(Re)packing Equal Disks into Rectangle
The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for...
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| Vydáno v: | Discrete & computational geometry Ročník 72; číslo 4; s. 1596 - 1629 |
|---|---|
| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.12.2024
Springer Nature B.V |
| Témata: | |
| ISSN: | 0179-5376, 1432-0444 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of
n
equal disks packed into a rectangle and integers
k
and
h
, we ask whether it is possible by changing positions of at most
h
disks to pack
n
+
k
disks. Thus the problem of packing equal disks is the special case of our problem with
n
=
h
=
0
. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for
h
=
0
. Our main algorithmic contribution is an algorithm that solves the repacking problem in time
(
h
+
k
)
O
(
h
+
k
)
·
|
I
|
O
(
1
)
, where |
I
| is the input size. That is, the problem is fixed-parameter tractable parameterized by
k
and
h
. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 Editor in Charge: Csaba D. Tóth |
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-024-00633-1 |