Advanced search options

Advanced Search Options 🞨

Browse by author name (“Author name starts with…”).

Find ETDs with:

in
/  
in
/  
in
/  
in

Written in Published in Earliest date Latest date

Sorted by

Results per page:

You searched for subject:(normal coordinated). One record found.

Search Limiters

Last 2 Years | English Only

No search limiters apply to these results.

▼ Search Limiters

1. Nayyeri, Amir. Combinatorial optimization on embedded curves.

Degree: PhD, 0112, 2013, University of Illinois – Urbana-Champaign

We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compute certain similarity measures for embedded curves on a surface. Finally, we present algorithms to solve computational problems for compactly presented curves. We describe the first algorithms to compute the shortest representative of a Z2-homology class. Given a directed graph embedded on a surface of genus g with b boundary cycles, we can compute the shortest single cycle Z2-homologous to a given even subgraph in 2O(g+b)nlog n time. As a consequence we obtain an algorithm to compute the shortest directed non-separating cycle in 2O(g)n log n time, which improves the previous best algorithm by a factor of O(√{n}) if the genus is a constant. Further, we can compute the shortest even subgraph in a given Z2-homology class if the input graph is undirected in the same asymptotic running time. As a consequence, we obtain the first near linear time algorithm to compute minimum (s, t)-cuts in surface embedded graphs of constant genus. We also prove that computing the shortest even subgraph in a Z2-homology class is in general NP-hard, which explains the exponential dependence on g. We also consider the corresponding optimization problem under Z-homology. Given an integer circulation Φ in a directed graph embedded on a surface of genus g, we describe algorithms to compute the minimum cost circulation that is Z-homologous to Φ in O(g8n log2 n log2 C) time if the capacities are integers whose sum is C or in gO(g)n3/2 time for arbitrary capacities. In particular, our algorithm improves the best known algorithm for computing the maximum (s, t)-flow on surface embedded graph after 20 years. The previous best algorithm, except for planar graphs, follow from general maximum flow algorithms for sparse graphs. Next, we consider two closely related similarity measures of curves on piecewise linear surfaces embedded in R3, called homotopy height and homotopic Frechét distance. These similarity measures capture the longest curve that appears and the longest length that any point travels in the best morph between two given curves, respectively. We describe the first polynomial-time O(log n)-approximation algorithms for both problems. Prior to our work no algorithms were known for the homotopy height problem. For the homotopic Frechét distance, algorithms were known only for curves on Euclidean plane with polygonal obstacles. Surprisingly, it is not even known if deciding if either the homotopy height or the homotopic Frechét distance is smaller that a given value is in NP. Finally, we consider normal curves on abstract triangulated surfaces. A curve is normal if it intersects any triangle in a finite set of arcs, each crossing between two different edges of the triangle.… Advisors/Committee Members: Erickson, Jeff G. (advisor), Erickson, Jeff G. (Committee Chair), Har-Peled, Sariel (committee member), Forsyth, David A. (committee member), Dey, Tamal (committee member).

Subjects/Keywords: Computational topology; combinatorial optimization; curves; maximum flow; minimum cut; curve similarity; normal coordinated

…Computational assumptions . . . . . . . . . . . . . . . . . . . . . 6.4 Normal coordinates vs. street… …complex . . . . . . . . . . . . . . . . . . . 6.4.1 Normal curves, normal isotopy, and normal… …connected normal curves . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Steps… …untracing . . . . . . . . 6.8 Normal coordinate algorithms . . . . . 6.8.1 One component… …6.8.2 Forward and reverse indexing . 6.8.3 Normal isotopy classes . . . . . 6.8.4 Isotopy… 

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Nayyeri, A. (2013). Combinatorial optimization on embedded curves. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/42333

Chicago Manual of Style (16th Edition):

Nayyeri, Amir. “Combinatorial optimization on embedded curves.” 2013. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed August 08, 2020. http://hdl.handle.net/2142/42333.

MLA Handbook (7th Edition):

Nayyeri, Amir. “Combinatorial optimization on embedded curves.” 2013. Web. 08 Aug 2020.

Vancouver:

Nayyeri A. Combinatorial optimization on embedded curves. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2013. [cited 2020 Aug 08]. Available from: http://hdl.handle.net/2142/42333.

Council of Science Editors:

Nayyeri A. Combinatorial optimization on embedded curves. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2013. Available from: http://hdl.handle.net/2142/42333

.