Abstract
Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.
| Original language | English |
|---|---|
| Pages (from-to) | 2242-2257 |
| Number of pages | 16 |
| Journal | SIAM Journal on Discrete Mathematics |
| Volume | 32 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2018 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2018 Society for Industrial and Applied Mathematics.
Funding
\ast Received by the editors November 17, 2016; accepted for publication (in revised form) April 18, 2018; published electronically September 6, 2018. http://www.siam.org/journals/sidma/32-3/M110407.html Funding: The first author's work was supported by People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n\circ [291734]. The second author's work was supported by the Doctoral Fellowships of Excellence at Bar-Ilan University, the Mordecai and Monique Katz Graduate Fellowship Program at Bar-Ilan University, and ISF grant 1083/13. \dagger Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria ([email protected]). \ddagger Computer Science Department, Ariel University, Ariel 40700, Israel, and Computer Science Department, Bar-Ilan University, Ramat-Gan 52900, Israel ([email protected]). The first author's work was supported by People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n\circ [291734]. The second author's work was supported by the Doctoral Fellowships of Excellence at Bar-Ilan University, the Mordecai and Monique Katz Graduate Fellowship Program at Bar-Ilan University, and ISF grant 1083/13.
| Funders | Funder number |
|---|---|
| Doctoral Fellowships of Excellence at Bar-Ilan University | |
| FP7/2007 | |
| Iowa Science Foundation | 1083/13 |
| Marie Curie | |
| Research Executive Agency | 291734 |
| Bar-Ilan University | |
| Seventh Framework Programme | FP7/2007-2013 |
Keywords
- Cake-cutting
- Convex objects
- Covering
- Packing
- Rectilinear polygons
- Redivision