Map folding

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2012

Author dc:creator

• Morgan, Thomas D., M. Eng. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Erik D. Demaine.

• Electrical Engineering and Computer Science.

Statement 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.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/77030

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/77030