Towards Combinatorial Characterizations and Algorithms for Bar-And-Joint Independence and Rigidity in 3D and Higher Dime...

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Material Information

Title:
Towards Combinatorial Characterizations and Algorithms for Bar-And-Joint Independence and Rigidity in 3D and Higher Dimensions
Physical Description:
1 online resource (134 p.)
Language:
english
Creator:
Cheng, Jialong
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
SITHARAM,MEERA
Committee Co-Chair:
UNGOR,ALPER
Committee Members:
THAI,MY TRA
DAVIS,TIMOTHY ALDEN
VINCE,ANDREW J

Subjects

Subjects / Keywords:
3d -- bound -- characterization -- combinatorial -- matroid -- rigidity
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre:
Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Combinatorial characterization of generic bar-and-joint rigidity in 3D and higher dimensions is a long open problem. This translates to a combinatorial characterization of the exact rank of generic rigidity matroids of graphs whose vertices represent joints and edges represent bars. A further question is the algorithmic complexity of determining rank (and rigidity) combinatorially. Our program of research is to iteratively provide increasingly better upper bounds on the rank, and corresponding algorithms: we start with an approximate characterization (and algorithm) to upper bound the rank; construct families of graphs that illuminate the gap between the upper bound and the true rank; give better characterizations, upper bounds, and algorithms that overcome some of these obstacles and iterate this process. Along the way, we obtain the following results. (a) The best polynomial time rank upper bound currently known for general graphs: i.e, the size of any maximal subgraph satisfying a counting condition given by Maxwell in the 1800's  (b) Another rank upper bound obtained as a simple inclusion-exclusion formula applied to subgraphs in a specific type of graph cover; and the construction of a subgraph satisfying Maxwell's counting condition, whose size meets this bound (c) First proofs or shortened proofs of correctness of  existing algorithms for detecting rigidity in certain classes of graphs (d) Systematic constructions that answer a 20 year old open problem about so-called rigidity circuits, and illuminate the  limitations of the bounds in (a) and (b); and finally (e) A method to overcome these obstacles and  obtain a combinatorial characterization that potentially captures 3D rigidity  provided an existing conjecture, concerning a well-studied structure  called abstract rigidity matroid, holds in 3 dimensions.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jialong Cheng.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: SITHARAM,MEERA.
Local:
Co-adviser: UNGOR,ALPER.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-12-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0046127:00001