Packing Unit Disks
dc.comment.hidden | This is the digital copy of my thesis. I also have a print copy, which leaves blank pages so that each chapter starts on the right side of a double page, and the margins are changed slightly to add extra gutter space (to accomodate the binding). | en |
dc.contributor.author | Lafreniere, Benjamin J. | |
dc.date.accessioned | 2008-08-27T19:19:50Z | |
dc.date.available | 2008-08-27T19:19:50Z | |
dc.date.issued | 2008-08-27T19:19:50Z | |
dc.date.submitted | 2008 | |
dc.description.abstract | Given a set of unit disks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Richard Rado conjectured 1/4 and proved 1/4.41. In this thesis, we consider a variant of this problem where the disjointness constraint is relaxed: selected disks must be k-colourable with disks of the same colour pairwise-disjoint. Rado's problem is then the case where k = 1, and we focus our investigations on what can be proven for k > 1. Motivated by the problem of channel-assignment for Wi-Fi wireless access points, in which the use of 3 or fewer channels is a standard practice, we show that for k = 3 we can cover at least 1/2.09 and for k = 2 we can cover at least 1/2.82. We present a randomized algorithm to select and colour a subset of n disks to achieve these bounds in O(n) expected time. To achieve the weaker bounds of 1/2.77 for k = 3 and 1/3.37 for k = 2 we present a deterministic O(n^2) time algorithm. We also look at what bounds can be proven for arbitrary k, presenting two different methods of deriving bounds for any given k and comparing their performance. One of our methods is an extension of the method used to prove bounds for k = 2 and k = 3 above, while the other method takes a novel approach. Rado's proof is constructive, and uses a regular lattice positioned over the given set of disks to guide disk selection. Our proofs are also constructive and extend this idea: we use a k-coloured regular lattice to guide both disk selection and colouring. The complexity of implementing many of the constructions used in our proofs is dominated by a lattice positioning step. As such, we discuss the algorithmic issues involved in positioning lattices as required by each of our proofs. In particular, we show that a required lattice positioning step used in the deterministic O(n^2) algorithm mentioned above is 3SUM-hard, providing evidence that this algorithm is optimal among algorithms employing such a lattice positioning approach. We also present evidence that a similar lattice positioning step used in the constructions for our better bounds for k = 2 and k = 3 may not have an efficient exact implementation. | en |
dc.identifier.uri | http://hdl.handle.net/10012/3907 | |
dc.language.iso | en | en |
dc.pending | false | en |
dc.publisher | University of Waterloo | en |
dc.subject | computational geometry | en |
dc.subject | disk packing | en |
dc.subject | covering | en |
dc.subject | colouring | en |
dc.subject | maximum independent set | en |
dc.subject | lower bounds | en |
dc.subject | discrete geometry | en |
dc.subject | algorithms | en |
dc.subject | complexity | en |
dc.subject | disk intersection graphs | en |
dc.subject.program | Computer Science | en |
dc.title | Packing Unit Disks | en |
dc.type | Master Thesis | en |
uws-etd.degree | Master of Mathematics | en |
uws-etd.degree.department | School of Computer Science | en |
uws.peerReviewStatus | Unreviewed | en |
uws.scholarLevel | Graduate | en |
uws.typeOfResource | Text | en |