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Map folding

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dc.contributor.advisor Erik D. Demaine. en_US Morgan, Thomas D., M. Eng. Massachusetts Institute of Technology en_US
dc.contributor.other Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. en_US 2013-02-14T15:40:11Z 2013-02-14T15:40:11Z 2012 en_US 2012 en_US
dc.description Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012. en_US
dc.description Cataloged from PDF version of thesis. en_US
dc.description Includes bibliographical references (p. 55). en_US
dc.description.abstract In 1997, Jack Edmonds posed as an open problem the computational complexity of deciding whether a given m x n map-rectangular paper with horizontal and vertical creases, each marked mountain or valley-has a flat folded state. This problem has remained open since then, even for the 2 x n case. This thesis presents several theoretical contributions to this problem. Most significantly, it presents an O(n⁹ ) time algorithm for deciding the flat foldability of a 2 x n map. To achieve this result, this thesis makes a sequence of reductions which ultimately lead to a new general hidden tree problem, where the goal is to construct a "valid" tree on a given polynomial set of candidate vertices, given oracles to navigate hypothetical partially constructed trees. To complete the algorithm, it is shown that the hidden tree problem can be solved in polynomial time using dynamic programming. Additionally, several faster algorithms are given for special cases of 2 x n map folding. This thesis goes on to extend this algorithm to optimization variants of the problem. In particular, by certain metrics it finds the simplest flat folded state achievable by a given 2 x n map in polynomial time. This thesis also provides results for the general m x n map folding problem. It presents a set of nontrivial necessary conditions for an m x n map to be flat foldable, that are checkable in polynomial. Additionally, this thesis presents a fixed parameter tractable algorithm for the m x n map folding problem, where the parameter is the entropy in the partial order induced by the mountain valley pattern on the cells of the map. en_US
dc.description.statementofresponsibility by Thomas D. Morgan. en_US
dc.format.extent 60 p. en_US
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
dc.rights M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. en_US
dc.rights.uri en_US
dc.subject Electrical Engineering and Computer Science. en_US
dc.title Map folding en_US
dc.type Thesis en_US M.Eng. en_US
dc.contributor.department Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. en_US
dc.identifier.oclc 825813101 en_US

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