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Hi

I am delving into the field of Computational Topology. I am aware of the books in this field, but

could anybody tell me a nice relevant paper in this field which tackles a "typical" Computational Topology problem?

Thank you

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What do you mean by "typical"? One which everyone is considering? One which everyone uses as an example? One which motivated the subject? One that your lecturer works on? etc. –  Yemon Choi Oct 27 '11 at 22:34
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One which everyone uses as an example –  user695652 Oct 27 '11 at 23:15
    
I don't know about tagging this computational-geometry, so I"m adding a computational-topology tag –  David Roberts Oct 27 '11 at 23:44
    
It really depends on the flavour of computational topology you're interested in. For example, computational 3-manifold topology -- things like the 3-sphere recognition algorithm -- have a very different flavour than say the Edelsbrunner / Harer text. –  Ryan Budney Oct 27 '11 at 23:49
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Oh thanks I wasn't aware of this. Yes I meant more the Edelsbrunner type. –  user695652 Oct 27 '11 at 23:50
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2 Answers

up vote 4 down vote accepted

Perhaps the paper by Jeff Erickson and Pratik Worah, "Computing the Shortest Essential Cycle," Discrete & Computational Geometry, Volume 44, Issue 4, December 2010 (PDF link), might serve your purposes. They compute the shortest "simple cycle that cannot be continuously deformed to a point or a single boundary." The input to their problem is "a combinatorial surface, which is an abstract topological surface $M$ together with an edge-weighted graph $G$ cellularly embedded on $M$." If $n$ is the complexity of the surface, their algorithm runs in $O(n^2 \log n)$ time, and faster, $O( n \log n)$, when the genus and number of boundaries are considered fixed. This paper is, in some sense, a culmination of a series of papers finding cycles on combinatorial surfaces, often to cut them along the cycles to produces simpler surfaces.
            Fig3

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This is nice. It harks back to the early days of Riemann surfaces, when 'connectedness' (=genus) was defined by curves along which one cut the surface. –  David Roberts Oct 28 '11 at 0:56
    
Thanks you for the nice paper suggestion, that was what I was looking for. –  user695652 Oct 28 '11 at 15:36
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The interesting books i know of are Edelsbrunner/Harer and Zomordian's thesis. On similar topics, the Comtop group at Stanford has very detailed information.

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