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Minimum Perimeter-Sum Partitions in the Plane [article]

Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, Ali D. Mehrabi
2017 arXiv   pre-print
We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i.  ...  Let P be a set of n points in the plane.  ...  This research was initiated when the first author visited the Department of Computer Science at TU Eindhoven during the winter 2015-2016.  ... 
arXiv:1703.05549v1 fatcat:kahih4brcnhmhowtnki3jeseky

Minimum Perimeter-Sum Partitions in the Plane

Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, Ali D. Mehrabi
2019 Discrete & Computational Geometry  
We consider the problem of partitioning P into two subsets P 1 and P 2 such that the sum of the perimeters of ch(P 1 ) and ch(P 2 ) is minimized, where ch(P i ) denotes the convex hull of P i .  ...  Let P be a set of n points in the plane.  ...  This research was initiated when the first author visited the Department of Computer Science at TU Eindhoven during the winter 2015-2016.  ... 
doi:10.1007/s00454-019-00059-0 fatcat:le5hvb7afzhtjeqwxieegcaatm

Squarepants in a tree

David Eppstein
2009 ACM Transactions on Algorithms  
within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters.  ...  Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into  ...  Approximate clustering for hyperbolic planes If a point set has all points at least constant distance apart Then convex hull perimeterminimum spanning tree length Proof idea: Show each of the following  ... 
doi:10.1145/1541885.1541890 fatcat:aiwsaw6ajveenmiohzwbxfr2ii

Tiling a Plane in a Dynamical Process and its Applications to Arrays of Quantum Dots, Drums, and Heat Transfer

O. Cybulski, R. Hołyst
2005 Physical Review Letters  
Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem  ...  The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large N, the stationary state becomes a periodic array of hexagonal cells.  ...  Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem  ... 
doi:10.1103/physrevlett.95.088304 pmid:16196909 fatcat:i2mv23khfvgarktfwjuko6un7a

Convex Regions and their 'Fairest' Equipartitioning Fans [article]

R. Nandakumar
2012 arXiv   pre-print
We discuss the partition of convex 2D regions into n (a positive integer) equal area convex pieces by fans with the following additional requirement: the perimeters of the resultant equal area pieces should  ...  A k-fan is a set of k half-lines (rays) all starting from the same point, called the origin of the fan.  ...  Let us count the changes in sign of the change in perimeter of each piece during this rotation. It can be seen that the sum over all n pieces of this count of 'switches' would be only O(m).  ... 
arXiv:1208.6508v1 fatcat:dgkfrokpvrhhbm2pjkymj67qke

Page 3044 of Mathematical Reviews Vol. , Issue 89F [page]

1989 Mathematical Reviews  
If a polyomino or a pseudo-polyomino is constructed in this lattice, then the associated value is the sum of assigned numbers in its cells.  ...  It is shown that, for skeletons L, this partition yields direct sum decompositions of the submodular polytope associated with f. An example of such an L is given by the collection of minimizers of /.  ... 

The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

S. J. Cox, E. Flikkema
2010 Electronic Journal of Combinatorics  
Candidates to the least perimeter partition of various polygonal shapes into $N$ planar connected equal-area regions are calculated for $N\le 42$, compared to partitions of the disc, and discussed in the  ...  Partitions of a square and a pentagon show greater disorder. Candidates to the least perimeter partition of the surface of the sphere into $N$ connected equal-area regions are also calculated.  ...  Table 2 : Perimeter E/R and topology of the candidates to the minimum perimeter partition into equal-area bubbles of the surface of the unit sphere.  ... 
doi:10.37236/317 fatcat:hxwterqfwzgg7drxhvv6bdt6xy

THE TRAVELING SALESMAN PROBLEM FOR LINES AND RAYS IN THE PLANE

ADRIAN DUMITRESCU
2012 Discrete Mathematics, Algorithms and Applications (DMAA)  
In the path variant, we seek a shortest path that visits each region.  ...  In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region.  ...  Let now Q * be an intersecting rectangle of L with minimum sum of the lengths of three sides.  ... 
doi:10.1142/s1793830912500449 fatcat:rhd7etjckvhytkmdyd2aba52py

On the honeycomb conjecture and the Kepler problem [article]

Fu-Gao Song, Francis Austin
2009 arXiv   pre-print
First, we proved that the regular hexagons are the only 2-dim blocks that have unit area and the least perimeter (or contain a unit circle and have the least area) that tile the plane.  ...  Finally, the Kepler conjecture can also be proved to be true by introducing the concept of the minimum 2-dim and 3-dim Kepler building blocks.  ...  The central point in the honeycomb conjecture is therefore to determine those suitable regular polygons of unit area and least perimeter that will tile the whole plane without leaving gaps or overlapping  ... 
arXiv:0906.1249v4 fatcat:a5rkrghn65fdrbjivjly5yfrfi

CONTENTS [chapter]

1954 Mathematics and Plausible Reasoning, Volume 1  
Triangle with minimum perimeter inscribed in a given triangle. 9. Traffic center of four points in space. 10. Traffic center of four points in a plane. 11.  ...  Minimum and maximum distances in plane geometry. 2. Minimum and maximum distances in solid geometry. 3. Level lines in a plane. 4. Level surfaces in space. 11.  ... 
doi:10.1515/9780691218304-toc fatcat:5ioz6wgwzvf6vhepm37zku2uty

The Least-Perimeter Partition of a Sphere into Four Equal Areas

Max Engelstein
2009 Discrete & Computational Geometry  
We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.  ...  In a perimeter-minimizing partition each region has a pressure, defined up to addition of a constant, so that the difference in pressures between regions A and B is the sum of the (signed) curvatures crossed  ...  plane into unit areas.  ... 
doi:10.1007/s00454-009-9197-8 fatcat:44oxtwllsbctnfatfu6rndwzke

The traveling salesman problem for lines and rays in the plane [article]

Adrian Dumitrescu
2012 arXiv   pre-print
Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.  ...  In the path variant, we seek a shortest path that visits each region.  ...  It can be verified that the perimeter of any other enclosing rectangle is larger (details in the Appendix), hence the rectangle Q enclosing γ constructed in our proof has minimum perimeter.  ... 
arXiv:1204.5828v1 fatcat:2r45nl25drbcjd6j25uj7wddh4

Page 7255 of Mathematical Reviews Vol. , Issue 2001J [page]

2001 Mathematical Reviews  
Under reasonable computa- tional models, we obtain lower bounds on the minimum amount of work required to maintain any binary space partition of mov- ing segments in the plane or any Steiner triangulation  ...  of moving points in the plane.  ... 

Page 6369 of Mathematical Reviews Vol. , Issue 87k [page]

1987 Mathematical Reviews  
There are some well-known elementary geometric statements concerning the points, triangles, polygons, etc., in the Euclidean plane, which are proved also in the hyperbolic plane if the points involved  ...  The problem is then to fill in the matrix of interpoint distances with a minimum error based on some norm. They propose using the distances of the points from same reference object as coordinates.  ... 

Least perimeter partition of the disc into N bubbles of two different areas

Francis Headley, Simon Cox
2019 The European Physical Journal E : Soft matter  
When the area ratio is significantly far from one, the least perimeter partitions tend to have a "mixed" configuration, in which bubbles of the same area are not adjacent to each other.  ...  We present conjectured candidates for the least perimeter partition of a disc into N ≤ 10 connected regions which take one of two possible areas.  ...  , and reproduction in any medium, provided the original work is properly cited.  ... 
doi:10.1140/epje/i2019-11857-0 fatcat:uq4jzm2zd5dqbpzuhvrqpommjq
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