinatorial and algorithmic frontiers.
Observing that trees with rank coincide exactly with skew trees (binary
trees where every internal node has at least one leaf as a child), we show the
following results in this frontier :
We present an time algorithm for computing . That is,
when the given trees are skew trees (we call this variant as skew rotation
distance problem) - where the intermediate trees are restricted to be skew as
well. In particular, our techniques imply that for any two skew trees
.
We show the following upper bound : for any two trees and of rank
at most and respectively, we have that: where . This bound is asymptotically
tight for .
En route our proof of the above theorems, we associate binary trees to
permutations and bivariate polynomials, and prove several characterizations in
the case of skew trees.Comment: 25 pages, 2 figures, Abstract shortened to meet arxiv requirement
On a Subposet of the Tamari Lattice
We explore some of the properties of a subposet of the Tamari lattice
introduced by Pallo, which we call the comb poset. We show that three binary
functions that are not well-behaved in the Tamari lattice are remarkably
well-behaved within an interval of the comb poset: rotation distance, meets and
joins, and the common parse words function for a pair of trees. We relate this
poset to a partial order on the symmetric group studied by Edelman.Comment: 21 page
Representing and retrieving regions using binary partition trees
This paper discusses the interest of Binary Partition Trees for image and region representation in the context of indexing and similarity based retrieval. Binary Partition Trees concentrate in a compact and structured way the set of regions that compose an image. Since the tree is able to represent images in a multiresolution way, only simple descriptors need to be attached to the nodes. Moreover, this representation is used for similarity based region retrieval.Peer ReviewedPostprint (published version
KP line solitons and Tamari lattices
The KP-II equation possesses a class of line soliton solutions which can be
qualitatively described via a tropical approximation as a chain of rooted
binary trees, except at "critical" events where a transition to a different
rooted binary tree takes place. We prove that these correspond to maximal
chains in Tamari lattices (which are poset structures on associahedra). We
further derive results that allow to compute details of the evolution,
including the critical events. Moreover, we present some insights into the
structure of the more general line soliton solutions. All this yields a
characterization of possible evolutions of line soliton patterns on a shallow
fluid surface (provided that the KP-II approximation applies).Comment: 49 pages, 36 figures, second version: section 4 expande
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