18,070 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
Clifford algebras, Spin groups and qubit trees
Representations of Spin groups and Clifford algebras derived from structure
of qubit trees are introduced in this work. For ternary trees the construction
is more general and reduction to binary trees is formally defined by deleting
of superfluous branches. Usual Jordan-Wigner construction also may be formally
obtained in such approach by bringing the process up to trivial qubit chain
("trunk"). The methods can be also used for effective simulations of some
quantum circuits corresponding to the binary tree structure. Modeling of more
general qubit trees and relation with mapping used in Bravyi-Kitaev
transformation are also briefly outlined.Comment: LaTeX 12pt, 36 pages, 9 figures; v5: updated, with two new
appendices. Comments are welcom
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
COSMICAH 2005: workshop on verification of COncurrent Systems with dynaMIC Allocated Heaps (a Satellite event of ICALP 2005) - Informal Proceedings
Lisboa Portugal, 10 July 200
Complexity dichotomy on partial grid recognition
Deciding whether a graph can be embedded in a grid using only unit-length
edges is NP-complete, even when restricted to binary trees. However, it is not
difficult to devise a number of graph classes for which the problem is
polynomial, even trivial. A natural step, outstanding thus far, was to provide
a broad classification of graphs that make for polynomial or NP-complete
instances. We provide such a classification based on the set of allowed vertex
degrees in the input graphs, yielding a full dichotomy on the complexity of the
problem. As byproducts, the previous NP-completeness result for binary trees
was strengthened to strictly binary trees, and the three-dimensional version of
the problem was for the first time proven to be NP-complete. Our results were
made possible by introducing the concepts of consistent orientations and robust
gadgets, and by showing how the former allows NP-completeness proofs by local
replacement even in the absence of the latter
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
Dynamics for holographic codes
We describe how to introduce dynamics for the holographic states and codes
introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the
definition of a continuous limit of the kinematical Hilbert space which we
argue may be achieved via the semicontinuous limit of Jones. Dynamics is then
introduced by building a unitary representation of a group known as Thompson's
group T, which is closely related to the conformal group in 1+1 dimensions. The
bulk Hilbert space is realised as a special subspace of the semicontinuous
limit Hilbert space spanned by a class of distinguished states which can be
assigned a discrete bulk geometry. The analogue of the group of large bulk
diffeomorphisms is given by a unitary representation of the Ptolemy group Pt,
on the bulk Hilbert space thus realising a toy model of the AdS/CFT
correspondence which we call the Pt/T correspondence.Comment: 40 pages (revised version submitted to journal). See video of related
talk: https://www.youtube.com/watch?v=xc2KIa2LDF
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